Controlling Sedimentation in Magnetorheological Fluids Through Ultrasound–Magnetic Field Coupling: Multiscale Analysis and Applications

Abstract

Magnetorheological fluids (MRFs) are multiphase materials whose viscosity can be controlled via magnetic fields. However, particle sedimentation undermines their long-term stability. This review examines stabilization strategies based on the interaction between ultrasonic waves and time-varying magnetic fields, analyzed through advanced mathematical models. The propagation of acoustic waves in spherical and cylindrical domains is studied, including effects such as cavitation, acoustic radiation forces, and viscous attenuation. The Biot–Stoll poroelastic model is employed to describe saturated granular media, while magnetic field modulation is investigated as a means to balance gravitational settling. The analysis highlights how acousto-magnetic coupling supports the design of programmable and self-stabilizing intelligent fluids for complex applications.

1. Introduction

Magnetorheological fluids (MRFs) represent an advanced class of multiphysical materials, characterized by their ability to rapidly and reversibly alter their rheological properties in response to an external magnetic field $\mathbf{H}$ [1,2,3]. Their structure is based on a colloidal suspension of micrometer- or submicrometer-sized ferromagnetic particles—typically composed of iron, iron oxide, or iron carbonyl—dispersed in a low-viscosity liquid matrix, such as silicone oils or engineered synthetic fluids [4,5]. When the magnetic field $\mathbf{H}$ is turned off, the particles are randomly distributed, and the fluid exhibits Newtonian or weakly viscoelastic behavior; in contrast, when $\mathbf{H}$ is applied, the particles align along the magnetic field lines, forming dipolar chains or columns that result in a semi-solid internal structure, drastically altering the apparent viscosity and resistance to flow [1,2,4,6,7].

From a theoretical standpoint, MRFs can be modeled through a coupling between the Navier–Stokes equations for fluid dynamics and Maxwell’s equations for the distribution of the magnetic field [8,9]. Moreover, the mechanical behavior is often described using constitutive models, such as the Bingham model or the Herschel–Bulkley model, which include a yield stress threshold ${\tau }_0$ [5,10], below which the material behaves as an elastic solid and above which it flows with a controllable viscosity [11,12]. The introduction of nonlinear formulations, viscoelastic models, and multiphysics computational approaches has enabled increasingly accurate simulations of dynamic response phenomena, magnetic hysteresis, and cyclic behavior under varying loads [13,14,15]. Naturally, more advanced dynamic formulations exist that consider nonlinear effects [5,16,17], the rate at which forces are applied, and the possibility that the fluid exhibits viscoelastic behavior [18,19,20,21,22]. Thanks to these characteristics, MRFs are used in a wide range of high-performance devices, including the following:

• Semi-active suspensions for vehicles, capable of real-time adjustment of damper stiffness to road conditions [23,24,25];
• Seismic isolation systems for bridges and smart buildings [26,27];
• Smart brakes and clutches with dynamically adjustable torque [28,29];
• Active prostheses and flexible biomedical devices [30,31];
• Actuators and joints with variable stiffness in advanced robotics [30,32,33,34].

However, the sedimentation of particles dispersed within the liquid matrix leads to a gradual phase separation, compromising both the structural stability and the performance of the MRF [35,36]. In fact, one of the main practical limitations hindering the prolonged and efficient use of MRFs is the gravitational sedimentation of magnetic particles, caused by the density difference between the solid phase and the liquid matrix [12,37,38]. Over time, this results in phase separation, with particle accumulation at the bottom, degradation of rheological properties, and loss of functionality. The phenomenon is exacerbated under extreme operating conditions (high/low temperatures, thermal cycles, long periods of inactivity) and necessitates frequent maintenance or regeneration of the fluid [39,40,41,42,43,44].

To address this challenge, various technological strategies have been explored:

• Use of stabilizing additives (polymers, surfactants, low-density resins) [45];
• Morphological and granulometric optimization of the particles [46];
• Use of physical technologies, including Ultrasonics (US), which are emerging as a dynamic, reversible, and non-invasive solution [47].

The propagation of US in MRFs induces acoustic radiation forces (ARFs), cavitation [47], and micro-vortices capable of keeping the particles suspended without altering the magnetic properties of the fluid [48]. The effectiveness depends on parameters such as frequency, wave intensity, particle size, and medium viscosity, allowing for fine modulation through active control [49,50].

The objective of this review is to provide an advanced, technology-oriented scientific overview of the role of US waves in mitigating sedimentation in MRFs, critically analyzing recent literature and integrating theoretical, modeling, and experimental knowledge. Three main strategies will be discussed in detail:

• Radial Acoustic Pressure (ARP): generated by spherical or cylindrical waves, it induces a non-uniform pressure field that, through the acoustic radiation force (ARF), stabilizes the particles [42];
• Variable Magnetic Fields: when combined with US, they enhance the suspension effect through synergistic acoustic–magnetic configurations;
• Biot–Stoll Model: this describes the propagation of elastic waves in saturated porous media, useful for advanced multiphysics analysis [51,52,53].

In addition to providing a solid theoretical foundation, this analysis has significant implications for technology transfer: from adaptive US diagnostics in the biomedical field to the development of intelligent acoustic metamaterials and high-performance industrial components. The synergy between acoustics and magnetism in MRFs can enable new generations of adaptive fluid materials for emerging applications. Figure 1 displays the flowchart of the entire content of the review.

Despite the growing interest in the combined use of magnetic fields and ultrasound for biomedical applications, the current literature still presents significant gaps. In particular, there is a lack of detailed theoretical studies that provide a unified description of the mechanical and magnetohydrodynamic interactions in magnetorheological fluids (MRFs) subjected to ultrasonic stimulation. Most existing works focus either on experimental results or treat the effects of acoustic and magnetic fields separately, without offering a coupled model capable of quantitatively predicting system behavior. This study aims to fill this gap by introducing a coupled physical–mathematical model that integrates viscoelastic, acoustic, and magnetic effects in describing the dynamic behavior of MRFs. Specifically, we derive and analyze systems of equations that govern ultrasound propagation in the presence of magnetic particles, taking into account controlled cavitation, forced transport, and variations in effective viscosity. This approach not only enhances the interpretation of existing experimental observations but also provides a predictive tool for the design of advanced systems for controlled drug delivery or localized stimulation. In this work, in addition to the mathematical derivation, a strong emphasis is placed on the physical understanding of the phenomena. Each formulation is accompanied by interpretations that directly link the equations to MRF stabilization mechanisms, such as acoustic radiation forces, cavitation, and magneto-hydrodynamic interactions. It is important to emphasize that this work is theoretical and analytical in nature and does not include original experimental data. The mathematical formulations presented, including wave propagation equations, radiation forces, and cavitation dynamics, are consistent with phenomena already documented in the existing literature. Their purpose is to provide predictive and interpretative tools for the design and optimization of intelligent fluid-based systems in both industrial and biomedical contexts. Experimental validation of the modeled configurations is considered outside the scope of this review and is addressed in previous studies.

Although the adopted approach offers a unified and coherent view of the interactions between magnetic field, ultrasonic waves, and suspension dynamics, each method used in the present work has specific limitations. Analytical models are based on simplifying hypotheses (such as linearization, radial symmetry, ideal conditions) necessary to obtain closed solutions, but not always representative of the real complexity. Numerical models, while expanding predictive capability, require experimental parameters that are difficult to determine precisely and can be sensitive to boundary conditions. Finally, the integration of physical models with application aspects, such as drug delivery, involves abstractions that do not include complex physiological effects. The identification of these limits is crucial to define the field of validity of the proposed solutions and outline future development directions.

The remainder of the review is structured as follows: After outlining the theoretical principles of RAP (see Section 2), the main innovations of this technique in both industrial and biomedical fields are reviewed (see Section 3), highlighting its advantages and disadvantages through an in-depth SWOT analysis. The same approach is applied to the technique based on variable magnetic fields (see Section 4 and Section 5) as well as to the approach based on the Biot–Stoll theory (see Section 6 and Section 7). Finally, additional discussions and an outline of future developments complete the review.

2. Radial Acoustic Pressure Theory

Radial acoustic pressure (RAP) is generated in fluids traversed by spherical or cylindrical US, creating a variable pressure field that interacts with suspended particles [42]. In MRFs, this force can effectively counteract the sedimentation of ferromagnetic particles, enhancing their long-term stability [42,54]. The pressure gradient induced by the US produces a net acoustic force that can balance gravitational force, provided that parameters such as frequency, wave amplitude, particle size, and fluid viscosity are properly optimized [9,42,54]. Recent studies indicate that specific frequency ranges promote micro-convective currents that help maintain a uniform particle distribution [9,42,54]. This technique represents an effective and non-invasive alternative for improving the long-term stability of MRFs, with potential applications in smart actuators [55], vibration isolation systems [56], and biomedical devices [57].

2.1. Propagation of Acoustic Waves in Fluids

Let $\Omega \subset {\mathbb{R}}^3$ be a sufficiently regular domain occupied by a fluid. In $\Omega$, we define an orthonormal Cartesian coordinate system $Oxyz$ with the origin located at the source. In this system, starting from the origin, the position of a single particle P is given by the vector coordinate $\mathbf{OP}$ [$\mathrm{m}$]. Denoting the time coordinate by t, $p(x,y,z,t)$ [$\mathrm{Pa}$] represents the acoustic pressure at a point $\mathbf{OP}=(x,y,z)$ at the time t [$\mathrm{s}$]. Then, US in a fluid obeys the well-known partial differential equation (PDE):

$${\nabla }^{2}p((x,y,z),t)-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^{2}p(x,y,z,t)}{\partial t^2}}=0$$

(1)

where c is the speed of sound in the fluid [$\mathrm{m} {\text{s}}^{-1}$]. Equation (1) represents the classical wave equation in a fluid medium. In this context, it describes how ultrasonic waves propagate within the MRF, generating pressure variations that can influence the motion of suspended particles.

2.1.1. Solution for Spherical Waves

In spherical coordinates $(r,\theta ,\varphi )$, the Laplacian is expressed in the form:

$$\begin{array}{c}\hfill {\nabla }^{2}p(r,\theta ,\varphi ,t)={\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(r^2{\displaystyle \frac{\partial p(r,\theta ,\varphi ,t)}{\partial r}}\right)+\\ \hfill +{\displaystyle \frac{1}{r^2\text{sin}\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(\text{sin}\theta {\displaystyle \frac{\partial p\left(\right(r,\theta ,\varphi ),t)}{\partial \theta }}\right)+{\displaystyle \frac{1}{r^2{\text{sin}}^2\theta }}{\displaystyle \frac{\partial p(r,\theta ,\varphi ,t)}{\partial {\varphi }^2}}\end{array}$$

(2)

In the case of radial symmetry, denoting the radial variable by r, the Laplacian takes the form

$${\nabla }^{2}p(r,t)={\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(r^2{\displaystyle \frac{\partial p(r,t)}{\partial r}}\right),$$

(3)

and Equation (1) becomes as follows:

$${\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(r^2{\displaystyle \frac{\partial p(r,t)}{\partial r}}\right)-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^{2}p(r,t)}{\partial t^2}}=0.$$

(4)

We look for a solution with separated variables, $p(r,t)=R\left(r\right)T\left(t\right)$, so that Equation (4) becomes

$${\displaystyle \frac{T\left(t\right)}{r^2}}{\displaystyle \frac{d}{dr}}\left(r^2{\displaystyle \frac{d\left(R\right(r\left)\right)}{dr}}\right)-{\displaystyle \frac{R\left(r\right)}{c^2}}{\displaystyle \frac{d^2\left(T\left(t\right)\right)}{d{t}^2}}=0,$$

(5)

which, divided by $R\left(r\right)T\left(t\right)$, yields

$${\displaystyle \frac{1}{r^{2}R\left(r\right)}}{\displaystyle \frac{d}{dr}}\left(r^2{\displaystyle \frac{dR\left(r\right)}{dr}}\right)={\displaystyle \frac{1}{c^{2}T\left(t\right)}}{\displaystyle \frac{d^{2}T\left(t\right)}{d{t}^2}}.$$

(6)

The term on the left-hand side of (6) depends solely on r, while the term on the right-hand side is a function of the variable t alone. Since r and t are independent variables, this equality can hold only if both sides are equal to the same constant, which must be negative to admit non-trivial solutions. We denote this constant by $-k^2$, where k is the wave number [$\text{rad} {\text{m}}^{-1}$]. This leads to two separate equations, the first of which is satisfied by the function $T\left(t\right)$:

$${\displaystyle \frac{d^{2}T\left(t\right)}{d{t}^2}}+k^2{c}^{2}T\left(t\right)=0,$$

(7)

and the second one satisfied by the function $R\left(r\right)$:

$${\displaystyle \frac{1}{r^2}}{\displaystyle \frac{d}{dr}}\left(r^2{\displaystyle \frac{dR\left(r\right)}{dr}}\right)+k^{2}R\left(r\right)=0.$$

(8)

The Equation (7) admits a general solution:

$$T\left(t\right)=C_1{e}^{ikct}+C_2{e}^{-ikct},$$

(9)

which represents a harmonic wave with angular frequency $\omega =kc$ [$\mathrm{rad} {\text{s}}^{-1}$]. Equation (8) yields

$$r^2{\displaystyle \frac{d^{2}R\left(r\right)}{d{r}^2}}+2r{\displaystyle \frac{dR\left(r\right)}{dr}}+k^2{r}^{2}R\left(r\right)=0,$$

(10)

which represents a special case of the spherical Bessel equation ($l=0$):

$$r^2{\displaystyle \frac{d^{2}R\left(r\right)}{d{r}^2}}+2(l+1)r{\displaystyle \frac{dR\left(r\right)}{dr}}+(k^2{r}^2-l(l+1))R\left(r\right)=0,$$

(11)

which the general solution is given by the following:

$$R\left(r\right)={\displaystyle \frac{A{e}^{ikr}+B{e}^{-ikr}}{r}}.$$

(12)

Thus, considering the contributions (9) and (12), we obtain the spherically symmetric solution:

$$\begin{array}{c}\hfill p(r,t)=\left({\displaystyle \frac{A{e}^{ikr}+B{e}^{-ikr}}{r}}\right)\left(C_1{e}^{ikct}+C_2{e}^{-ikct}\right)=\\ \hfill ={\displaystyle \frac{1}{r}}\left[A(C_1{e}^{ik(r+ct)}+C_2{e}^{ik(r-ct)})+B(C_1{e}^{-ik(r-ct)}+C_2{e}^{-ik(r+ct)})\right].\end{array}$$

(13)

In Equation (13), we observe that the term $e^{ik(r+ct)}$ represents an incoming wave originating from infinity and propagating toward the origin. This is physically meaningless, as we are considering a source located at the origin. Similarly, the term containing $e^{-ik(r-ct)}$ represents a wave propagating outward toward $r\to +\infty$ but generated by an imaginary source at infinity, which is also not acceptable in the present context. Therefore, the solution (13) takes the following form:

$$p(r,t)={\displaystyle \frac{1}{r}}\left[\overline{A}{e}^{ik(r-ct)}+\overline{B}{e}^{-ik(r+ct)}\right]$$

(14)

where $\overline{A}=A{C}_2$ and $\overline{B}=B{C}_2$. The solution obtained in Equation (14) describes a spherical wave emerging from the center of the domain, consistent with the hypothesis of an acoustic source located at the origin. The attenuation with $1/r$ reflects the natural dissipation of energy with distance and anticipates the radial action on the particles, a key element in counteracting sedimentation.

2.1.2. Solution for Cylindrical Waves

In cylindrical coordinates, the Laplacian is expressed in the form

$$\begin{array}{c}\hfill {\nabla }^{2}p(r,\theta ,z,t)=\\ \hfill ={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial p(r,\theta ,z,t)}{\partial r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^{2}p(r,\theta ,z,t)}{\partial {\theta }^2}}+{\displaystyle \frac{{\partial }^{2}p(r,\theta ,z,t)}{\partial z^2}}.\end{array}$$

(15)

In the case of axial symmetry, the pressure does not depend on $\theta$, so that Equation (15) becomes

$${\nabla }^{2}p(r,z,t)={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial p(r,z,t)}{\partial r}}\right)+{\displaystyle \frac{{\partial }^{2}p(r,z,t)}{\partial z^2}},$$

(16)

and Equation (1) takes the form

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial p(r,z,t)}{\partial r}}\right)+{\displaystyle \frac{{\partial }^{2}p(r,z,t)}{\partial z^2}}-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^{2}p(r,z,t)}{\partial t^2}}=0$$

(17)

As with spherical waves, we proceed by assuming a separable-variable solution of the form $p(r,z,t)=R\left(r\right)Z\left(z\right)T\left(t\right)$, which, when substituted into (15), yields

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\partial \left(R\right(r\left)Z\right(z\left)T\right(t\left)\right)}{\partial r}}\right)+{\displaystyle \frac{{\partial }^2\left(R\left(r\right)Z\left(z\right)T\left(t\right)\right)}{\partial z^2}}-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2\left(R\left(r\right)Z\left(z\right)T\left(t\right)\right)}{\partial t^2}}=0,$$

(18)

from which

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{r}}{\displaystyle \frac{d}{dr}}\left(r{\displaystyle \frac{dR\left(r\right)}{dr}}\right)Z\left(z\right)T\left(t\right)+{\displaystyle \frac{d^{2}Z\left(z\right)}{d{z}^2}}R\left(r\right)T\left(t\right)-{\displaystyle \frac{1}{c^2}}R\left(r\right)Z\left(z\right){\displaystyle \frac{d^{2}T\left(t\right)}{d{t}^2}}=0.\end{array}$$

(19)

Then, dividing by $R\left(r\right)Z\left(z\right)T\left(t\right)$, we obtain

$${\displaystyle \frac{1}{R\left(r\right)r}}{\displaystyle \frac{d}{dr}}\left(r{\displaystyle \frac{dR\left(r\right)}{dr}}\right)+{\displaystyle \frac{1}{Z\left(t\right)}}{\displaystyle \frac{d^{2}Z\left(z\right)}{d{z}^2}}={\displaystyle \frac{1}{c^{2}T\left(t\right)}}{\displaystyle \frac{d^{2}T\left(t\right)}{d{t}^2}}.$$

(20)

The variable separation requires that

$${\displaystyle \frac{1}{c^{2}T\left(t\right)}}{\displaystyle \frac{d^{2}T\left(t\right)}{d{t}^2}}=-{\displaystyle \frac{{\omega }^2}{c^2}},$$

(21)

$${\displaystyle \frac{1}{R\left(r\right)r}}{\displaystyle \frac{d}{dr}}\left(r{\displaystyle \frac{dR\left(r\right)}{dr}}\right)+{\displaystyle \frac{{\omega }^2}{c^2}}=-{\displaystyle \frac{1}{Z\left(t\right)}}{\displaystyle \frac{d^{2}Z\left(z\right)}{d{z}^2}}$$

(22)

Again, the separation of variables requires that

$${\displaystyle \frac{1}{Z\left(z\right)}}{\displaystyle \frac{d^{2}Z\left(z\right)}{d{z}^2}}=-k^2$$

(23)

and consequently

$${\displaystyle \frac{1}{R\left(r\right)r}}{\displaystyle \frac{d}{dr}}\left(r{\displaystyle \frac{dR\left(r\right)}{dr}}\right)-k^2+{\displaystyle \frac{{\omega }^2}{c^2}}=0.$$

(24)

Remark 1.

