Slip-Driven Interaction of Dual Spheres in Couple Stress Fluids Within a Permeable Medium

Abstract

This study investigates the consistent and uniform movement of two spherical particles within an infinite porous medium saturated with a couple stress fluid, with a particular focus on the effects of surface slippage. The research reveals that surface slippage significantly reduces the drag force experienced by the particles, thereby influencing their hydrodynamic interactions. Conversely, increases in permeability and particle size similarity tend to enhance both the drag force and the inter-particle interaction forces, affecting the overall dynamics of particle motion. The analysis is conducted within the low-Reynolds-number regime, characteristic of laminar flow dominated by viscous forces, and employs boundary collocation methodologies to derive semi-analytical solutions to the governing differential equations. This approach enables a detailed characterization of the flow behavior and inter-particle forces in intricate fluid environments, including those with porous matrices and complex rheological properties. The findings from this investigation are consistent with prior numerical analyses, notably those conducted by Alotaibi and El-Sapa (2025), and corroborate earlier studies by Shehadeh and Ashmawy (2019), which examined cases of no slippage and permeability effects. Additionally, the results align with earlier research by Shreen et al. (2018) concerning viscous fluids, thereby reinforcing the validity of the conclusions. Overall, the study enhances the understanding of particle-fluid interactions in porous, couple stress-rich media, providing valuable insights into the roles of surface slippage, permeability, and particle size in determining hydrodynamic forces.

1. Introduction

The foundational theories relevant to couple stress and micropolar fluids are extensively discussed by Stokes [1,2], who introduced the concept of couple stresses and analyzed the behavior of micropolar fluids, highlighting their significance in flows involving microstructures such as suspensions and biological fluids. Happel and Brenner [3] provided a comprehensive treatment of low-Reynolds-number hydrodynamics, particularly focusing on particulate media, which serves as a theoretical basis for modeling slow viscous flows around spherical objects. In the context of flow through porous media, Ochoa-Tapia and Whitaker [4,5] developed and experimentally validated models for momentum transfer across the interface between a porous medium and a free fluid, establishing critical boundary conditions that influence velocity and stress continuity. Their two-part study forms a cornerstone for understanding fluid behavior at porous boundaries. Furthermore, Breugem [6] refined the understanding of effective viscosity in channel-type porous media through both computational and empirical approaches, improving the accuracy of the Brinkman equation over a wide range of porosities. Building upon these foundational works, the current study integrates couple stress theory with porous media modeling and slip boundary conditions to analyze the hydrodynamic interactions of two translating spheres in a more generalized and realistic physical framework.

The interaction between two spheres in a fluid encompasses various elements influenced by the physical characteristics of the spheres and the fluid, including hydrodynamic forces such as the following. Drag force: Each sphere encounters a drag force while traversing the fluid, which is contingent upon the fluid’s viscosity, the spheres’ velocity, their dimensions, and their morphology. Several studies have investigated the hydrodynamic interaction between spherical particles under various fluid conditions. Tsuji et al. [7] conducted experimental work to measure the drag force between two spheres in a dense-phase fluid flow, introducing a pendulum-based method to quantify interaction effects over varying distances. Ekiel-Jeżewska et al. [8] examined microhydrodynamic interactions between nearby spheres in low-Reynolds-number flows, emphasizing conditions that could lead to mechanical contact. Cichocki et al. [9] analyzed the influence of boundaries, such as flat free surfaces and rigid walls, on hydrodynamic interactions among spheres, highlighting how proximity to these surfaces modifies drag behavior. Sulaymon et al. [10] explored interactions in both Newtonian and non-Newtonian fluids, demonstrating that drag coefficients in power-law fluids depend strongly on particle separation and Reynolds number. Goddard et al. [11] provided analytical solutions for time-dependent viscous flow induced via two approaching spheres, offering insights across all inter-sphere distances. While these foundational works have advanced our understanding of sphere–sphere interactions, they primarily focus on Newtonian or non-porous media. Our study extends this literature by examining the interaction of dual spheres within a Brinkman–couple stress fluid in a porous medium, incorporating both slippage and microstructural fluid effects—features not addressed comprehensively in the previous studies. In the work of El-Sapa [12], a quasi-analytical study of a micropolar fluid that is not actively moving between a hard sphere and a nonconcentric spherical boundary condition with a slip regime is accomplished. In the first scenario, the inner sphere experiences a hydrodynamic drag force and a translational wall correction factor (TWCF) that are directly proportional to the translational velocities. In the second scenario, the inner sphere experiences a hydrodynamic torque and an RWCF that are directly proportional to the angular velocities.

The analytical investigation of a fluid flow through a porous medium between a concentric rigid sphere and a spherical cavity including an incompressible axisymmetric steady coupling stress. Within the porous zone, the flow field is controlled via Brinkman’s equation. The conditions of slip and spin slip are applied by Alotaibi and El-Sapa [13] on the surfaces of the sphere and the cavity. Couple stress fluids are a category of complicated fluids that incorporate supplementary stress components (couple stresses) that consider the dimensions of particles inside the fluid. This idea is crucial in fluids exhibiting microstructural effects, such as suspensions, polymers, or biological fluids. In an incompressible pair stress fluid, Ashmawy [14] investigated the hydrodynamic interaction of two revolving spheres. It is believed that the two spheres, although having different angular velocities, are gradually rotating about their center lines. Shehadah and Ashmawy [15] examined the uniform translational movement of two collinear stiff spheres in an incompressible pair stress fluid. The two spheres are presumed to be of dissimilar dimensions and are traversing at distinct velocities along a linear axis that connects their centers. Al-Hanaya and El-Sapa [16] clarified the axisymmetric spinning of two eccentric spheres submerged in an incompressible couple stress fluid across a porous annular domain. The inquiry includes boundary conditions imposed on the surfaces of both spheres, which spin axially at varying angular velocities. An incompressible pair stress fluid was used to study the rotating oscillation of a rigid sphere by Shehadeh and Ashmawy [17]. The spherical boundary was subjected to the traditional no-slip boundary constraints. In addition, the pair stresses at the sphere’s edge are presumed to disappear. In this investigation, the motion is produced via a rapid angular velocity-dependent rotary oscillation of the rigid sphere along an axis passing through its center.

The intricate behaviors of fluids with microstructural effects when they interact with or flow through porous structures must be considered while studying the interaction of couple stress fluids with porous media. This combination has several practical uses in the fields of biology, industry, and ecology. The complex issue of evaluating the