The term ${\omega }^2/c^2$ in (24) arises from the separation of the temporal component. However, from an engineering perspective, since $\omega \ll c$, the term ${\omega }^2/c^2$ in Equation (24) can be considered negligible [58].

By virtue of Remark 1, the following equations are obtained:

$${\displaystyle \frac{d^{2}T\left(t\right)}{d{t}^2}}+{\omega }^{2}T\left(t\right)=0,$$

(25)

that describes the temporal oscillation of the wave, and

$${\displaystyle \frac{1}{R\left(r\right)r}}{\displaystyle \frac{d}{dr}}\left(r{\displaystyle \frac{dR\left(r\right)}{dr}}\right)-k^2=0,$$

(26)

$${\displaystyle \frac{d^{2}Z\left(z\right)}{d{z}^2}}+k^{2}Z\left(z\right)=0,$$

(27)

which represent the radial and axial dependence of the solution. From (25), we easily obtain the general solution:

$$T\left(t\right)=A_T{e}^{i\omega t}+B_T{e}^{-i\omega t}$$

(28)

where $A_T$ and $B_T$ are arbitrary constants to be determined based on the initial conditions. Moreover, by substituting Euler’s formulas, $e^{i\omega t}=\text{cos}\left(\omega t\right)+i \text{sin}\left(\omega t\right)$ and $e^{-i\omega t}=\text{cos}\left(\omega t\right)-i \text{sin}\left(\omega t\right)$, in (28), we obtain

$$T\left(t\right)=A_T(\text{cos}\left(\omega t\right)+i \text{sin}\left(\omega t\right))+B_T(\text{cos}\left(\omega t\right)-i \text{sin}\left(\omega t\right)).$$

(29)

Moreover, by separating the real and imaginary parts, it makes sense to write

$$T\left(t\right)=(A_T+B_T)\text{cos}\left(\omega t\right)+i(A_T-B_T)\text{sin}\left(\omega t\right).$$

(30)

Since $T\left(t\right)$ must be a real function, the coefficients of the trigonometric functions must be real. Therefore, by setting $C=A_T+B_T$ and $D=i(A_T-B_T)$, in order for C and D to be real, $A_T$ and $B_T$ must be complex conjugates:

$$A_T={\displaystyle \frac{C}{2}}+i{\displaystyle \frac{D}{2}},B_T={\displaystyle \frac{C}{2}}-i{\displaystyle \frac{D}{2}}.$$

(31)

Thus, the solution, which can be rewritten in terms of trigonometric functions, is as follows:

$$T\left(t\right)=C \text{cos}\left(\omega t\right)+D \text{sin}\left(\omega t\right).$$

(32)

The solution (32) describes a harmonic oscillation with angular frequency $\omega$, while the amplitude and phase are determined by C and D.

Similarly, the solution of the longitudinal component (along z) takes the form $Z\left(z\right)=A_Z{e}^{ikz}+B_Z{e}^{-ikz}$, from which $Z\left(z\right)=\overline{C}cos\left(kz\right)+\overline{D}sin\left(kz\right)$ with

$$\overline{C}=A_Z+B_Z,\overline{D}=i(A_Z-B_Z).$$

(33)

As before, in order for $\overline{C}$ and $\overline{D}$ to be real, it is necessary that

$$A_Z={\displaystyle \frac{\overline{C}}{2}}+i{\displaystyle \frac{\overline{D}}{2}},B_Z={\displaystyle \frac{\overline{C}}{2}}-i{\displaystyle \frac{\overline{D}}{2}}.$$

(34)

We can now determine the solution of (26). After a few simple steps, we obtain the Bessel equation of order zero,

$$r^2{\displaystyle \frac{d^{2}R\left(r\right)}{d{r}^2}}+r{\displaystyle \frac{dR\left(r\right)}{dr}}-k^2{r}^{2}R\left(r\right)=0$$

(35)

which admits the following solution

$$R\left(r\right)=A{J}_0\left(kr\right)+B{Y}_0\left(kr\right)$$

(36)

in which (Bessel function of the first kind)

$$J_0\left(kr\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{{(-1)}^m}{{(m!)}^2}}{\left({\displaystyle \frac{kr}{2}}\right)}^{2m},$$

(37)

and (Bessel function of the second kind)

$$Y_0\left(kr\right)={\displaystyle \frac{2}{\pi }}\left[J_0\left(kr\right)ln{\displaystyle \frac{kr}{2}}+\sum _{m=0}^{\infty }{\displaystyle \frac{{(-1)}^m}{{(m!)}^2}}{\left({\displaystyle \frac{kr}{2}}\right)}^{2m}\left(\psi (m+1)-ln2\right)\right],$$

(38)

where $\psi (m+1)=-\gamma +{\sum }_{n=1}^m{\displaystyle \frac{1}{n}}$, with $\gamma$ denoting the Euler–Mascheroni constant. Ultimately, the solution of Equation (17) by separation of variables is the following:

$$\begin{array}{c}\hfill p(r,z,t)=\\ \hfill =\left(A{J}_0\left(kr\right)+B{Y}_0\left(kr\right)\right)\left(\overline{C} \text{cos}\left(kz\right)+\overline{D} \text{sin}\left(kz\right)\right)\left(C \text{cos}\left(\omega t\right)+D \text{sin}\left(\omega t\right)\right).\end{array}$$

(39)

The condition that the solution does not diverge, as $r\to 0$ implied $B=0$, thus yielding

$$p(r,z,t)=A{J}_0\left(kr\right)\left(\overline{C} \text{cos}\left(kz\right)+\overline{D} \text{sin}\left(kz\right)\right)\left(C \text{cos}\left(\omega t\right)+D \text{sin}\left(\omega t\right)\right).$$

(40)

Remark 2.

From an engineering standpoint, the elimination of the Bessel function of the second kind $Y_0\left(kr\right)$ (due to $B=0$) from the radial solution is justified by the fact that it diverges as $r\to 0$. Such a divergence is incompatible with the physical conditions of the problem, as it would imply a non-finite pressure field at the center of the domain, which lacks physical meaning in a real fluid medium. To ensure a physically acceptable solution, only the Bessel function of the first kind $J_0\left(kr\right)$ is retained, as it is regular at $r=0$. In this way, a continuous and bounded pressure field is obtained throughout the domain, a necessary condition to ensure that the resulting acoustic forces are distributed in a regular and controllable manner. This configuration is particularly advantageous in engineering applications requiring active control of dispersed particles, as it enables the generation of radial acoustic forces free from discontinuities or local instabilities. Moreover, a well-behaved pressure field allows for more effective and reliable use of US in MRFs, enhancing the system’s ability to counteract sedimentation and promoting a uniform distribution of particles over time.

Remark 3.

When a US wave propagates through an MRF, it generates a periodic variation in pressure. The acoustic pressure in a region of the MRF is given by the solution to the wave Equation (40), which describes the pressure field generated by the US, but does not directly explain how this affects the particles suspended in the MR fluid. As US passes through the MR fluid, they induce additional forces on the suspended particles due to the acoustic pressure gradient (pressure differences cause attraction or repulsion of the particles) and the acoustic radiation force (a net force that tends to push particles toward regions of minimum or maximum pressure).

Remark 4.

When a US wave interacts with an MRF, acoustic cavitation plays a key role in keeping the particles suspended and enhancing the fluid’s stability. This phenomenon occurs when the pressure of the wave drops below the vapor pressure of the liquid, causing the formation and rapid collapse of microbubbles. The collapse generates microjets and shock waves that break up particle agglomerates and reduce sedimentation. The intensity of cavitation depends on the US parameters and the MRF conditions, affecting its viscosity and rheological behavior. When properly controlled, this interaction promotes a uniform distribution of particles and optimizes the fluid’s properties for advanced applications.

2.2. Pressure Gradient and ARF for Spherical Waves

Starting from (13), the radial pressure gradient can be expressed as

$${\displaystyle \frac{\partial p(r,t)}{\partial r}}={\displaystyle \frac{\partial }{\partial r}}\left[{\displaystyle \frac{\overline{A}{e}^{ik(r-ct)}+\overline{B}{e}^{-ik(r+ct)}}{r}}\right],$$

(41)

namely

$${\displaystyle \frac{\partial p(r,t)}{\partial r}}=-{\displaystyle \frac{\overline{A}{e}^{ik(r-ct)}(1-ikr)+\overline{B}{e}^{-ik(r+ct)}(1+ikr)}{r^2}}.$$

(42)

The ARF, which is proportional to the pressure gradient, is given by the relation $F_r=-V_p{\displaystyle \frac{\partial p}{\partial r}},$ where $V_p$ is the volume of the particle immersed in the MRF. By substituting the obtained pressure gradient, we obtain the following expression for the ARF:

$$F_r={\displaystyle \frac{V_p}{r^2}}\left[\overline{A}{e}^{ik(r-ct)}(1-ikr)+\overline{B}{e}^{-ik(r+ct)}(1+ikr)\right].$$

(43)

Equation (43) explains how the acoustic pressure gradient gives rise to a net force (ARF) on the suspended particles. This force acts in the opposite direction to gravity and can therefore balance or reduce the sedimentation speed, depending on the parameters of the acoustic wave and the position relative to the source. The term $1/r^2$ indicates that both the pressure gradient and the ARF decrease with the square of the distance from the source, which means that particles farther away will experience a weaker acoustic effect compared to those nearby. Moreover, the term $(1\pm ikr)$ shows that the force depends on both the real part and an imaginary component, implying a phase variation in the particle’s response to the acoustic pressure. Finally, the presence of complex exponentials indicates that the radial force oscillates over time, following the propagation of the US. Clearly, the ARF counteracts particle sedimentation in an MRF, as it creates a suspending action that opposes gravity. If the intensity of the US is adequate, the particles will remain uniformly distributed in the fluid, thereby improving its stability.

2.3. Effect of Acoustic Cavitation: The Case of Spherical US Waves

The effect of acoustic cavitation in MRFs can be studied by analyzing the behavior of $p(r,t)$ and the ARF. Cavitation occurs when the acoustic pressure drops below the vapor pressure of the liquid, leading to the formation of microbubbles, which, upon subsequent collapse, generate strong pressure gradients, fluid microjets, and turbulence.

To determine the onset of cavitation, we consider the acoustic pressure (14). Let us $P_0$ denote the atmospheric pressure; then, the total acoustic pressure in the MRF must fall below the vapor pressure $P_v$ of

$$P_0+p(r,t)\le P_v.$$

(44)

Since $P_0\approx 101325$ Pa at sea level and $P_v\approx 3169$ Pa, we have $P_v<P_0$. However, since we require that ${max}_r\left|p(r,t)\right|$ be at least equal to the difference between $P_0$ and $P_v$, it is appropriate to write

$$\underset{r}{max}\left|p(r,t)\right|\ge P_0-P_v$$

(45)

Therefore, by substituting (14) into (44), we obtain

$${\displaystyle \frac{\overline{A}{e}^{ik(r-ct)}+\overline{B}{e}^{-ik(r+ct)}}{r}}\le P_v-P_0.$$

(46)

We now define the amplitude of the acoustic pressure as follows:

$$p_a=\underset{r}{max}\left|p(r,t)\right|=\underset{r}{max}\left|{\displaystyle \frac{\overline{A}{e}^{ik(r-ct)}+\overline{B}{e}^{-ik(r+ct)}}{r}}\right|.$$

(47)

We use Euler’s formula for exponentials

$$e^{ik(r-ct)}=\text{cos}\left(k(r-ct)\right)+i \text{sin}\left(k(r-ct)\right),$$

(48)

$$e^{-ik(r+ct)}=\text{cos}\left(k(r+ct)\right)-i \text{sin}\left(k(r+ct)\right).$$

(49)

Substituting into the acoustic pressure and separating the real and imaginary parts, we obtain

$$\begin{array}{c}\hfill \left|p\right(r,t\left)\right|=\\ \hfill ={\displaystyle \frac{\sqrt{{[\overline{A} \text{cos}\left(k(r-ct)\right)+\overline{B} \text{cos}\left(k(r+ct)\right)]}^2+{[\overline{A} \text{sin}\left(k(r-ct)\right)-\overline{B} \text{sin}\left(k(r+ct)\right)]}^2}}{r}}.\end{array}$$

(50)

Using the trigonometric identity and exploiting $\text{cos}(\alpha +\beta )=\text{cos} \alpha \text{cos} \beta - \text{sin} \alpha \text{sin} \beta$, relation (50) becomes

$$\begin{array}{c}\hfill \left|p(r,t)\right|={\displaystyle \frac{\sqrt{{\overline{A}}^2+{\overline{B}}^2+2\overline{A}\overline{B} \text{cos}\left(2kr\right)}}{r}}.\end{array}$$

(51)

Therefore,

$$\begin{array}{c}\hfill p_a=\underset{r}{max}\left|p(r,t)\right|={\displaystyle \frac{\sqrt{{\overline{A}}^2+{\overline{B}}^2+2\overline{A}\overline{B}}}{r}}={\displaystyle \frac{|\overline{A}+\overline{B}|}{r}}.\end{array}$$

(52)

Finally, to obtain the cavitation condition, we use the maximum amplitude of the acoustic pressure, resulting in

$${\displaystyle \frac{|\overline{A}+\overline{B}|}{r}}\ge P_0-P_v.$$

(53)

If the amplitude of the acoustic pressure exceeds the threshold $P_0-P_v$, then the local fluid pressure drops below the vapor pressure, leading to the formation of cavitation microbubbles. These nonlinear phenomena affect the propagation of the sound wave by introducing dissipation and dispersion. The microbubbles act as centers of absorption and scattering of acoustic energy, converting part of the sound energy into heat and reducing the wave amplitude as it propagates through the fluid.

To mathematically represent this attenuation, an exponential damping term $e^{-\overline{\beta }r}$ is introduced, where $\overline{\beta }$ is the acoustic attenuation coefficient, which depends on the properties of the fluid, the US frequency, and the bubble concentration. This term describes the reduction in wave amplitude with distance due to energy dissipation caused by cavitation. Without cavitation, the acoustic wave would undergo only the intrinsic attenuation of the fluid, but in the presence of cavitation, the damping effect becomes predominant, influencing the spatial distribution of the acoustic pressure. Starting from the radial acoustic force calculated without cavitation, the introduction of the term $e^{-\beta r}$ modifies (43) as follows:

$$F_r^{\mathrm{cav}}={\displaystyle \frac{V_p}{r^2}}\left[\overline{A}{e}^{ik(r-ct)}(1-ikr)+\overline{B}{e}^{-ik(r+ct)}(1+ikr)\right]e^{-\overline{\beta }r},$$

(54)

where $\overline{\beta }$ [$1/\mathrm{m}$], representing the rate at which the wave loses energy, directly influences the force exerted on the particles. In the presence of cavitation, this force decreases more rapidly compared to a bubble-free fluid, making the stabilization of particles at large distances from the US source more challenging. The introduction of this attenuation factor is therefore essential to accurately describe the interaction between US and MRF under cavitating conditions. The inclusion of the exponential attenuation $e^{-\overline{\beta }r}$ allows you to realistically model the dissipative effect of cavitation. This is a fundamental step to understand how the action of ultrasound can vary with distance from the source and how this influences the local stability of the MRF.

Cavitation has a significant impact on the stability of MRFs. Here, we analyze the interaction between cavitation, the acoustic pressure generated by spherical US waves, and the distribution of suspended magnetic particles, with the aim of understanding how cavitation may contribute to fluid stability by preventing particle sedimentation.

The formation of microbubbles under the action of spherical waves and their violent collapse generate highly localized pressure impulses. The dynamics of these bubbles are described by the Rayleigh–Plesset equation [59]:

$$\begin{array}{c}\hfill R_b\left(t\right){\displaystyle \frac{d^2{R}_b\left(t\right)}{d{t}^2}}+{\displaystyle \frac{3}{2}}{\left({\displaystyle \frac{d{R}_b\left(t\right)}{dt}}\right)}^2=\\ \hfill ={\displaystyle \frac{1}{\rho }}\left(P_0-P_v+p_a \text{cos}\left(\omega t\right)-{\displaystyle \frac{2\sigma }{R_b\left(t\right)}}-{\displaystyle \frac{4\mu }{R_b\left(t\right)}}{\displaystyle \frac{d{R}_b\left(t\right)}{dt}}-\rho {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2\sigma }{R_b\left(t\right)}}+P_v\right)\right),\end{array}$$

(55)

in which $R_b\left(t\right)$ is the bubble radius [$\mathrm{m}$], $\rho$ is the density of the MRF [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^2$], $P_0$ is the atmospheric pressure [$\mathrm{Pa}$], $P_v$ is the vapor pressure [$\mathrm{Pa}$], $p_a=max\left|p(r,t)\right|$ is the amplitude of $p(r,t)$ [$\mathrm{Pa}$], $\omega$ is the angular frequency [$\mathrm{rad}/\mathrm{s}$], $\sigma$ is the surface tension [$\mathrm{N}\mathrm{m}$], and $\mu$ is the dynamic viscosity [$\mathrm{Pa}\mathrm{s}$].

Equation (55) is a fundamental tool for studying the stability of an MRF, as it describes the dynamics of cavitation bubbles in the fluid under the action of US waves. Starting from (55), we introduce a small perturbation $\delta R_b\left(t\right)$ around the equilibrium radius $R_{b0}$:

$$R_b\left(t\right)=R_{b0}+\delta R_b\left(t\right)\left|\delta R_b\left(t\right)\right|\ll R_{b0}.$$

(56)

Remark 5.

$R_{b0}$ constant in time represents an equilibrium configuration of the system. In this situation, the bubble is not subject to external variations or instabilities: the external pressure is balanced by the internal pressure, the bubble’s expansion velocity is zero, and the time derivatives of the radius vanish. In the context of linearization, we assume that the variation of the radius arises solely from a small, time-dependent perturbation, $\delta R_b\left(t\right)$, while the value around which the expansion is performed, namely $R_{b0}$, remains fixed. This allows the equation to be expanded into a Taylor series, retaining only the first-order terms in $\delta R_b\left(t\right)$, which greatly simplifies the analysis.

Remark 6.

Equation (55) represents the full nonlinear Rayleigh–Plesset equation, which describes the radial dynamics of a gas bubble in a viscous and compressible fluid. To analyze the system’s behavior, it is useful to consider a linearized version around the equilibrium radius $R_b=R_0$, assuming small radial oscillations. In this approximation, the equation reduces to a forced, damped harmonic oscillator, where effective mass (due to fluid inertia), damping (due to viscosity), and stiffness (due to surface tension and internal pressure) terms can be identified. The dynamic response of the system can then be characterized by the dimensionless damping ratio $\zeta ={\displaystyle \frac{c}{2\sqrt{mk}}}$, where m, c, and k are the effective mass, damping, and stiffness coefficients, respectively. Depending on the value of ζ, the system can be underdamped ($\zeta <1$), critically damped ($\zeta =1$), or overdamped ($\zeta >1$). Under the physical conditions considered in this work, and using realistic fluid parameters (e.g., for water or magnetorheological oils), the response is typically underdamped, which is consistent with oscillatory bubble behavior. This implies that the system is stable, exhibiting damped oscillations around the equilibrium radius.

We expand each term of Equation (55) to first order. The inertial term, using Equation (56), yields

$$\begin{array}{c}\hfill R_b\left(t\right){\displaystyle \frac{d^2{R}_b\left(t\right)}{d{t}^2}}=(R_{b0}+\delta R_b\left(t\right))\left({\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}\right)=\\ \hfill =R_{b0}{\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}+\delta R_b\left(t\right){\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}\end{array}$$

(57)

The term $\delta R_b\left(t\right){\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}$ in (57) is neglected because it is of higher order regarding the linearization (a quadratic term in the perturbation $\delta R_b\left(t\right)$). Therefore, Equation (57) takes the following form:

$$\begin{array}{c}\hfill R_b\left(t\right){\displaystyle \frac{d^2{R}_b\left(t\right)}{d{t}^2}}=(R_{b0}+\delta R_b\left(t\right))\left({\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}\right)\approx R_{b0}{\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}\end{array}$$

(58)

Moreover, the kinetic term in Equation (55) is negligible since there

$${\left({\displaystyle \frac{d{R}_b\left(t\right)}{dt}}\right)}^2={\left({\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}\right)}^2$$

(59)

are terms of order $\mathcal{O}\left({\delta }^2\right)$.

To expand the term due to capillary pressure,

$${\displaystyle \frac{2\sigma }{R_b\left(t\right)}}={\displaystyle \frac{2\sigma }{R_{b0}+\delta R_b\left(t\right)}}.$$

(60)

Let us consider the function $f\left(x\right)={\displaystyle \frac{1}{a+x}}$, with $\left|x\right|\ll \left|a\right|$, whose Taylor series expansion around $x=0$ is $f\left(x\right)={\displaystyle \frac{1}{a}}-{\displaystyle \frac{x}{a^2}}+{\displaystyle \frac{x^2}{a^3}}-···$, which, for the first order, yields ${\displaystyle \frac{1}{a+x}}\approx {\displaystyle \frac{1}{a}}-{\displaystyle \frac{x}{a^2}}.$ Therefore, by setting $a=R_{b0}$, $x=\delta R_b\left(t\right)$, we can write the following:

$${\displaystyle \frac{1}{R_b\left(t\right)}}={\displaystyle \frac{1}{R_{b0}+\delta R_b\left(t\right)}}\approx {\displaystyle \frac{1}{R_{b0}}}-{\displaystyle \frac{\delta R_b\left(t\right)}{R_{b0}^2}},$$

(61)

from which

$${\displaystyle \frac{2\sigma }{R_b\left(t\right)}}\approx 2\sigma \left({\displaystyle \frac{1}{R_{b0}}}-{\displaystyle \frac{\delta R_b\left(t\right)}{R_{b0}^2}}\right)={\displaystyle \frac{2\sigma }{R_{b0}}}-{\displaystyle \frac{2\sigma }{R_{b0}^2}}\delta R_b\left(t\right).$$

(62)

To derive the expansion of the viscous term, we first observe that, since $R_{b0}$ is constant,

$${\displaystyle \frac{d{R}_b\left(t\right)}{dt}}={\displaystyle \frac{d\delta R_b\left(t\right)}{dt}},$$

(63)

so that

$$\begin{array}{c}\hfill {\displaystyle \frac{4\mu }{R_b\left(t\right)}}{\displaystyle \frac{d{R}_b\left(t\right)}{dt}}=4\mu \left({\displaystyle \frac{1}{R_b\left(t\right)}}\right){\displaystyle \frac{d{R}_b\left(t\right)}{dt}}\approx \\ \hfill \approx 4\mu \left({\displaystyle \frac{1}{R_{b0}}}-{\displaystyle \frac{\delta R_b\left(t\right)}{R_{b0}^2}}\right){\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}={\displaystyle \frac{4\mu }{R_{b0}}}{\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}-{\displaystyle \frac{4\mu }{R_{b0}^2}}\delta R_b\left(t\right){\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}\end{array}$$

(64)

Obviously, the second term is of order $\mathcal{O}\left({\delta }^2\right)$ and is therefore negligible in the first-order linearization. Thus, Equation (64) yields the following:

$${\displaystyle \frac{4\mu }{R_b\left(t\right)}}{\displaystyle \frac{d{R}_b\left(t\right)}{dt}}\approx {\displaystyle \frac{4\mu }{R_{b0}}}{\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}.$$

(65)

Let us focus on the elastic term. Since $P_v$ is constant over time, the following expression is valid:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2\sigma }{R_b\left(t\right)}}+P_v\right)={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2\sigma }{R_b\left(t\right)}}\right).$$

(66)

Then, considering the Equation (16), and differentiating Equation (66) regarding time, we obtain [59]

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2\sigma }{R_b\left(t\right)}}\right)\approx {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2\sigma }{R_{b0}}}-{\displaystyle \frac{2\sigma }{R_{b0}^2}}\delta R_b\left(t\right)\right)$$

(67)

Finally, considering that $\sigma$ does not depend on time, we can write

$$\rho {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{2\sigma }{R_b\left(t\right)}}+P_v\right)\approx -{\displaystyle \frac{2\sigma \rho }{R_{b0}^2}}{\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}.$$

(68)

The negative sign reflects the fact that an increase in the perturbation $\delta R_b\left(t\right)$ reduces the capillary pressure, since the latter is inversely proportional to the radius. Therefore, taking all contributions into account, Equation (55) takes the following form:

$$\begin{array}{c}\hfill R_{b0}{\displaystyle \frac{d^2\delta R_b\left(t\right)}{d{t}^2}}=\\ \hfill ={\displaystyle \frac{1}{\rho }}(P_0-P_v+P_a \text{cos}\left(\omega t\right)-\left({\displaystyle \frac{2\sigma }{R_{b0}}}-{\displaystyle \frac{2\sigma }{R_{b0}^2}}\delta R_b\left(t\right)\right)-\\ \hfill -{\displaystyle \frac{4\mu }{R_{b0}}}{\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}+{\displaystyle \frac{2\sigma \rho }{R_{b0}^2}}{\displaystyle \frac{d\delta R_b\left(t\right)}{dt}}),\end{array}$$

(69)

which makes it possible to analyze how small changes in the bubble radius affect its dynamics. By studying the solution of this equation, one can determine whether a small perturbation grows or decays over time: a decay implies that the MRF is stable. Stability is achieved when the amplitude of the acoustic pressure $\left|p\right(r,t\left)\right|$ and the viscosity $\mu$ counteract destabilizing forces such as surface tension $\sigma$, thereby keeping the particles suspended and preventing their sedimentation. The linearized form of the Rayleigh–Plesset Equation (69) allows for the dynamic response of microbubbles in the perturbative regime to be evaluated. The stability of the MRF is linked to the fact that the growth of these perturbations is controlled by the viscosity of the medium and the amplitude of the acoustic pressure.

2.4. Effect of Acoustic Cavitation: The Case of Cylindrical US Waves

Acoustic cavitation generated by cylindrical US waves is less significant compared to that induced by spherical waves. This is because the pressure field for cylindrical waves is described by a solution involving Bessel functions, which decay slowly with distance, thereby distributing the acoustic energy over a wider geometry. In contrast, spherical waves concentrate energy in a more localized region. This explains why the pressure field of cylindrical waves is less focused and thus less effective at generating strong pressure gradients and powerful microjets, limiting the effectiveness of cavitation in keeping particles suspended compared to spherical waves. As a result, the stability of an MRF achieved through cylindrical waves is weaker than in the case of spherical waves, as the less-concentrated distribution of acoustic pressure reduces the ability to counteract particle sedimentation over extended time scales.

Remark 7.

The difference between spherical and cylindrical acoustic waves in generating cavitation in MRFs lies in the distribution of acoustic energy and pressure gradients. Spherical waves concentrate energy around the source, producing stronger pressure gradients that more easily trigger cavitation, promoting the formation and collapse of microbubbles that help break up particle aggregates and prevent sedimentation. In contrast, cylindrical waves spread energy over a broader wavefront, resulting in weaker pressure gradients and less pronounced cavitation effects. For this reason, the present study gives preference to spherical waves, as they are considered physically more effective in promoting MRF stability through induced cavitation [60,61].

Remark 8.

In the stabilization of MRFs using ultrasound, three main physical phenomena play a crucial role: acoustic radiation force, cavitation, and micro-vortices. The acoustic radiation force, generated by the pressure gradient of the wave, exerts a net push on the particles, directing them toward regions of minimum or maximum pressure and helping to counteract sedimentation. Cavitation occurs when the fluid pressure drops below its vapor pressure, leading to the formation of microbubbles that collapse violently, producing microjets and turbulence capable of breaking up particle aggregates and promoting a more uniform distribution. Micro-vortices are rotational currents induced by non-uniform pressure fields, which keep particles suspended through continuous mixing of the fluid. The combined action of these three mechanisms ensures dynamic stability and a uniform distribution of magnetic particles within the fluid [62].

The SWOT analysis associated with the application of the radial acoustic pressure theory is summarized in

Table 1. SWOT analysis of the technique based on radial acoustic pressure theory.

Strengths	Highly effective technology for preventing particle sedimentation and maintaining uniform distribution, featuring flexible magnetic modulation and versatile applications while optimizing energy efficiency.
Weaknesses	The system’s effectiveness relies on accurate selection of operational parameters, as suboptimal choices may promote particle aggregation and complicate analysis due to the coexistence of stationary and oscillating components.
Opportunities	The development prospects for this technology are promising, with potential improvements in efficiency through the use of higher-susceptibility materials and real-time optimization. Its versatility makes it suitable for both industrial and biomedical applications.
Threats	The threats include sensitivity to environmental variations, mechanical degradation from cyclic stress, and competition from more widely adopted alternative technologies.

.

3. Industrial and Biomedical Applications: Recent Innovations in Particle Stabilization Through RAP Techniques

3.1. Optimization of the Rheological Response of MRF: Dynamic Control for Enhanced Stability and Performance in Damping Systems

Optimizing the rheological response of MRFs is fundamental for the development of intelligent vibration control devices, especially in high-performance applications. Although MRFs offer effective semi-active control thanks to their rapid viscosity change under a magnetic field, their efficiency strongly depends on microstructural stability and the ability to maintain a uniform particle distribution, avoiding sedimentation and aggregation that would compromise their response [63,64].

Recent studies have shown that magnetic fields varying in intensity and frequency help maintain a uniform suspension of particles, improving stability even under dynamic conditions [65]. This enables more precise control of the viscoelastic response, reducing hysteresis and improving mechanical reversibility, with significant benefits for damping devices in intelligent vehicles, seismic protection systems, and active prostheses [66].

The use of multiphysics models that integrate magnetic, fluid dynamic, and mechanical aspects has improved the understanding of the internal dynamics of MRFs, including interparticle effects and local field gradients. Recent studies strongly suggest that real-time feedback strategies, supported by magnetic sensors and predictive algorithms, allow for the adaptation of operating conditions, ensuring optimal performance even in complex environments [67,68].

Rheological optimization of MRFs enhances energy efficiency and system durability, reducing the need for maintenance and ensuring consistent performance even under prolonged stress. To this end, improving fluid properties through stabilizing additives, particle coatings, and advanced carrier fluids represents a promising research direction [69].

3.2. Adaptive Materials for Acoustic Control: The Integration of US and Magnetic Fields to Optimize the Performance of MRFs

The interaction between US and magnetic fields represents an innovative approach to optimizing the performance of MRFs. The acoustic–magnetic combination enables precise modulation of the fluid’s internal microstructure, leveraging not only the static forces of magnetic fields but also the mechanical action of high-frequency acoustic waves, which generate local pressures and vibrations useful for controlling particle distribution [70,71,72,73].

This multidisciplinary approach improves the stability, homogeneity, and operational efficiency of MRFs, especially in contexts requiring a fast, reversible, and locally controlled fluid response. US breaks up particle clusters, preventing compact structures that would increase viscosity and reduce acoustic transparency, while the magnetic field aligns the particles along preferred directions, enhancing the fluid’s controlled anisotropy [74,75]. The harmonization of these effects enables precise control over the material’s mechanical and acoustic properties.

One of the most promising applications of the integration between US and magnetic fields lies in advanced biomedical devices, where high-resolution US imaging is essential while preserving fluid dynamics. This combination helps maintain particles in suspension, preventing sedimentation and ensuring good acoustic transmission—also useful for therapies such as targeted drug delivery or acoustic–magnetic stimulation of deep tissues [76,77]. Recent studies have shown that simultaneous acoustic–magnetic stimulation produces nonlinear effects in particle distribution, fostering the development of smart fluids sensitive to combined stimuli [78], consistent with earlier observations on sedimentation reduction via oscillating fields [79]. The integration of these technologies thus represents a new design paradigm, where the synergistic interaction between mechanical and magnetic forces produces emergent properties, opening new perspectives for advanced technological and biomedical applications.

3.3. Tunable Acoustic Metamaterials: The Use of Programmable Acoustic Response for Advanced Sound Wave Control

Tunable acoustic metamaterials are among the most promising innovations for advanced control of sound waves, due to their ability to overcome the limitations of natural materials in manipulating acoustic propagation. Composed of sub-wavelength periodic structures, their mechanical response can be actively modulated, enabling spatial and temporal control over acoustic properties. Compared to conventional materials, these programmable metamaterials offer independent control over refractive index, impedance, and phase velocity, unlocking advanced applications such as reconfigurable acoustic lenses, dynamic focusing, and adaptive sound-absorbing barriers.

The tunability of metamaterials can be achieved through electromechanical actuators, magnetic fields, thermal variations, or, increasingly, through acoustic or electroacoustic modulation. By integrating these structures with smart fluids, such as MRF or ferrofluids, fluidic metamaterials with real-time reconfigurable acoustic properties can be obtained. The interaction between the material and the external field enables a dynamic and reversible acoustic response to varying stimuli [51,52,53].

Programmable acoustic metasurfaces enable dynamic control of reflected or transmitted wavefronts, allowing for the synthesis of planar, conical, or focused waves without volumetric structures. This technology is applicable not only to US beam forming but also to broadband underwater communication, high-resolution non-invasive imaging, and acoustic manipulation of suspended particles—such as in levitation or contactless transport of cells and biological materials.

Recent studies have shown that digital acoustic programming can be achieved using phase-controllable electroacoustic components, paving the way for intelligent acoustic metamaterials with precise control at the metamolecular level. In this field, hybrid architectures are emerging that combine rigid and fluid elements, integrating geometric precision with rheological adaptability. Key contributions include Zhang et al., who developed a reconfigurable-topology metamaterial for three-dimensional control of pressure fields in liquids [80], and the introduction of adaptive analog circuits in metamaterial cells by Popa et al. [81].

The integration of tunable acoustic metamaterials into biomedical devices opens up innovative solutions for diagnostics and therapy. Emerging applications include flexible US probes that conform to the body surface and systems for dynamic focusing of US energy. Moreover, these materials can be employed in US-guided non-invasive surgery, enabling precise control over beam intensity and direction for high-precision targeted treatments while minimizing damage to healthy tissues [82].

3.4. ARF and US Techniques for Improving Concrete Properties

The use of advanced physical techniques, such as radio frequency (RF) and US, has opened new possibilities for improving the mechanical properties and durability of concrete. RF, through controlled volumetric heating, accelerates cement hydration and promotes a more homogeneous matrix, reducing micro-cracks and increasing early strength [83,84]. US, on the other hand, enhances the dispersion of components, especially in reactive mixes or those containing nanoparticles, reducing porosity and strengthening the bond between paste and aggregates, both in the fresh state and during curing [85,86].

The combination of RF and US amplifies these effects, reducing permeability and improving durability, particularly in aggressive environments, such as marine structures or those exposed to freeze–thaw cycles [87,88]. These technologies are suitable for in situ applications, thanks to their adjustability based on the environment and type of concrete. In particular, US treatments during curing in humid environments improve saturation and adhesion between the matrix and aggregates, proving advantageous for 3D printing and high-performance precast elements [89,90].

Two key studies are [85], on the effect of US on cement crystallization and microstructure, and [83], on the use of RF for rapid curing and the enhancement of early mechanical strength and resistance to chloride penetration.

3.5. MRF and US: Advanced US Diagnostics

The integration of MRF and US represents an innovative approach in the design of high-performance biomedical US systems, particularly useful in point-of-care settings, emergencies, and resource-limited environments [91,92]. The MRF-US combination enables the development of intelligent fluid interfaces capable of real-time modulation of properties such as viscosity, acoustic impedance, and particle stability [91,92].

In US devices, image quality depends on the acoustic transparency of the coupling medium. MRFs, used as active rheological layers, can adapt to both tissue and signal but suffer from sedimentation of the ferromagnetic particles, which leads to signal artifacts [93,94,95].

US waves effectively address this issue: RAP generates an ARF that keeps the particles suspended [96,97]. Additionally, acoustic cavitation—triggered when the wave pressure exceeds the vapor threshold—breaks down aggregates through microjets and local turbulence, thereby improving the medium’s transparency [98,99].

The combined use of US and magnetic fields allows for the formation of oriented structures compatible with sound propagation, reducing scattering and attenuation [100]. This is particularly beneficial for portable or wearable devices, ensuring efficiency even under variable operating conditions [101].

In practical applications, US probes with MRF-US interfaces conform to the patient’s anatomy, optimizing transmission and reducing the required mechanical pressure. The use of acoustically stabilized MRFs minimizes the need for chemical additives, improving compatibility and lowering costs [102].

Probes for endocavitary imaging are under development [103], along with flexible patches for continuous vascular monitoring [104], and mobile devices for rapid diagnosis in military or humanitarian emergency scenarios [105]. Integration with active control technologies and predictive algorithms also enables real-time automatic calibration of the visco-acoustic response [106].

3.6. Effects of Acoustic Transparency on Phase Coherence and Scattering

In systems based on MRFs stabilized by ultrasound, acoustic transparency is a key property for ensuring proper wave propagation and maintaining signal integrity. It refers to the medium’s ability to transmit ultrasonic waves without causing significant attenuation, unwanted reflections, or phase distortion. This condition depends on the uniform distribution of suspended particles and the structural stability of the fluid. When the fluid contains aggregates, local density gradients, or microscopic viscosity variations, the waves encounter discontinuities that lead to scattering and local refraction. These effects disrupt phase coherence—i.e., the synchronicity between wave components—resulting in degraded signal or image quality. Scattering may also cause unwanted energy absorption and reduced wave intensity. A well-designed combination of magnetic field and ultrasound can minimize these inhomogeneities. Ultrasonic waves, through acoustic pressure and controlled cavitation, act as a “dynamic mixer” that keeps the particles dispersed and prevents the formation of dense or sedimented structures. Simultaneously, a modulated magnetic field inhibits the static alignment of particles, which would otherwise create local anisotropies. This synergistic interaction produces an acoustically homogeneous fluid medium, with reduced local variations in impedance and density. As a result, acoustic transparency improves, allowing coherent wave propagation and significantly enhancing spatial resolution, image quality, and the effectiveness of acoustic therapies based on focusing or controlled aberration [107].

3.7. MRF and US: US-Mediated Drug Delivery

The integration of MRF and US represents an innovative approach to targeted drug delivery, with great potential for precision medicine. MRFs, due to their dynamic response to magnetic fields, can act as active carriers for the localized release of therapeutic agents, while focused US generates acoustic pressure gradients that selectively guide particles toward specific anatomical targets [70,108,109,110].

Mechanical effects such as acoustic radiation and controlled cavitation facilitate active drug transport and permeation through sonoporation, which can be modulated to avoid tissue damage [111,112]. The external magnetic field ensures the suspension and localization of particles, even in deep regions [65,113].

This combination has shown effectiveness in preclinical models for tumors, infections, and neurodegenerative diseases, with the development of implantable or portable miniaturized devices based on magnetic micro-coils and piezoelectric transducers [114,115].

Beyond targeted drug delivery and imaging, the proposed magneto–ultrasound systems also show promising potential in other clinical domains. In particular, applications in orthopedic surgery and regenerative medicine have gained increasing attention. Magnetically controlled scaffolds and ultrasound-assisted stimulation have been proposed to enhance bone repair, modulate cell behavior, and accelerate tissue regeneration [116,117]. These technologies support patient-specific therapies and may significantly reduce recovery times by enabling minimally invasive, spatiotemporally controlled interventions [118,119].

The use of biocompatible particles with multi-stimuli-responsive coatings enhances selectivity [120], while the integration of sensors and predictive models based on artificial intelligence enables real-time adaptation of operating conditions [121,122].

Overall, the MRF–US synergy enables advanced theranostic systems for controlled drug release and simultaneous imaging, with concrete prospects for technology transfer toward personalized therapies with low systemic impact [123,124,125].

3.8. Potential Energy Losses and Transducer Limitations

The integration of MRF and US represents an innovative approach to targeted drug delivery, with great potential for precision medicine. MRFs, due to their dynamic response to magnetic fields, can act as active carriers for the localized release of therapeutic agents, while focused US generates acoustic pressure gradients that selectively guide particles toward specific anatomical targets [70,108,110]. Mechanical effects, such as acoustic radiation and controlled cavitation, facilitate active drug transport and permeation through sonoporation, which can be modulated to avoid tissue damage [111,112]. The external magnetic field ensures the suspension and localization of particles, even in deep regions [65,113]. This combination has shown effectiveness in preclinical models for tumors, infections, and neurodegenerative diseases, with the development of implantable or portable miniaturized devices based on magnetic micro-coils and piezoelectric transducers [114,115]. The use of biocompatible particles with multi-stimuli-responsive coatings enhances selectivity [120], while the integration of sensors and predictive models based on artificial intelligence enables real-time adaptation of operating conditions [121,122].

However, some limitations and energy losses must be taken into account when designing MRF–US-based delivery systems. One critical issue is the efficiency of piezoelectric ultrasound transducers, which may experience mechanical or dielectric losses, especially under continuous or high-frequency operation. These losses can reduce the effective acoustic pressure delivered to the target and generate local heating, which may compromise safety or therapeutic precision [121,122]. Acoustic attenuation in biological tissues (e.g., bone, fat, or air-filled regions) further contributes to energy loss, reducing penetration depth and focal accuracy. This can limit the application of ultrasound-based actuation in deep-seated targets unless compensated by power control or phased-array focusing. In the magnetic domain, eddy current losses and magnetic hysteresis can arise in embedded coils or in surrounding conductive tissues, especially at high frequencies. These effects reduce the net magnetic force acting on the MRF and increase power consumption. Moreover, miniaturized coils may suffer from reduced field gradients, limiting actuation performance. To address these challenges, current strategies include optimizing transducer design through impedance matching and thermal management, using resonant actuation modes, and developing hybrid transducers with integrated feedback. Smart control systems can dynamically adapt power delivery to tissue properties and geometry, reducing energy waste while maintaining effective stimulation.

Overall, despite these limitations, the MRF–US synergy enables the development of advanced theranostic systems for controlled drug release and simultaneous imaging, with concrete prospects for clinical translation toward personalized therapies with minimal systemic exposure [123,124,125].

3.9. MRF and US: Localized Ablation (US-Assisted Magnetic Hyperthermia)

Localized thermal ablation based on the interaction between MRF and US represents a promising non-invasive therapeutic technique for the treatment of deep-seated tumors [108,112].

In this context, radial acoustic pressure plays a key role in energy focusing and hyperthermia optimization [126,127].

MRFs, composed of ferromagnetic nanoparticles suspended in biocompatible liquid matrices, generate heat under alternating magnetic fields through Néel and Brownian relaxation mechanisms [65]. The concentration of particles in the tumor area—also achieved through acoustic compression—allows for the therapeutic threshold (42–45 degrees Celsius) to be reached more rapidly [113].

Concentric or spherical US transducers generate acoustic pressure that concentrates magnetic particles, increasing local density and thermal efficiency under a magnetic field [111]. Acoustic energy also directly contributes to tissue heating [128].

Synchronization between the US and the magnetic field, supported by integrated HIFU and imaging systems (US or MRI), enables real-time thermal control [129]. Acoustophoresis techniques further enhance the dynamic localization of particles [125].

This technology is applicable in the treatment of liver, brain, pancreatic, and kidney tumors, offering targeted thermal efficacy with high safety [110]. It can also be integrated with controlled release of thermally activated drugs and localized immune stimulation [113,126].

Thanks to functionalized materials and adaptive control systems, the approach based on radial acoustic pressure in the presence of MRFs is emerging as a customizable and minimally invasive therapeutic option for proximity oncology [129].

3.10. Limitations of Acousto-Magnetic Models in Biomedical Implants

Despite the theoretical effectiveness of the acousto-magnetic approach in stabilizing MRFs, its application in biomedical implants presents several structural and operational limitations. First, the mathematical models often assume a homogeneous and isotropic medium for ultrasonic wave propagation—a condition rarely met in real human tissues, which are inherently heterogeneous, anisotropic, and layered. This discrepancy can lead to significant deviations between the predicted theoretical behavior and the actual system response once implanted. Second, the equations governing acoustic wave propagation and magnetic field distribution are typically solved in idealized domains (such as spherical or cylindrical geometries) under stationary or quasi-linear conditions. However, biomedical implants operate in dynamic environments characterized by local physiological variations—such as blood flow, temperature changes, and tissue micro-movements—that are not fully captured by these models. Another important limitation is the difficulty in accurately quantifying local parameters, such as medium viscosity, the actual distribution of magnetic particles, and the effective intensity of the magnetic field within the human body. These variables are essential for reliable predictions but are often estimated indirectly or approximately, reducing the model’s accuracy. Lastly, from a safety perspective, the long-term interactions between ultrasonic waves, magnetic fields, and sensitive biological tissues are not always fully addressed. Although some models account for phenomena like cavitation and acoustic radiation force, biothermal and cellular effects are still not comprehensively integrated into the modeling framework [130].

3.11. MRF and US: Tissue Engineering

In the biomedical field, the ability to modulate the physical properties of MRF in real time offers a significant advantage, particularly in ensuring diagnostic precision, high resolution, and acoustic stability [79,131]. The use of dynamic magnetic configurations allows for the optimal suspension of particles during US imaging and reduces energy consumption during idle phases. This makes MRFs suitable not only for fixed devices but also for portable and wearable “point-of-care” systems, ideal for emergencies, military settings, or areas with limited healthcare resources [79,131]. Beyond diagnostics, magnetically controlled MRFs are also used in US-guided therapies, such as targeted drug delivery, where viscosity modulation enhances targeting [132], or in ablative treatments like US-assisted hyperthermia, where they serve as multifunctional mediators [133]. In tissue engineering, they can also act as fluid scaffolds to stimulate cells and tissues through combined magnetic and US fields [68]. In all these applications, maintaining the acoustic transparency of the fluid is essential: precise control of the microstructure prevents inhomogeneities and scattering, improving phase coherence and the quality of the US signal. Optimized through advanced magnetic control techniques, MRFs thus emerge as versatile platforms for high-performance fluid interfaces in biomedical imaging systems, enabling integrated diagnostic and therapeutic functionalities [79].

3.12. Challenges in the Implementation of the RAP Technique

Despite recent advancements in the application of the Radial Acoustic Pumping (RAP) technique with radial configuration, its practical implementation still presents significant challenges. One of the main issues is energy efficiency: the continuous operation of acoustic generators entails considerable power consumption, which is particularly problematic in long-term applications or in energy-constrained environments, such as portable or autonomous devices [71,134]. Moreover, thermal dissipation can lead to undesirable effects on system stability [135]. Another critical issue concerns the integration of US transducers into existing systems, especially in complex or miniaturized geometries, where structural modifications are required, increasing both costs and complexity, as well as introducing compatibility issues with other components [136,137]. The optimization of transducer geometry, piezoelectric materials, and their coupling with the medium remains an active area of research [77,138]. For the effective deployment of the radial RAP technique, it is therefore necessary to address energy and design challenges in an integrated manner, developing scalable, compatible, and low-impact solutions [139].

Further examples of industrial and biomedical applications based on the radial acoustic pressure technique are reported in

Table 2. RAP for MRF and US: further industrial applications.

Ref.	Domain	Methods	Conclusions
[108,110,111,112,113,125,126,127,128,129,140,141]	Field-guided microfabrication and assembly	Combined use of acoustic and magnetic fields to guide the controlled self-assembly of particles into functional three-dimensional structures.	$\mathbf{H}$ and RAP enables precise, rapid, and reconfigurable particle assembly for advanced 3D structures in micro-robotics and bioengineering.
[142,143,144,145,146,147]	Intelligent lubrication in complex environments	MRF and US integration enables adaptive lubrication with improved stability and tribological performance.	Smart MRF-US lubrication adapts dynamically to improve efficiency and reduce wear.
[148,149,150,151,152,153,154,155,156]	Active control of vibrations and noise	MR devices with predictive models and algorithms enable active vibration control in industrial structures.	Actively controlled MR materials improve vibration damping, structural efficiency, and durability in advanced industries.
[146,157,158,159,160,161]	Advanced liquid treatment and magneto-acoustic separations	Magnetic fields and US enable precise control of MRF for imaging, localized heating, and liquid separation applications.	MRF and US coupling enables advanced liquid separation and fluid control for diagnostics and therapy.

and

Table 3. RAP for MRF and US: further biomedical applications.

Ref.	Domain	Methods	Conclusions
[162,163,164,165,166,167]	Non-invasive focused neuro-stimulation	3D-guided focused US enables precise, non-invasive deep brain stimulation.	Precise and safe technique with promising clinical applications in neurology and advanced neuro-therapy.
[168,169,170,171,172]	Dynamic control of the tumor microenvironment for immunotherapy	US-activated magnetic nano-carriers enhance tumor immunotherapy by locally modulating the microenvironment.	Localized tumor microenvironment modulation enhances immunotherapy efficacy in preclinical models.
[173,174,175,176,177,178]	Acceleration of bone and muscle healing	US, magnetic fields, and injectable hydrogels promote tissue regeneration via mechanotransduction and inflammation control.	Enhances bone and muscle healing by promoting cell growth and tissue repair.
[179,180,181,182,183,184,185]	Intracranial navigation and delivery across the blood-brain barrier	Focused US with microbubbles and AI enables precise blood-brain barrier opening for targeted drug delivery.	Safe, targeted blood-brain barrier opening improves drug delivery precision and efficacy.

, respectively, which illustrate the emerging technological contexts in which this approach proves particularly effective.

4. Effect of the Time-Varying Magnetic Field

The sedimentation of particles, caused by gravity, can be mitigated by applying a time-varying magnetic field, $\mathbf{H}\left(t\right)$ [$\mathrm{A}/{\mathrm{m}}^2$].

Let us first observe that each particle is subject to the gravitational force ${\mathbf{F}}_g$ [$\mathrm{N}$], with magnitude

$$|{\mathbf{F}}_g|=V_p{\rho }_{p}g$$

(70)

where ${\rho }_p$ is the particle density [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$], g is the gravitational acceleration [$\mathrm{m}/{\mathrm{s}}^2$], and $V_p$ is the volume of a single particle (assumed spherical) [${\mathrm{m}}^3$]. From a dimensional point of view, the volume $V_p$ has dimension $L^3$, the density ${\rho }_p$ has dimension $M{L}^{-3}$, and the gravitational acceleration g has dimension $L{T}^{-2}$. Multiplying the three terms yields $\left[L^3\right]·\left[M{L}^{-3}\right]·\left[L{T}^{-2}\right]=ML{T}^{-2}$, which corresponds exactly to the dimension of force in the International System of Units, that is, $ML{T}^{-2}$, equivalent to a Newton ($\mathrm{kg}·\mathrm{m}/{\mathrm{s}}^2$).

$$V_p={\displaystyle \frac{4}{3}}\pi R^3$$

(71)

where R represents the radius of the particle [$\mathrm{m}$].

The particle is also subjected to the buoyant force ${\mathbf{F}}_b$ [$\mathrm{N}$], with magnitude

$$|{\mathbf{F}}_b|=V_p{\rho }_{f}g$$

(72)

where ${\rho }_f$ is the density of the fluid [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$]. Since $V_p$ is the particle volume $\left[{\mathrm{m}}^3\right]$, ${\rho }_f$ is the fluid density $[\mathrm{kg}/{\mathrm{m}}^3]$, and g is the gravitational acceleration $[\mathrm{m}/{\mathrm{s}}^2]$. The product $V_p{\rho }_{f}g$ therefore has units $\left[{\mathrm{m}}^3\right]·[\mathrm{kg}/{\mathrm{m}}^3]·[\mathrm{m}/{\mathrm{s}}^2]=[\mathrm{kg}·\mathrm{m}/{\mathrm{s}}^2]=\left[\mathrm{N}\right]$, which confirms the dimensional consistency of the equation.

Moreover, due to the dynamic viscosity of the fluid, $\eta$ [$\mathrm{Pa}\mathrm{s}$], a Stokes force ${\mathbf{F}}_v\left(t\right)$ [$\mathrm{N}$] acts on the particle, which is moving with a speed v [$\mathrm{m} {\mathrm{s}}^{-1}$], with magnitude

$$|{\mathbf{F}}_v\left(t\right)|=6\pi \eta Rv\left(t\right).$$

(73)

In Equation (73), the term $|{\mathbf{F}}_v\left(t\right)|$ is the magnitude of the force, which has the dimensions of force, that is, newtons $\left[\mathrm{kg}·{\mathrm{m}/\mathrm{s}}^2\right]$. On the right-hand side, $\eta$ is the dynamic viscosity of the fluid, with dimensions of pascal-seconds $[\mathrm{kg}/(\mathrm{m}·\mathrm{s}\left)\right]$; R is the radius of the sphere, with dimensions $\left[\mathrm{m}\right]$; and $v\left(t\right)$ is the speed of the sphere, with dimensions $\left[\mathrm{m}/\mathrm{s}\right]$. Multiplying the units $\left[{\displaystyle \frac{\mathrm{kg}}{\mathrm{m}·\mathrm{s}}}\right]·\left[\mathrm{m}\right]·\left[{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}\right]={\displaystyle \frac{\mathrm{kg}·{\mathrm{m}}^2}{{\mathrm{s}}^2}}=\left[\mathrm{N}\right]$. Thus, both sides of the equation have consistent units, confirming that the equation is dimensionally correct. Finally, denoting by ${\mu }_0$ [$\mathrm{H} {\mathrm{m}}^{-1}$] the magnetic permeability of vacuum and by $\chi$ the (dimensionless) magnetic susceptibility of the particle, the particle is subject to a time-varying magnetic force ${\mathbf{F}}_m(x,y,z,t)$ [$\mathrm{N}$], whose magnitude can be quantified as

$$|{\mathbf{F}}_m(x,y,z,t)|={\mu }_0\chi V_p\left|\mathbf{H}(x,y,z,t)\right||\nabla \mathbf{H}(x,y,z,t)|.$$

(74)

To avoid any ambiguity, we clarify the physical dimensions of the parameters involved in Equation (74). The force $F_m$ has dimensions of Newtons, that is, $\left[ML{T}^{-2}\right]$. The parameter ${\mu }_0$, representing the magnetic permeability of vacuum, has units of henries per meter, namely $[H/\mathrm{m}]=\left[N·A^{-2}\right]$. The magnetic susceptibility $\chi$ is dimensionless. The particle volume $V_p$ is expressed in cubic meters $\left[{\mathrm{m}}^3\right]$. The magnetic field magnitude $\left|H\right|$ is measured in amperes per meter $[A/\mathrm{m}]$, and its spatial gradient $|\nabla H|$ has dimensions $[A/{\mathrm{m}}^2]$. By combining the dimensions of all the terms on the right-hand side of the equation, we obtain the following:

$$[{\mu }_0\chi V_p\left|\mathbf{H}(x,y,z,t)\right||\nabla \mathbf{H}(x,y,z,t)|]=\left[{\displaystyle \frac{N}{A^2}}\right]·\left[1\right]·\left[{\mathrm{m}}^3\right]·\left[{\displaystyle \frac{A}{\mathrm{m}}}\right]·\left[{\displaystyle \frac{A}{{\mathrm{m}}^2}}\right]=\left[N\right],$$

(75)

which confirms that the entire expression has the physical dimension of force. Therefore, Equation (74) is dimensionally consistent. It is worth noting that the gravitational force is taken as positive under the convention of a downward vertical axis; the buoyant force acts in the opposite direction to gravity, opposing sedimentation, just as the component of the Stokes drag force is negative, opposing the motion of the particles; therefore, if the particle moves downward, the viscous force is directed upward. Finally, the component of the magnetic force is negative because the time-varying magnetic field is applied in such a way as to counteract sedimentation, thus acting in the same direction defined as positive. To derive the equation of motion for the particle, we apply the second law of dynamics, obtaining

$$m{\displaystyle \frac{dv\left(t\right)}{dt}}=|{\mathbf{F}}_g|-|{\mathbf{F}}_b|-|{\mathbf{F}}_v\left(t\right)|-|{\mathbf{F}}_m(x,y,z,t)|,$$

(76)

From which, considering that $m={\rho }_p{V}_p$, we can write

$${\rho }_p{V}_p{\displaystyle \frac{dv\left(t\right)}{dt}}=V_p({\rho }_p-{\rho }_f)g-6\pi \eta Rv\left(t\right)-{\mu }_0\chi V_p\left|\mathbf{H}\right||\nabla \mathbf{H}|.$$

(77)

Finally, by dividing by ${\rho }_p{V}_p$, we obtain

$${\displaystyle \frac{dv\left(t\right)}{dt}}+{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}v\left(t\right)=g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)-{\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}.$$

(78)

The equation of motion (78) shows how the velocity of the particle depends on the balance between the gravitational push, the viscous resistance, and the magnetic force. This model offers a useful quantitative framework for designing magnetic configurations that prevent sedimentation. The sedimentation of particles is mitigated when their velocity $v\left(t\right)$ decreases or tends to zero over time. The absence of sedimentation corresponds to a steady-state condition, in which the terminal velocity is null, i.e., $v_s=0$. To characterize this regime, it is sufficient to consider the limit as $t\to \infty$:

$${\displaystyle \frac{dv\left(t\right)}{dt}}=0.$$

(79)

This condition indicates that the net force acting on the particle is zero, implying dynamic equilibrium. In this context, setting the time derivative of velocity to zero is justified, as it defines the state in which gravitational, magnetic, and viscous forces are exactly balanced. So, from Equation (78), it makes sense to write

$$v_s={\displaystyle \frac{{\rho }_p{V}_p}{6\pi \eta R}}\left[g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)-{\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}\right].$$

(80)

To verify that the right-hand side of the Equation (80) has the dimensions of a velocity, let us first analyze the prefactor ${\rho }_p{V}_p/\left(6\pi \eta R\right)$. The particle density ${\rho }_p$ has dimensions $[\mathrm{kg}/{\mathrm{m}}^3]$, and the volume $V_p$ has dimensions $\left[{\mathrm{m}}^3\right]$, so their product yields mass $\left[\mathrm{kg}\right]$. The denominator contains the dynamic viscosity $\eta$, with dimensions $[\mathrm{kg}/(\mathrm{m}·\mathrm{s}\left)\right]$, multiplied by a length R with dimensions $\left[\mathrm{m}\right]$, resulting in $[\mathrm{kg}/\mathrm{s}]$. Therefore, the entire prefactor has dimensions $\left[\mathrm{kg}\right]/[\mathrm{kg}/\mathrm{s}]=\left[\mathrm{s}\right]$, i.e., time. Next, we analyze the terms inside the square brackets. The first term is $g(1-{\rho }_f/{\rho }_p)$, where g is the gravitational acceleration with dimensions $[\mathrm{m}/{\mathrm{s}}^2]$, and the quantity in parentheses is dimensionless. Thus, this term has dimensions $[\mathrm{m}/{\mathrm{s}}^2]$. The second term is magnetic in nature, ${\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}$. The magnetic susceptibility $\chi$ is dimensionless. The magnetic field $\left|\mathbf{H}\right|$ has dimensions $[\mathrm{A}/\mathrm{m}]$, and its gradient $|\nabla \mathbf{H}|$ has dimensions $[\mathrm{A}/{\mathrm{m}}^2]$, so their product gives $[{\mathrm{A}}^2/{\mathrm{m}}^3]$. The magnetic permeability of free space ${\mu }_0$ has dimensions $[\mathrm{kg}·\mathrm{m}/\left({\mathrm{A}}^2·{\mathrm{s}}^2\right)]$. Multiplying all these terms together yields $[{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|]=\left[{\displaystyle \frac{\mathrm{kg}·\mathrm{m}}{{\mathrm{A}}^2·{\mathrm{s}}^2}}\right]·\left[{\displaystyle \frac{{\mathrm{A}}^2}{{\mathrm{m}}^3}}\right]=\left[{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^2·{\mathrm{s}}^2}}\right]$. Dividing by ${\rho }_p$ with dimensions $[\mathrm{kg}/{\mathrm{m}}^3]$ gives $\left[{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^2·{\mathrm{s}}^2}}\right]·\left[{\displaystyle \frac{{\mathrm{m}}^3}{\mathrm{kg}}}\right]=\left[{\displaystyle \frac{\mathrm{m}}{{\mathrm{s}}^2}}\right]$, which again corresponds to an acceleration. Therefore, the entire bracketed term has dimensions of acceleration $[\mathrm{m}/{\mathrm{s}}^2]$, and when multiplied by the prefactor with dimensions $\left[\mathrm{s}\right]$, the result is a velocity, $\left[\mathrm{s}\right]·[\mathrm{m}/{\mathrm{s}}^2]=[\mathrm{m}/\mathrm{s}]$.

To eliminate sedimentation, it is necessary that $v_s=0$, so that

$${\displaystyle \frac{{\rho }_p{V}_p}{6\pi \eta R}}\left[g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)-{\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}\right]=0.$$

(81)

Therefore, the condition $v_s=0$ reduces to the following:

$$g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)-{\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}=0,$$

(82)

We observe that the term $g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)$ in Equation (82) represents the contribution of the net gravitational force that causes the particles to settle downward. In particular, in an MRF, it holds that ${\rho }_f<{\rho }_p$ (the fluid is less dense than suspended particles), so that

$$g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)>0.$$

(83)

Therefore, when the effective weight exceeds the magnetic force, we have

$$g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)>{\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}$$

(84)

so that $v_s>0$ and the particle settles under gravity. Conversely, when the magnetic force is stronger than the effective weight, we obtain

$${\displaystyle \frac{{\mu }_0\chi \left|\mathbf{H}\right||\nabla \mathbf{H}|}{{\rho }_p}}\ge g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)$$

(85)

and the particle rises, that is, it moves upward through the fluid. Condition (85) represents an engineering criterion for sizing the magnetic field: if the magnetic force exceeds the gravitational resultant, the particle does not sediment. This constraint guides the design of magnetic control systems in real devices.

4.1. Sinusoidally Modulated $\mathbf{H}$

The magnetic field $\mathbf{H}$ in an MRF can be formulated in different ways depending on practical requirements and system characteristics, but the choice of a sinusoidal modulation of $\mathbf{H}$ has proven to be by far the most efficient in many practical applications. This type of modulation, based on a periodic variation of the field strength according to a sinusoidal function, maintains a homogeneous distribution of particles, effectively counteracting sedimentation as it balances the gravitational force and prevents particle aggregation (static fields tend to promote the formation of particle chains) by reducing the formation of magnetic chains. In applications such as adaptive suspension systems, active vibration control, and seismic isolation devices, sinusoidal modulation provides an optimal balance between energy efficiency and dynamic performance.

4.2. Stability Criteria and Physical Interpretation

The effectiveness of time-varying magnetic fields in stabilizing MRF depends on their ability to balance the forces acting on suspended particles, counteracting sedimentation. The magnetic field must generate a force sufficient to offset gravity—related to the density difference between particles and fluid—and overcome the viscous resistance to movement. Stability is achieved when these forces remain balanced over time, keeping the particles uniformly distributed and preventing the formation of aggregates that would compromise the fluid’s properties. Static magnetic fields tend to promote particle alignment and aggregation, whereas sinusoidal modulation of the field proves more effective: a mean component counteracts gravity, while an oscillating component disrupts the formation of stable structures. Maintaining a homogeneous suspension depends on several design parameters: the intensity of the mean field, the amplitude and frequency of modulation, and the spatial gradient of the magnetic field. The intensity must match the gravitational force, the modulation should break up particle chains before they stabilize, and the frequency must effectively interfere with sedimentation dynamics. The spatial gradient is paramount for properly orienting the magnetic force. When these parameters are well balanced, particles remain suspended and their velocity approaches zero, indicating that sedimentation has been effectively suppressed. This condition represents the fundamental stability criterion for using time-varying magnetic fields and is essential for applications requiring long-term suspension stability, such as biomedical devices, seismic isolation systems, and smart fluidic materials [186].

Remark 9.

In addition to sinusoidal modulation, there are other formulations, such as step fields, sawtooth fields, and random modulation, which present limitations in achieving uniform particle distribution. Step fields, useful for rapid rheological adjustments, tend to promote the formation of aggregates. Sawtooth fields, employed in cooling systems, are less effective at maintaining a stable suspension. Randomly modulated fields, while preventing the formation of stable structures, often result in higher energy consumption.

Remark 10.

Although the magnetic field is, in general, a vector quantity that depends on both space and time, in this section, it is represented as $\mathbf{H}$ with a slight abuse of notation: only its magnitude is considered, and the field is treated as a scalar function of time. This simplification is justified under the assumption that the magnetic field is unidirectional and approximately uniform over the spatial domain of interest—a common and valid approximation in many compact MRF systems. The temporal variation of $\mathbf{H}$ is retained, as it represents the main control parameter influencing sedimentation dynamics. Any spatial variation is introduced through the gradient term $\nabla \mathbf{H}$, which is treated as a known, externally imposed design parameter. This scalar treatment of the magnetic field allows for simplified yet physically meaningful models.

The choice of a sinusoidally modulated magnetic field effectively counteracts particle settling in an MRF, as it balances the gravitational force, prevents particle aggregation (static fields tend to promote chain formation). Usually, sinusoidal modulation of $\mathbf{H}$ consists of a fixed component to ensure stability and a time-varying component to prevent aggregate formation. Specifically,

$$\mathbf{H}={\mathbf{H}}_0\left[1+\zeta \text{sin}\left(\omega t\right)\right]$$

(86)

where ${\mathbf{H}}_{\mathbf{0}}$ is the mean (constant) component of $\mathbf{H}$, while $\zeta$ represents the amplitude modulation coefficient ($0<\zeta <1$). Obviously, $\omega$ is the angular frequency of $\mathbf{H}$, which can be adjusted to match different operating conditions. In Equation (86), ${\mathbf{H}}_0$ ensures a magnetic force sufficient to balance the average gravitational force, while ${\mathbf{H}}_0\zeta \text{sin}\left(\omega t\right)$ disrupts particle chains and prevents aggregation. Clearly, to maximize effectiveness, it is important to optimize $\omega$, which should be selected close to the characteristic sedimentation frequency to avoid the formation of stable structures. Additionally, values of $\zeta$ between 0.4 and 0.6 are generally effective. Finally, ${\mathbf{H}}_0$ must be strong enough to generate a magnetic force that balances gravity.

Although ${\mathbf{H}}_{\mathbf{0}}$ is conceptually constant, it is not treated as such because, in the context of the sedimentation phenomenon, what matters is not its temporal stability, but its interaction with the gradient of the magnetic field. The gradient is, in fact, responsible for generating the magnetic force capable of counteracting sedimentation. The presence of a sinusoidally modulated field implies dynamic variations that prevent the formation of particle aggregates, thereby avoiding the development of stable structures that would promote sedimentation. Therefore, the choice to treat it as a quasi-stationary quantity is motivated by the need to consider the spatial and temporal distribution of the field to maximize the anti-sedimentation effect.

To obtain the behavior of $v\left(t\right)$ considering Equation (86), we write Equation (78) in the following form:

$${\displaystyle \frac{dv\left(t\right)}{dt}}+{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}v\left(t\right)=C\left(t\right)$$

(87)

where

$$C\left(t\right)=g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0(1+\zeta \text{sin}\left(\omega t\right))\left|\right|\nabla \left[{\mathbf{H}}_0(1+\zeta \text{sin}\left(\omega t\right))\right]|}{{\rho }_p}}.$$

(88)

We analyze the dimensional consistency of each term on the right-hand side. The first term involves the gravitational acceleration g, which has dimensions of $[L/T^2]$, multiplied by the factor $\left(1-{\rho }_f/{\rho }_p\right)$, which is dimensionless, since both ${\rho }_f$ and ${\rho }_p$ are densities with dimensions $[M/L^3]$. Therefore, this term has the dimensions of acceleration. The second term involves the vacuum magnetic permeability ${\mu }_0$, with dimensions $[M·L/\left(T^2·I^2\right)]$; the magnetic susceptibility $\chi$, which is dimensionless; the magnetic field $|{\mathbf{H}}_0|$, with dimensions $[I/L]$; and the spatial gradient $\nabla {\mathbf{H}}_0$, with dimensions $[I/L^2]$. The product $|{\mathbf{H}}_0|·\nabla {\mathbf{H}}_0$ then has dimensions $[I^2/L^3]$. Multiplying by ${\mu }_0$ yields $\left[{\displaystyle \frac{M·L}{T^2·I^2}}\right]·\left[{\displaystyle \frac{I^2}{L^3}}\right]=\left[{\displaystyle \frac{M}{L^2·T^2}}\right]$. Dividing by the particle density ${\rho }_p$, with dimensions $[M/L^3]$, we obtain ${\displaystyle \frac{[M/\left(L^2·T^2\right)]}{[M/L^3]}}=[L/T^2]$, which confirms that the second term also has the dimensions of acceleration. In conclusion, both terms on the right-hand side of the equation have dimensions $[L/T^2]$, and thus, $C\left(t\right)$ is dimensionally an acceleration. Let us now expand the term $C\left(t\right)$. In particular, recalling that

$${\text{sin}}^2\left(\omega t\right)={\displaystyle \frac{1-\text{cos}\left(2\omega t\right)}{2}},$$

(89)

so that (88) becomes

$$\begin{array}{c}\hfill C\left(t\right)=g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0(1+\zeta \text{sin}\left(\omega t\right))\left|\right|\nabla \left[{\mathbf{H}}_0(1+\zeta \text{sin}\left(\omega t\right))\right]|}{{\rho }_p}}=\\ \hfill =g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}}\left[1+2\zeta \text{sin}\left(\omega t\right)+{\displaystyle \frac{{\zeta }^2}{2}}(1-\text{cos}\left(2\omega t\right))\right].\end{array}$$

(90)

We observe that $\zeta <1$ (small modulation amplitude); hence, ${\zeta }^2\ll 1$. As a result, the oscillating contribution at frequency $2\omega$ is a secondary effect (or a second-order harmonic residual) that can be neglected in a first-order modeling of the dynamic behavior. Therefore, we can approximate $C\left(t\right)$ as follows:

$$\begin{array}{c}\hfill C\left(t\right)\approx g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}}\left(1+2\zeta \text{sin}\left(\omega t\right)\right).\end{array}$$

(91)

To solve the Equation (87), we consider the associated homogeneous equation:

$${\displaystyle \frac{dv\left(t\right)}{dt}}+{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}v\left(t\right)=0$$

(92)

which provides the solution

$$v_h\left(t\right)=A{e}^{-{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}t}.$$

(93)

We look for a particular solution $v_p\left(t\right)$ that depends on the shape of $C\left(t\right)$. Since $\mathbf{H}\left(t\right)$ is sinusoidally modulated, we try a solution of the form:

$$v_p\left(t\right)=k_0+k_1 \text{sin}\left(\omega t\right)+k_2 \text{cos}\left(\omega t\right).$$

(94)

For the purpose of determining the constants $k_0$, $k_1$ and $k_2$, we substitute the time derivative of (94) into Equation (87), thus obtaining

$$k_1\omega \text{cos}\left(\omega t\right)-k_2\omega \text{sin}\left(\omega t\right)+{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}{k}_0+{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}[k_1 \text{sin}\left(\omega t\right)+k_2 \text{cos}\left(\omega t\right)]=C\left(t\right).$$

(95)

By now equating the terms of Equation (91) with those of Equation (95), we obtain, regarding the constant terms,

$${\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}{k}_0=g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}},$$

(96)

while the equality of the coefficients in $\text{sin}\left(\omega t\right)$ and $\text{cos}\left(\omega t\right)$ yield

$$-k_2\omega +{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}{k}_1={\displaystyle \frac{2\zeta {\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}}$$

(97)

and

$$k_1\omega +{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}{k}_2=0,$$

(98)

respectively. The system formed by conditions (96)–(98) can now be solved to determine $k_0,k_1,k_2$ as functions of the physical parameters, considering a first-order dynamics in $\zeta$, with sufficient accuracy for most practical applications. In particular, Equation (96) immediately provides

$$k_0={\displaystyle \frac{{\rho }_p{V}_p}{6\pi \eta R}}\left[g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}}\right].$$

(99)

From (98), we obtain

$$k_2=-{\displaystyle \frac{{\rho }_p{V}_p\omega }{6\pi \eta R}}{k}_1$$

(100)

which, when substituted into (97), yields

$$\left({\displaystyle \frac{{\rho }_p{V}_p{\omega }^2}{6\pi \eta R}}+{\displaystyle \frac{6\pi \eta R}{{\rho }_p{V}_p}}\right)k_1={\displaystyle \frac{2\zeta {\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}},$$

(101)

from which we found

$$k_1={\displaystyle \frac{12\zeta {\mu }_0\chi \pi \eta R{V}_p|{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{\left({\rho }_p^2{V}_p^2{\omega }^2+{\left(6\pi \eta R\right)}^2\right)}},$$

(102)

which, when substituted into (100), yields

$$k_2=-{\displaystyle \frac{2\omega {\rho }_p\zeta {\mu }_0\chi V_p^2|{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{\left({\rho }_p^2{V}_p^2{\omega }^2+{\left(6\pi \eta R\right)}^2\right)}}.$$

(103)

Lastly, the particular solution (94) becomes

$$\begin{array}{c}\hfill v_p\left(t\right)=k_0+k_1 \text{sin}\left(\omega t\right)+k_2 \text{cos}\left(\omega t\right)=\\ \hfill ={\displaystyle \frac{{\rho }_p{V}_p}{6\pi \eta R}}\left[g\left(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}\right)+{\displaystyle \frac{{\mu }_0\chi |{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{{\rho }_p}}\right]+\\ \hfill +{\displaystyle \frac{12\zeta {\mu }_0\chi \pi \eta R{V}_p|{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{\left({\rho }_p^2{V}_p^2{\omega }^2+{\left(6\pi \eta R\right)}^2\right)}}\text{sin}\left(\omega t\right)-{\displaystyle \frac{2\omega {\rho }_p\zeta {\mu }_0\chi V_p^2|{\mathbf{H}}_0\left|\right|\nabla {\mathbf{H}}_0|}{\left({\rho }_p^2{V}_p^2{\omega }^2+{\left(6\pi \eta R\right)}^2\right)}}\text{cos}\left(\omega t\right).\end{array}$$

(104)

The solution (104) represents the velocity profile of a particle subjected to the combined action of gravitational, viscous, and time-modulated magnetic forces. It consists of a quasi-stationary component and two oscillatory terms, each explicitly dependent on the physical and design parameters of the system. This structure highlights the possibility of controlling the sedimentation behavior through the appropriate selection of the amplitude and frequency of the magnetic field, thereby offering an operational framework useful for optimizing the suspension’s stability in engineering applications. Unlike more general formulations that include a transient exponential decay, this solution represents the asymptotic behavior at long times, where the transient dynamics have already vanished due to viscous damping. The quasi-stationary component of the particle motion represents the slow and nearly constant part of their dynamic behavior within the MRF, resulting from the balance between the gravitational force, which tends to pull them downward, and the magnetic force generated by an externally applied field. This component explicitly depends on parameters such as the size and volume of the particles, the viscosity of the MRF, and the density difference between the particles and the hosting fluid. When the applied magnetic field is static but exhibits a spatial gradient, the particles can experience an upward thrust capable of partially or completely counteracting gravity. In the limiting case, where the magnetic and gravitational forces exactly cancel out, the average particle velocity drops to zero, and net sedimentation disappears. In addition to this quasi-stationary component, the motion of the particles may include oscillatory contributions due to the sinusoidal modulation of the magnetic field over time. The intensity of these oscillations directly depends on the modulation amplitude and the magnetic susceptibility of the particles and can significantly affect the spatial and temporal distribution of the MRF. The $\text{sin}\left(\omega t\right)$ term represents a component of the particle motion in phase with the field modulation, and its amplitude increases with the viscosity, radius, and volume of the particle. The $\text{cos}\left(\omega t\right)$ term, on the other hand, is proportional to the modulation frequency, and its negative sign indicates a phase shift related to the viscous response of the fluid. These periodic oscillations prevent the formation of stable sedimented structures and promote a more uniform dispersion of the particles. The frequency-dependent structure of the solution shows that there exists an optimal range of modulation frequencies: frequencies that are too low are ineffective in counteracting gravity, while frequencies that are too high are attenuated by viscous damping. Similarly, a modulation amplitude that is too small results in a magnetic response over time that is too weak to effectively support the particles. Overall, the explicit solution (104) demonstrates that sedimentation in a magnetorheological fluid can be strongly reduced or even eliminated through the application of a non-uniform, sinusoidally modulated magnetic field. The effectiveness of this strategy depends on the interaction between the rheological properties of the system, the geometry of the particles, and the spatiotemporal characteristics of the magnetic field, which must be designed to balance the forces acting on the particles and maintain a homogeneous distribution over time.

The SWOT analysis of this approach is summarized in

Table 4. SWOT Analysis of the technique based on variable magnetic field theory.

Strengths	The theory of variable magnetic fields enables real-time, non-invasive modulation of material properties like viscosity and stiffness, offering energy-efficient and adaptable control in both industrial and biomedical systems, especially when integrated with smart materials.
Weaknesses	The theory of variable magnetic fields enables real-time, non-invasive modulation of material properties like viscosity and stiffness, offering energy-efficient and adaptable control in both industrial and biomedical systems, especially when integrated with smart materials.
Opportunities	Variable magnetic field-based systems offer strong potential for future applications, enabling precise control in targeted therapies, soft robotics, advanced manufacturing, and real-time structural health monitoring.
Threats	Despite their potential, magnetic field-based systems face challenges such as electromagnetic interference, strict biomedical regulations, high material and integration costs, and competition from more scalable alternative technologies.

.

5. Innovations and Industrial and Biomedical Applications of the Effect of Variable Magnetic Field on MRFs

5.1. Adaptive Rheological Control in High-Frequency Hydraulic Circuits

MRFs are fundamental in high-frequency hydraulic systems due to their ability to dynamically modulate viscosity under the influence of magnetic fields, enabling adaptive rheological control essential for advanced actuators and robotic applications. Their effectiveness has been demonstrated in devices such as dampers and actuators, where viscosity regulation helps attenuate vibrations and improve stability [187,188]. The integration of MRF valves into hydraulic circuits has shown rapid response and precise flow modulation, which is crucial in dynamic environments [189].

A significant limitation is the sedimentation of magnetic particles, which can compromise performance over time. The use of US has been proposed as a solution to maintain a uniform particle distribution and ensure the rheological stability of the fluid [190]. Additionally, optimizing the composition of MRFs—particularly particle size and concentration—has shown significant effects on performance in high-frequency flow, with larger particles promoting greater viscosity variation under a magnetic field [146,191].

Finally, the adoption of advanced control systems, such as fuzzy logic and adaptive control, has further enhanced the performance of MRFs, allowing fine and real-time adjustment of the fluid’s properties [188].

5.2. Intelligent and Reconfigurable Assembly of Microcomponents

The synergistic integration of MRF and US fields is emerging as a promising strategy for intelligent and reconfigurable micro-component assembly. This combination enables precise manipulation of particles by leveraging the tunable rheological properties of MRFs and the localized acoustic forces generated by US. Recent studies have shown that simultaneously applied magnetic and acoustic fields can induce the self-organization of particles into complex three-dimensional configurations, useful for the fabrication of microdevices [192,193]. In “pinch” mode, MRFs can create controllable flow channels to guide the assembly of micro-components [146]. These techniques, when applied in microfluidic environments, have led to the development of adaptable and precise micro-assembly systems, with applications in microelectronics and biomedicine [194]. However, challenges remain regarding the stability of MRFs and the need for fine calibration of acoustic parameters, which hinder full industrial adoption [12].

5.3. Selective Hyperthermia and Non-Invasive Ablation of Tumor Tissues

Magnetic hyperthermia (MHT) is an emerging therapeutic strategy for the non-invasive treatment of solid tumors. It relies on the use of magnetic nanoparticles, such as SPIONs, which generate localized heat under an alternating magnetic field, inducing cellular apoptosis [195,196]. The use of advanced geometries, such as magnetic nanorings, enables more precise heating, thereby reducing damage to healthy tissues [197]. Integration with magnetic resonance imaging (MRI) has enhanced treatment precision and enabled real-time thermal monitoring [198].

In parallel, magnetic resonance-guided high-intensity focused US (MRgFUS) has proven effective in thermal ablation of tumors, allowing for targeted treatments with minimal impact on surrounding tissues [199]. MRgFUS has shown particular effectiveness in brain tumors, such as glioblastoma, also improving blood–brain barrier permeability and drug efficacy [165].

The combination of MHT and MRgFUS offers a synergistic approach, merging localized magnetic heating with US precision, with the potential to enhance therapeutic efficacy and reduce side effects in resistant tumors [200].

5.4. Targeted Drug Delivery Mediated by US

Recently, US-mediated targeted drug delivery has emerged as a non-invasive technique for the controlled release of therapeutic agents, leveraging the mechanical and thermal effects of US to enhance cellular and tissue permeability. A promising strategy involves the use of US-sensitive nano-carriers, such as microbubbles and nano-droplets, which can be activated in the correct location by focused US to achieve precise and time-controlled release; in particular, perfluorocarbon-based nano-droplets have shown efficacy with low-frequency US [201].

In oncology, the combination of US and microbubbles has improved drug penetration in solid tumors due to increased vascular permeability induced by cavitation [202]. In the case of neurological diseases, the use of focused US with microbubbles has enabled the temporary and reversible opening of the blood–brain barrier (BBB), facilitating drug delivery to the brain, as demonstrated in the targeted release of propofol in non-human primates [203].

Further opportunities arise from the use of smart materials, such as US-sensitive hydrogels, which allow for on-demand drug release in response to acoustic stimuli [204]. Overall, US-mediated drug delivery represents a promising therapeutic frontier for oncological, neurological, and infectious diseases, although further studies are needed to optimize parameters, carrier selectivity, and long-term safety.

Further examples of industrial and biomedical applications based on the variable magnetic field technique are reported in

Table 5. Variable magnetic field theory: further industrial applications.

Ref.	Domain	Methods	Conclusions
[121,204,205,206,207]	3D printing of rheologically adaptive materials	3D printing and US activation enable controlled rheology and targeted release of functional materials.	3D printing of rheologically adaptive materials enables precise control of shape and function through rheological tuning and external activation, such as US.
[121,146,208,209,210,211]	Selective industrial filtration and separation	US and magnetic gradients enhance particle separation and flow control in MRF-based industrial systems.	US combined with MRF enhances efficient, real-time industrial filtration.
[212,213,214,215,216]	Smart actuators for soft robotics in hazardous environments	US-controlled soft actuators with self-healing enable safe, precise operation in harsh environments.	US-guided soft actuators enable precise, safe operation in extreme conditions.
[217,218,219,220]	Tunable vibration absorption systems in construction	Adaptive vibration absorption with magnetorheological dampers for construction.	MRF-based tunable vibration systems enhance seismic protection in construction.

and

Table 6. Variable magnetic field theory: further biomedical applications.

Ref.	Domain	Methods	Conclusions
[221,222,223,224,225]	Cell engineering for organoids-on-a-chip	Advanced technologies improve organoid-on-a-chip growth, control, and automated analysis.	Organoids-on-a-chip offer improved cell function and precision for personalized research and therapy.
[226,227,228,229,230]	Magneto-acoustic liquid biopsy	Magneto-US methods improve liquid biopsy sensitivity by isolating tumor biomarkers.	Magneto-acoustic technologies improve liquid biopsy for early cancer detection.
[108,231,232,233,234]	Assisted intracranial neuro-transport	Focused US and magnetic agents enable targeted blood–brain barrier crossing.	Focused US and magnetic nanoparticles enable safe, targeted drug delivery across the blood–brain barrier.
[235,236,237,238]	Systems for selective biofilm disruption	Magnetic microstructures and US selectively disrupt biofilms, boosting antimicrobial efficacy.	US and magneto-active materials enhance selective biofilm removal and antimicrobial therapy.

, respectively, which illustrate the advanced technological contexts in which this approach proves particularly promising for long-term safety.

6. MRF and US: Biot–Stoll

The propagation of US in MRFs under the influence of fields can be modeled using the Biot–Stoll theory, which describes the propagation of elastic waves in porous materials saturated with a compressible fluid. Based on this approach, an MRF can be regarded as a porous medium (with a skeleton formed by the suspended particles) saturated by the fluid [239].

6.1. Fundamental Equations

Biot developed a general model for wave propagation in porous media, which was later extended by Stoll to include dissipation due to intergranular friction. In MRF, the solid structure (skeleton of ferromagnetic particles) and the carrier fluid are treated as two distinct but coupled phases. If we denote the displacement of the solid frame by $\mathbf{u}(x,y,z,t)$ and the relative displacement of the fluid in the pores by $\mathbf{U}(x,y,z,t)$, then the quantity $e=\nabla ·\mathbf{u}$ represents the dilatation of the frame, while $\vartheta =\beta \nabla ·\left(\mathbf{u}\right(x,y,z,t)-\mathbf{U}(x,y,z,t\left)\right)$ quantifies the relative dilatation between the frame and the fluid. Denoting by ${\rho }_f$ and ${\rho }_r$ the density of the fluid and solid particles (grains), respectively, and by $\eta$ and $\beta$ the viscosity of the fluid and the porosity, respectively, the longitudinal wave propagation equations according to Biot are given by [51,239]

$$\left\{\begin{array}{c}{\nabla }^2(He-C\vartheta )={\displaystyle \frac{{\partial }^2}{\partial t^2}}(\rho e-{\rho }_f\vartheta )\hfill \\ {\nabla }^2(Ce-M\vartheta )={\displaystyle \frac{{\partial }^2}{\partial t^2}}({\rho }_{f}e-{\rho }_c\vartheta )-{\displaystyle \frac{\eta }{B_0}}{\displaystyle \frac{\partial \vartheta }{\partial t}}\hfill \end{array}\right.$$

(105)

where $B_0$ is the dynamic permeability of the frame, and ${\rho }_c={\displaystyle \frac{{\rho }_f}{\beta }}\tilde{\gamma }$ is the added mass factor, with $\tilde{\gamma }$ representing the structure factor that accounts for the increase in the apparent inertia of the fluid caused by the tortuosity of the pores. Note that ${\rho }_c$ has the dimensions of a density, i.e., mass per unit volume $[\mathrm{kg}/{\mathrm{m}}^3]$, since ${\rho }_c={\displaystyle \frac{{\rho }_f}{\beta }}\tilde{\gamma }$ where ${\rho }_f$ is the fluid density $[\mathrm{kg}/{\mathrm{m}}^3]$, and both $\beta$ and $\tilde{\gamma }$ are dimensionless parameters. Therefore, multiplying ${\rho }_c$ by the particle volume yields a quantity with the dimensions of mass, consistent with the physical interpretation of ${\rho }_c$ as an added mass term.

Finally, $\rho =(1-\beta ){\rho }_r+\beta {\rho }_f$ represents the density of the fluid-saturated material. The parameters $H,C,M$ are the elastic moduli of the porous medium, defined as follows [51,239]:

$$H={\displaystyle \frac{{(K_r-K_b)}^2}{D-K_b}}+K_b+{\displaystyle \frac{4}{3}}{G}_b$$

(106)

$$C={\displaystyle \frac{K_r(K_r-K_b)}{D-K_b}}$$

(107)

$$M={\displaystyle \frac{K_r^2}{D-K_b}}$$

(108)

$$D=K_r\left(1+{\displaystyle \frac{K_r-K_f}{K_f}}\beta \right),$$

(109)

where $K_f$ is the bulk modulus of the fluid; $K_r=K_r^{\prime }-i{K}_r^{″}$ is the bulk modulus of the solid material composing the porous frame (the mass modulus of the individual particle in the case of granular media); and $K_b=K_b^{\prime }-i{K}_b^{″}$ and $G_b=G_b^{\prime }-i{G}_b^{″}$ are the bulk and shear moduli of the particle assembly measured under constant pore fluid pressure. To describe the MRF, the compressibility modulus of the fluid, $K_f$, can be considered entirely elastic. In contrast, the compressibility modulus of the particles, $K_r$, is different, since it is not composed of a compact siliceous material, but of carbonyl iron powder directly dispersed in silicone oil. As a result, it is unlikely that the particles have a compact structure; more likely, they are formed by long intertwined molecules that structurally reorganize stress. This behavior is represented here by a complex compressibility modulus for the granular material.

6.2. Stoll’s Viscous Correction

Below a certain frequency—determined by pore size and viscosity—only Poiseuille flow occurs in the composite material. According to Hovem and Ingram [52], the pore size parameter $a_p$, denoted by $d_m$ as the mean particle diameter, can be computed as [51,239]

$$a_p={\displaystyle \frac{d_m}{3}}{\displaystyle \frac{\beta }{\beta -1}}$$

(110)

whereas, if $k^{\prime }$ considers the shape of the pores by defining

$$B_0={\displaystyle \frac{d_m^2}{36k^{\prime }}}{\displaystyle \frac{{\beta }^3}{{(\beta -1)}^2}},$$

(111)

we can apply Stoll’s viscous correction to ${\rho }_c$, obtaining [51]

$${\rho }_c={\displaystyle \frac{{\rho }_f}{\beta }}\left(1+{\displaystyle \frac{\eta \beta }{B_0{\rho }_f}}{\displaystyle \frac{F^{″}\left(\mathcal{K}\right)}{\omega }}\right).$$

(112)

By means of (112), denoting by

$$\mathcal{K}=a_p\sqrt{{\displaystyle \frac{\omega {\rho }_f}{\eta }}},$$

(113)

Stoll analyzed the laminar flow of a fluid within a cylindrical duct by developing the function $F\left(\mathcal{K}\right)$ (which describes the magnetic microstructure), accounting for deviations from Poiseuille flow at high frequencies. To this end, in Equation (105), the term $\eta$ must be replaced with $\eta F\left(\mathcal{K}\right)$. This implies the introduction of an additional—or even artificial—structural constant. Several authors have adopted an alternative approach, directly incorporating the corrective flow function $F\left(\mathcal{K}\right)=F^{\prime }\left(\mathcal{K}\right)+i{F}^{″}\left(\mathcal{K}\right)$ into Equation (105), thereby obtaining [51,52,53,239].

Remark 11.

Equation (113) represents a dimensionless quantity that arises in the analysis of the response of a particle immersed in a fluid subjected to an oscillatory field. In this expression, $a_p$ denotes the particle radius, ω is the angular frequency of oscillation, ${\rho }_f$ is the fluid density, and η is the dynamic viscosity of the fluid. Dimensional analysis shows that the term under the square root has the dimension of an inverse length, and thus, when multiplied by $a_p$, the result is dimensionless. Physically, $\mathcal{K}$ quantifies the balance between inertial and viscous effects in an oscillatory context. Small values of $\mathcal{K}$ indicate a viscosity-dominated regime, where the effects of oscillation penetrate deeply into the fluid. Conversely, large values of $\mathcal{K}$ correspond to an inertia-dominated regime, where oscillatory effects are confined to a thin layer around the particle. This quantity is closely related to the Womersley number or modified versions of the Stokes number and is relevant in applications involving particle dynamics in viscous media under oscillatory forcing, the Basset history force, and fluid–structure interactions in unsteady regimes.

$${\nabla }^2(Ce-M\vartheta )={\displaystyle \frac{{\partial }^2({\rho }_{f}e-{\rho }_c\vartheta )}{\partial t^2}}-{\displaystyle \frac{\eta F\left(\mathcal{K}\right)}{B_0}}{\displaystyle \frac{\partial \vartheta }{\partial t}}$$

(114)

where

$$F\left(\mathcal{K}\right)={\displaystyle \frac{\mathcal{K}T\left(\mathcal{K}\right)}{4\left[1+{\displaystyle \frac{2iT\left(\mathcal{K}\right)}{\mathcal{K}}}\right]}}$$

(115)

which represents the viscosity correction factor, in which $T\left(\mathcal{K}\right)$ denotes the well-known Kelvin complex function

$$T\left(\mathcal{K}\right)={\displaystyle \frac{{\text{ber}}_0^{\prime }\left(\mathcal{K}\right)+i{\text{bei}}_0^{\prime }\left(\mathcal{K}\right)}{{\mathrm{ber}}_0\left(\mathcal{K}\right)+i{\mathrm{bei}}_0}}$$

(116)

in which ${ber}_0$ and ${bei}_0$ are the Kelvin functions of the first kind, defined in terms of the Bessel function of the first kind $J_n\left(x\right)$ with order $n=0$. In particular, considering that

$${ber}_0\left(x\right)=\Re \left(J_0\left(x{e}^{i\pi /4}\right)\right),$$

(117)

and

$${bei}_0\left(x\right)=\Im \left(J_0\left(x{e}^{i\pi /4}\right)\right),$$

(118)

so that

$${ber}_0\left(x\right)+i{bei}_0\left(x\right)=J_0\left(x{e}^{i\pi /4}\right),$$

(119)

where

$$J_0\left(x{e}^{i\pi /4}\right)=\sum _{m=0}^{\infty }{\displaystyle \frac{{(-1)}^m}{{(m!)}^2}}{\left({\displaystyle \frac{x}{2}}\right)}^{4m}+\sum _{m=0}^{\infty }{\displaystyle \frac{{(-1)}^m}{{(m!)}^2}}{\left({\displaystyle \frac{x}{2}}\right)}^{4m+2}.$$

(120)

We observe that $\mathcal{K}$ quantifies the ratio between the fluid’s inertia and the viscous resistance within the porous medium. In other words, if $\mathcal{K}\gg 1$, then the fluid’s inertia dominates the wave propagation behavior. Conversely, if $\mathcal{K}\ll 1$, the fluid is strongly damped by viscous effects and behaves more diffusively than oscillatory. Since $\mathcal{K}$ depends on the frequency $\omega$, we can say that at high frequencies, $\mathcal{K}$ increases, and thus, the effect of inertia becomes more significant; on the other hand, at low frequencies, $\mathcal{K}$ is small, which means that viscous dissipation plays a predominant role in the wave behavior [239].

Remark 12.

The expression for $T\left(\mathcal{K}\right)$ represents a ratio between the derivative of the Kelvin functions and the functions themselves and is used to describe the correction of viscous flow in the wave propagation equations in porous materials and MRFs.

Remark 13.

The function $J_0\left(x{e}^{i\pi /4}\right)$, which appears in the analytical solution, is a Bessel function of the first kind evaluated at a complex argument. This results in a combination of oscillatory and exponentially decaying behavior. Such behavior reflects the interplay between wave-like and diffusive phenomena and is typical in time-harmonic viscoelastic or magneto-mechanical systems. While the detailed plot is omitted for brevity, the qualitative behavior of this function is well characterized and consistent with the expected physical response.

6.3. Physical Interpretation and Comparative Perspective

The Biot–Stoll model is an advanced tool for analyzing the mechanical and acoustic properties of saturated composite materials, such as magnetorheological fluids. Unlike approaches that focus solely on acoustic waves or magnetic fields, this model simultaneously describes the response of both the solid and fluid phases, making it particularly suitable for studying wave propagation in multiphase systems where elastic deformations and filtration motions coexist. From an acoustic perspective, it captures wave dispersion and attenuation—phenomena not addressed by models based solely on radial acoustic propagation. Mechanically, it allows for the evaluation of dynamic stiffness, complex impedance, and viscoelastic response, which are essential in designing high-performance systems. Thanks to its ability to integrate structural and dynamic aspects, the model offers a comprehensive description of the medium’s behavior. Compared to the other two approaches analyzed—the radial acoustic pressure theory and the use of time-varying magnetic fields—the Biot–Stoll model stands out for its capacity to represent the overall response of the composite material, including phase interactions. While the first two methods focus on localized effects or particle distribution, neglecting internal structure, the Biot–Stoll model is better suited for applications requiring a in-depth understanding of dynamic behavior, such as fluidic metamaterials, multiphysics sensors, or advanced biomedical devices. Although it involves greater theoretical and computational complexity, the SWOT analysis highlights that this model enables accurate and predictive simulations even under complex operating conditions, making it particularly advantageous in contexts where mechanical, acoustic, and functional aspects must be coherently integrated [240].

6.4. Frequency Equation

From (122), to obtain the equation for the frequency of the US waves, for e and $\vartheta$, we assume the following formulations,

$$e=A_1{e}^{i(\omega t-lx)},$$

(121)

and

$$\vartheta =A_2{e}^{i(\omega t-lx)},$$

(122)

where

$$l=l_r+i{l}_i$$

(123)

denotes the complex wavenumber of the longitudinal US wave, with $l_i$ as the attenuation coefficient, while $\omega$ represents the angular frequency, and x is the distance traveled by the wave during propagation.

Remark 14.

The parameter l provides an immediate indication of the dynamic regime: real values close to unity indicate resonance, while complex deviations reflect dissipative effects. This formulation is useful for normalizing the system equations and allows for a direct assessment of the stability and efficiency of the process, facilitating comparison between different operating regimes.

Restricting ourselves to a one-dimensional analysis, we rewrite the first equation of system (105) and Equation (114) as follows:

$$\left\{\begin{array}{c}H{\displaystyle \frac{{\partial }^{2}e}{\partial x^2}}-C{\displaystyle \frac{{\partial }^2\vartheta }{\partial x^2}}-\rho {\displaystyle \frac{{\partial }^{2}e}{\partial t^2}}+{\rho }_f{\displaystyle \frac{{\partial }^2\vartheta }{\partial t^2}}=0\hfill \\ C{\displaystyle \frac{{\partial }^{2}e}{\partial x^2}}-M{\displaystyle \frac{{\partial }^2\vartheta }{\partial x^2}}-{\rho }_f{\displaystyle \frac{{\partial }^{2}e}{\partial t^2}}+{\rho }_c{\displaystyle \frac{{\partial }^2\vartheta }{\partial t^2}}+\eta {\displaystyle \frac{F\left(\mathcal{K}\right)}{B_0}}{\displaystyle \frac{\partial \vartheta }{\partial t}}=0\hfill \end{array}\right.$$

(124)

from which, considering both (121) and (122), we matricially can write

$$\left[\begin{array}{cc}H{l}^2-\rho {\omega }^2& {\rho }_f{\omega }^2-C{l}^2\\ C{l}^2-{\rho }_f{\omega }^2& {\rho }_c{\omega }^2-M{l}^2-i\omega {\displaystyle \frac{\eta F\left(\mathcal{K}\right)}{B_0}}\end{array}\right]\left[\begin{array}{c}{A}_1\\ A_2\end{array}\right]=0,$$

(125)

which implies the following dispersion relation:

$$(H{l}^2-\rho {\omega }^2)\left({\rho }_c{\omega }^2-M{l}^2-i\omega {\displaystyle \frac{\eta F\left(\mathcal{K}\right)}{B_0}}\right)+{(C{l}^2-{\rho }_f{\omega }^2)}^2=0,$$

(126)

To solve Equation (126), with the aim of obtaining the longitudinal wave velocities and the attenuation coefficient as a function of frequency, we use a numerical technique based on the Newton–Raphson procedure. Based on the experimental data of an MRF-132DG [239,241] reported in

Table 7. Physical parameters of the magnetorheological fluid MRF-132DG used in the model. Data from Parker Lord technical datasheet.

Symbol	Value	Description
${\rho }_r$	3160 kg ${\mathrm{m}}^{-3}$	Density of solid particles
${\rho }_f$	2.95–3.15 g ${\mathrm{cm}}^{-3}$	Density of the base fluid
$\beta$	0.30	Porosity (particle volume fraction)
c	1.5	Structural coefficient
$\eta$	0.112 Pa·s	Off-state dynamic viscosity (at 40°)
$K_r^{\prime }$	$1.26\times {10}^{11}$ Pa	Real part of bulk modulus of particles
$K_r^{″}$	$5\times {10}^9$ Pa	Imaginary part of bulk modulus of particles
$K_f$	$9.8\times {10}^8$ Pa	Bulk modulus of the fluid
$K_b$	$2\times {10}^8$ Pa	Bulk modulus of the porous skeleton
$G_b$	$4\times {10}^6$ Pa	Shear modulus of the skeleton
$B_0$	$4\times {10}^{-12}$ ${\mathrm{m}}^2$	Intrinsic permeability
$F\left(\mathcal{K}\right)$	$1+0.1i$	Dynamic correction function

, we separate the real part of Equation (126) from its imaginary part.

From (123), it is legitimate to write

$$l^2={(l_r+i{l}_i)}^2=l_r^2-l_i^2+2i{l}_r{l}_i=a+ib$$

(127)

where

$$a=l_r^2-l_i^2,b=2l_r{l}_i.$$

(128)

Then, the expansion of the first term in Equation (126) becomes

$$H{l}^2-\rho {\omega }^2=H(a+ib)-\rho {\omega }^2=(Ha-\rho {\omega }^2)+iHb,$$

(129)

while the second term, considering that $F\left(\mathcal{K}\right)=1+0.1i$, becomes

$$\begin{array}{c}\hfill {\rho }_c{\omega }^2-M{l}^2-i\omega {\displaystyle \frac{\eta F(\mathcal{K}}{B_0}}={\rho }_c{\omega }^2-M(a+ib)-i\omega {\displaystyle \frac{\eta (1+0.1i)}{B_0}}=\end{array}$$

(130)

$$\begin{array}{c}\hfill =\left({\rho }_c{\omega }^2-Ma+\omega {\displaystyle \frac{0.1\eta }{B_0}}\right)-i\left(Mb+\omega {\displaystyle \frac{\eta }{B_0}}\right).\end{array}$$

(131)

Finally, regarding $C{l}^2-{\rho }_f{\omega }^2$, we can write

$$C{l}^2-{\rho }_f{\omega }^2=C(a+ib)-{\rho }_f{\omega }^2=(Ca-{\rho }_f{\omega }^2)+iCb,$$

(132)

from which

$${(Ca-{\rho }_f{\omega }^2+iCb)}^2={(Ca-{\rho }_f{\omega }^2)}^2-C^2{b}^2+2iCb(Ca-{\rho }_f{\omega }^2).$$

(133)

Then, by substituting everything into (126), we obtain

$$\begin{array}{c}\hfill \left((Ha-\rho {\omega }^2)+iHb\right)\left[\left({\rho }_c{\omega }^2-Ma+\omega {\displaystyle \frac{0.1\eta }{B_0}}\right)-i\left(Mb+\omega {\displaystyle \frac{\eta }{B_0}}\right)\right]+\\ \hfill +{(Ca-{\rho }_f{\omega }^2)}^2-C^2{b}^2+2iCb(Ca-{\rho }_f{\omega }^2)=0.\end{array}$$

(134)

We define

$$A:=Ha-\rho {\omega }^2,B:=Hb,$$

(135)

$$C_1:={\rho }_c{\omega }^2-Ma+\omega {\displaystyle \frac{0.1\eta }{B_0}},C_2:=-Mb-\omega {\displaystyle \frac{\eta }{B_0}},$$

(136)

and

$$D_1:=+{(Ca-{\rho }_f{\omega }^2)}^2-C^2{b}^2,D_2:=2Cb(Ca-{\rho }_f{\omega }^2)$$

(137)

so that Equation (134) takes the following compact form:

$$(A+iB)(C_1+i{C}_2)+D_1+i{D}_2=0,$$

(138)

from which

$$\left\{\begin{array}{c}A{C}_1-B{C}_2+D_1=0\hfill \\ A{C}_2+B{C}_1+D_2=0,\hfill \end{array}\right.$$

(139)

which provides

$$\left\{\begin{array}{c}(Ha-\rho {\omega }^2)\left({\rho }_c{\omega }^2-Ma+\omega {\displaystyle \frac{0.1\eta }{B_0}}\right)-Hb\left(Mb+\omega {\displaystyle \frac{\eta }{B_0}}\right)+{(Ca-{\rho }_f{\omega }^2)}^2-C^2{b}^2=0\hfill \\ (Ha-\rho {\omega }^2)\left(-Mb-\omega {\displaystyle \frac{\eta }{B_0}}\right)+Hb\left({\rho }_c{\omega }^2-Ma+\omega {\displaystyle \frac{0.1\eta }{B_0}}\right)+2Cb(Ca-{\rho }_f{\omega }^2)=0,\hfill \end{array}\right.$$

(140)

with $a=l_r^2-l_i^2$, and $b=2l_r{l}_i$.

To apply the well-known Newton–Raphson numerical procedure to system (140), we rewrite it in the following form:

$$\left\{\begin{array}{c}{f}_1(l_r,l_i)=0\hfill \\ f_2(l_r,l_i)=0.\hfill \end{array}\right.$$

(141)

The vector of unknowns is defined as

$$\mathbf{x}=\left[\begin{array}{c}{l}_r\\ l_i\end{array}\right],$$

(142)

with

$$\mathbf{f}\left(\mathbf{x}\right)=\left[\begin{array}{c}{f}_1(l_r,l_i)\\ f_2(l_r,l_i)\end{array}\right]$$

(143)

The Newton–Raphson rule requires

$${\mathbf{x}}^{(k+1)}={\mathbf{x}}^{\left(k\right)}-J^{-1}\left({\mathbf{x}}^{\left(k\right)}\right)·\mathbf{f}\left({\mathbf{x}}^{\left(k\right)}\right),$$

(144)

where J is the Jacobian matrix of $\mathbf{f}$ regarding $\mathbf{x}$:

$$J=\left[\begin{array}{cc}{\displaystyle \frac{\partial f_1(l_r,l_i)}{\partial l_r}}& {\displaystyle \frac{\partial f_1(l_r,l_i)}{\partial l_i}}\\ {\displaystyle \frac{\partial f_2(l_r,l_i)}{\partial l_r}}& {\displaystyle \frac{\partial f_2(l_r,l_i)}{\partial l_i}}\end{array}\right]$$

(145)

Obviously, each partial derivative is approximated using finite differences. Once an initial vector for (142) is set,

$${\mathbf{x}}^{\left(0\right)}=\left[\begin{array}{c}{l}_r^{\left(0\right)}\\ l_i^{\left(0\right)}\end{array}\right].$$

(146)

At each iteration k, we evaluate $f_1^{\left(k\right)}$ and $f_2^{\left(k\right)}$, and by computing the Jacobian matrix $J^{\left(k\right)}$, we solve the system

$$J^{\left(k\right)}·{\delta }^{\left(k\right)}={\mathbf{f}}^{\left(k\right)}$$

(147)

obtaining ${\delta }^{\left(k\right)}$, which, when substituted into (144), yields

$${\mathbf{x}}^{(k+1)}={\mathbf{x}}^{\left(k\right)}-{\delta }^{\left(k\right)}$$

(148)

until $|{\mathbf{f}}^{\left(k\right)}|<\epsilon ={10}^{-6}$. Once the procedure converges to the solution $(l_r,l_i)$, both the phase velocity, $v_p={\displaystyle \frac{\omega }{l_r}}$, and the spatial attenuation coefficient, $\alpha =l_i$, can be easily obtained. The aforementioned numerical procedure was implemented in MATLAB R2024b, yielding, as the frequency increases, the semi-logarithmic trends of the phase velocity (Figure 2) and the attenuation coefficient (Figure 3).

Both Figure 2 and Figure 3 illustrate the response of the biphasic magneto-viscoelastic medium to the propagation of ultrasonic waves, within the theoretical framework provided by the Biot–Stoll model, appropriately extended to include the effect of a magnetorheological fluid. The first figure highlights the trend of the phase velocity, which increases monotonically with frequency, indicating a strongly dispersive behavior of the medium. This result is consistent with the predictions of Biot’s model for saturated porous media, where the presence of a fluid phase elastically coupled to the solid matrix generates a frequency-dependent variation in propagation speed. From an engineering perspective, such dispersion can be exploited to design acoustically selective structures, such as mechanical frequency filters or adaptive-response metamaterials. The second figure shows the spatial attenuation in decibels, displaying a clear increase with rising frequency. This effect is mainly due to the viscosity of the fluid phase and the internal dissipation of the solid phase, modeled through a complex bulk modulus $K_r=K_r^{\prime }-i{K}_r^{″}$. This formulation accounts for internal mechanical losses and provides a realistic description of the medium’s dynamic behavior under high-frequency excitation. From an application standpoint, the frequency-dependent attenuation is critically important for the design of active damping systems, vibroacoustic barriers, or ultrasonic sensors in controlled environments. Moreover, the theoretical framework provided by the extended Biot–Stoll model enables quantitative predictions of the material’s acoustic behavior as a function of controllable parameters, such as the fluid viscosity, the stiffness of the matrix, or the intensity of the applied magnetic field. This direct link between microstructural properties and the medium’s dynamic response paves the way for the rational design of smart materials and high-performance devices, in which wave propagation can be modulated in real time according to operational or environmental requirements. These features make the analyzed system highly attractive for applications in aerospace, biomedical engineering, adaptive vibration isolation, and non-invasive diagnostics.

Remark 15.

It is worth noting that the presence of an externally applied field $\mathbf{H}$ contributes, to some extent, to increasing the viscosity of the MRF, which acts as a retardant to the sedimentation of the particles. In particular, [4,5],

$$\eta \left(\right|\mathbf{H}\left|\right)={\eta }_0+{\eta }_{1}tanh{{\displaystyle \frac{\left|\mathbf{H}\right|}{\left|\mathbf{H}\right|}}}_{\mathrm{sat}}.$$

(149)

Il parametro ${\left|\mathbf{H}\right|}_{\mathrm{sat}}$ rappresenta l’intensità caratteristica del campo magnetico oltre la quale l’aumento di η del MRF tende a saturarsi. Esso modula il passaggio da una risposta lineare a una risposta satura nel comportamento reologico del MRF, ed è un parametro chiave per la progettazione e il controllo di sistemi intelligenti basati su fluidi magnetici. Inoltre, ${\eta }_0$ is the intrinsic dynamic viscosity of the MRF (i.e., when $\mathbf{H}=\mathbf{0}$); ${\eta }_1$ is the maximum increase in viscosity due to $\mathbf{H}$, indicating how much the viscosity can increase relative to the initial value ${\eta }_0$ when $\left|\mathbf{H}\right|$ is applied. The term $tanh{\displaystyle \frac{\left|\mathbf{H}\right|}{{\left|\mathbf{H}\right|}_{\mathrm{sat}}}}$ modulates the effect of $\mathbf{H}$ on the viscosity. In particular, for weak fields (i.e., $\left|\mathbf{H}\right|\ll {\left|\mathbf{H}\right|}_{\mathrm{sat}}$), it follows that $tanh\left(x\right)\approx x$, so the viscosity increases almost linearly with $\left|\mathbf{H}\right|$. In the presence of strong fields (i.e., $\left|\mathbf{H}\right|\gg {\left|\mathbf{H}\right|}_{\mathrm{sat}}$), $tanh\left(x\right)\approx 1$, resulting in saturation at ${\eta }_0+{\eta }_1$, reaching the maximum allowed value.

Remark 16.

It is worth noting that the integration of machine learning models for the prediction and optimization of operational parameters, such as ultrasonic frequency, magnetic field intensity, container geometry, and carrier fluid viscosity, could ensure the dynamic stability of the system. Such an approach would allow for real-time identification of suboptimal configurations that lead to instability or particle aggregation, dynamic adaptation of control parameters in response to environmental changes or variations in the fluid’s rheological properties, and a reduction in the computational cost of coupled numerical simulations through the use of predictive models supporting traditional algorithms. In particular, the use of deep neural networks, convolutional neural networks applied to spatial domains, and hybrid models combining experimental data with physical constraints could prove useful for learning instability patterns from simulations and experiments, improving numerical convergence in acoustic–magnetic models through data-driven regularization, and constructing risk maps for sedimentation based on operating conditions. This type of development would contribute to strengthening the concept of magnetorheological fluids as intelligent materials, capable of dynamically and controllably adapting to complex application contexts, offering new perspectives for advanced biomedical devices and high-performance engineering systems.

Table 8. SWOT analysis for the Biot–Stoll approach.

Strengths	The Biot–Stoll model with viscous correction reveals that higher frequencies boost viscosity and energy dissipation in MRFs, improving suspension stability and reducing sedimentation, aided by magnetic field-induced thickening.
Weaknesses	The model is analytically complex due to its use of PDEs and matrices, relies on experimentally calibrated variables, assumes MRF homogeneity, and includes empirical corrections that may lack general validity across frequencies or compositions.
Opportunities	Integrating US and magnetic fields allows real-time control of suspensions for applications like electronic cooling and drug delivery, with neural networks aiding predictive modeling and adaptability to nanoparticle-based microfluidic systems.
Threats	Experimental validation is difficult due to complex measurements, nonlinear field–wave interactions, and sensitivity to material property variations.

summarizes the SWOT analysis of the Biot–Stoll approach, highlighting its key strengths, limitations, opportunities for development, and potential implementation challenges in the context of MRF modeling.

7. Industrial and Biomedical Applications of the Biot–Stoll Theory in MRF and US

7.1. Advanced Diagnostics in Composite Systems and Sandwich Structures

The integration of the Biot–Stoll theory with the use of MRF and US represents an innovative approach for advanced diagnostics of sandwich structures and composite materials. The theory, developed to describe wave propagation in poroelastic media, provides a solid theoretical foundation for analyzing the interactions between elastic waves and complex materials such as MRFs, whose mechanical response varies under the influence of magnetic fields. The incorporation of MRFs into sandwich structures, particularly through impregnated porous fabrics, has demonstrated improvements in damping and acoustic insulation, as well as enhanced stability against sedimentation. The use of the Biot–Stoll theory also enables more precise characterization of the mechanical and acoustic properties of composite materials [242]. The MRF–US combination, modeled according to Biot–Stoll, enables advanced structural health monitoring systems. For example, the use of Lamb waves has shown remarkable effectiveness in detecting defects in honeycomb structures without the need for complex multimodal analyses [243]. The integration of MRFs into such systems further enhances their sensitivity and adaptability. From a technological standpoint, these advanced solutions support predictive maintenance and safety in critical sectors such as aerospace, automotive, and civil engineering. Real-time monitoring and adaptability to operational conditions make infrastructures more efficient and secure, promoting the development of smart materials and adaptive structures.

7.2. Adaptive Control of Sound Transmission in Industrial Ducts

The control of sound transmission in industrial ducts is particularly complex in the presence of high sound pressure levels and variable frequencies. The integration of MRF and US, described through the Biot–Stoll theory, represents an innovative solution. Recent studies strongly suggest that an increase in the magnetic field enhances the viscosity of the MRF, increasing acoustic attenuation and improving sound insulation [244]. This effect enables the development of adaptive noise control devices capable of modulating their acoustic properties in real time [245]. From an application perspective, the adoption of MRF+US systems in industrial ducts can improve energy efficiency and acoustic comfort. In HVAC systems, for instance, such technologies reduce noise transmission, enhancing indoor environmental quality [246], while in aerospace and automotive sectors, they can reduce perceived noise in exhaust or ventilation ducts, improving user experience [247]. To ensure long-term reliability, it is essential to prevent the sedimentation of magnetic particles through the use of stable MRFs and optimized compositions. Finally, predictive models based on the Biot–Stoll theory allow for the tailored design of high-performance acoustic systems for specific applications [248].

7.3. Production and Quality Control in Multiphase 3D Printing

Multiphase 3D printing represents an evolution in additive manufacturing, enabling the production of advanced components through the combined deposition of different materials. It is applied in sectors such as aerospace, biomedical, and electronics, where the integration of distinct mechanical, thermal, or electrical properties is critical. The process requires precise control of parameters to ensure material compatibility, interfacial cohesion, and the absence of defects. Real-time monitoring systems, equipped with optical, thermal, and acoustic sensors, have been developed to detect anomalies, including through convolutional neural networks achieving 94% accuracy in print quality classification [249]. A further advancement is the use of multisensory digital twins, which optimize and correct the process in real time, improving efficiency and reducing waste [153]. Industrial adoption requires standardization and training, as demonstrated by initiatives such as FLEETWERX, promoted by the US Navy, which develops mobile multiphase additive manufacturing units for remote or hostile environments [250].

7.4. Non-Invasive Diagnosis of Pathological Soft Tissues

The early and non-invasive diagnosis of soft tissue pathologies, particularly the distinction between benign and malignant lesions, represents a significant clinical challenge. The integration of MRF and US, modeled according to the Biot–Stoll poroelastic theory, has recently offered new opportunities to enhance diagnostic sensitivity and specificity [251]. This theory describes the propagation of acoustic waves in saturated poroelastic media, accounting for the interaction between the solid matrix and the interstitial fluid. When applied to soft tissues infiltrated with MRFs, it enables accurate modeling of the system’s acoustic response as a function of local changes in stiffness and viscosity induced by external magnetic fields, thereby improving anomaly detection [251]. The use of MRFs as magnetically tunable acoustic contrast agents allows for the local alteration of the mechanical properties of tissues. The analysis of reflected and transmitted waves enables more precise identification of pathological areas, facilitating the distinction between healthy tissue and suspicious lesions [252]. From an application standpoint, this technology is promising for the development of portable and low-cost diagnostic devices, which are also useful in resource-limited settings. Moreover, the real-time modulation of MRFs paves the way for adaptive imaging systems capable of optimizing images based on tissue characteristics [253,254]. Finally, the remote induction of shear waves via transcranial magnetic stimulation, combined with poroelastic analysis, suggests new non-invasive applications for assessing soft tissue stiffness, with potential impacts in oncological diagnostics and musculoskeletal monitoring [255].

7.5. Real-Time Monitoring of Tissue Regeneration

Tissue regeneration is one of the most complex and promising challenges in regenerative medicine, and real-time monitoring of healing processes is essential to optimize therapies. In this context, the integration of Biot–Stoll theory with the combined use of MRF and US offers an innovative and non-invasive method for monitoring tissue regeneration. The Biot–Stoll theory, which describes acoustic wave propagation in saturated poroelastic media, enables accurate modeling of the response of biological tissues treated with MRF. These fluids, due to their magneto-acoustic sensitivity, can function as intelligent contrast agents to detect mechanical and structural changes during healing. Recent studies have highlighted that the combined use of MRF and US can promote tissue regeneration: for instance, US-activated hydrogels have enabled the controlled release of growth factors [256], while low-intensity US has stimulated cell proliferation and extracellular matrix synthesis [257]. From a technological standpoint, this integration paves the way for advanced medical devices, such as implantable sensors capable of monitoring regeneration in real time and providing immediate feedback on the effectiveness of therapy. These devices have applications in clinical fields such as orthopedic surgery and regenerative medicine, improving the quality of care and reducing recovery times.

7.6. Optimization of Targeted Delivery in Complex Tissues

The targeted delivery of drugs within complex biological tissues, hindered by physiological barriers such as the extracellular matrix or fibrosis, represents a significant challenge in medicine. The integration of Biot–Stoll poroelastic theory with the combined use of MRF and US offers an innovative approach to model and control drug diffusion in tissues. The Biot–Stoll theory, which describes the propagation of acoustic waves in poroelastic media, enables the prediction of tissue responses to the simultaneous activation of MRF and US. MRFs can locally alter tissue permeability, while US facilitates drug penetration through cavitation and acoustic streaming [258,259], thereby improving delivery efficiency and reducing systemic side effects. Advanced fluid–structure interaction models, such as those proposed by Ambartsumyan et al. [260], enhance the prediction of drug distribution, supported also by studies on hydrogels as analogs of soft tissues [261]. Furthermore, modeling the non-Newtonian behavior of MRFs is essential for applications such as targeted blood transport [262]. From a technological standpoint, the clinical application of this approach requires to be advanced medical devices integrating magnetic generators and US transducers, controlled by intelligent systems based on Biot–Stoll theory. Such systems could be employed for localized treatments, for example, in oncology or targeted brain drug delivery, offering real-time control and improving the efficacy and personalization of therapies.

Further examples of industrial and biomedical applications based on the Biot–Stoll multiphysics approach are presented in

Table 9. Biot–Stoll formulation for MRF and US: further industrial applications.

Ref.	Domain	Methods	Conclusions
[263,264,265,266,267,268,269,270,271]	Optimization of transport in porous pipelines for technical fluids	Advanced poroelastic modeling optimizes fluid transport in porous systems.	Easily implementable computational solutions
[272,273,274]	Acoustic monitoring of smart structures coated with MRF materials	US and poroelastic modeling enhance structural monitoring with MRF materials.	MRFs with acoustic sensors enable adaptive US structural monitoring.
[275,276,277,278,279,280]	Systems for acoustic printing of gradient composite materials	US enables controlled 3D printing of gradient composite materials.	Acoustic-assisted 3D printing enables precise gradient composites with directional control.

and

Table 10. Biot–Stoll Formulation for MRF & US: Further Biomedical Applications.

Ref.	Domain	Methods	Conclusions
[281,282,283,284,285,286,287,288]	Low-intensity neuro-stimulation in porous environments	Low-intensity US and Biot–Stoll poroelastic models enable noninvasive neural modulation by targeting brain tissue mechanics.	Biot–Stoll theory enhances the safe, precise modulation of brain activity using low-intensity US.
[289,290,291,292,293,294,295,296,297,298]	Predictive bio-acoustics in acoustically active prosthetic implants	Smart sensors and predictive acoustics enhance the performance of active prosthetic devices.	Intelligent acoustics improve prosthetic implant performance monitoring.
[299,300,301,302]	Poro-acoustic control of hydrogels in intra-articular drug delivery US-activated hydrogels enable controlled drug release in joints through poroelastic and biophysical mechanisms.	US-sensitive poroelastic hydrogels enable targeted, sustained drug delivery for improved osteoarthritis treatment.

, respectively, which illustrate the technological contexts in which this model proves particularly effective in describing and controlling the behavior of multiphase fluids. The entire mathematical treatment is aimed at understanding and quantifying the physical phenomena that determine the stability of MRFs. The equations are not presented in isolation, but as tools for modeling real mechanisms, experimentally verifiable and applicable in concrete technological contexts.

8. Conclusions and Perspectives

This study conducted an in-depth analysis of the potential integration between ultrasound (US) and magnetorheological fluids (MRFs) as a strategy to counteract the sedimentation of ferromagnetic particles and improve the long-term rheological stability of such systems. The focus was placed on specific physical mechanisms, including radial acoustic pressure, cavitation, and induced microvortices, as well as on the interaction between acoustic forces and varying magnetic fields. These phenomena were examined from a multiphysics perspective, combining fluid dynamics, acoustics, and magnetostatics modeling to more accurately interpret fluid behavior under complex dynamic conditions. The application of Biot–Stoll’s poroelastic theory further enhanced the predictive capabilities of the models, proving particularly useful in contexts where pressure waves interact with deformable saturated media, such as biological tissues or hybrid materials employed in advanced sensing technologies.

The theoretical approach presented is therefore relevant not only for the fundamental study of MRFs but also for their potential applications in a growing range of smart devices. From an application standpoint, the synergistic use of US and MRF opens promising scenarios in the development of active vibration control technologies, semi-adaptive damping systems, soft robotics, implantable biomedical devices, and targeted drug delivery systems. These applications require fluid materials capable of dynamically responding to multiple stimuli while maintaining stable and repeatable operating characteristics. The possibility of integrating miniaturized acoustic transducers with magnetic field generators, potentially connected to intelligent electronic circuits, represents a crucial step toward the technological transfer from research to the production of commercial devices.

For these solutions to be effectively adopted in industrial or clinical settings, several technical challenges must be addressed, including the optimization of operating parameters (US frequency and amplitude, magnetic field intensity, properties of the carrier fluid), energy efficiency management, biochemical compatibility of the materials used, and the scalability of production processes. Furthermore, validating the reliability of predictive models requires systematic comparison with experimental evidence, to be conducted through targeted campaigns in high-fidelity real or simulated environments, using acoustic diagnostics, imaging, and real-time monitoring techniques.

In this context, strengthening the dialogue between numerical modeling and experimentation is essential. While theoretical models allow for rapid exploration of a wide range of configurations and operating scenarios, experiments provide the necessary feedback to validate hypotheses, refine physical parameters, and extend the reliability of models to realistic domains. An iterative approach, alternating between simulation and experimental verification, is therefore crucial to reduce design uncertainty and accelerate the development of functional prototypes. It is worth highlighting the limitations of the different approaches used in this study. Analytical models, while useful for compactly describing dominant phenomena, are only valid in simplified and symmetric regimes. Numerical simulations, although more flexible, strongly depend on the quality of the input data and can be affected by non-negligible numerical effects in the presence of discontinuities or complex geometries. Furthermore, the physical model underlying the magneto-acoustic description does not include some biological interactions (such as metabolism, local inflammation, or immune response) that can significantly alter the effectiveness of the delivery mechanisms. These aspects represent possible extensions of the work in an experimental and multidisciplinary direction. In future developments, it will also be necessary to undertake comparative analyses between acousto-magnetic stabilization methods and alternative approaches such as electrostatic stabilization via surfactants, both in terms of stabilizing efficacy and compatibility across different application domains. Such benchmarks could help clarify the advantages and limitations of each technique under specific physical, chemical, and functional conditions, guiding informed design choices in multiphysics systems. The technological transfer of this knowledge can occur through collaborations between academic research groups, clinical centers, and industrial partners specialized in the production of mechatronic components, biomedical devices, or adaptive control systems. The development of prototypes based on acoustically modulated MRFs could give rise to new product lines in the fields of smart suspensions, acoustic–magnetic sensing, microrobots for minimally invasive procedures, and controlled-release therapeutic systems. At the same time, the growing power of multiphysics simulation tools and the evolution of dedicated software create a favorable environment for rapid iteration between virtual design and experimental validation. The accumulation of theoretical and experimental evidence supporting the effectiveness of integrating ultrasound and MRFs helps expand application possibilities and solidify the scientific foundations for future developments. As actuation and control technologies advance, it will become increasingly feasible to design materials and devices capable of adapting in real time to their operating context, with significant impacts on advanced automation, personalized medicine, and the design of interconnected intelligent systems.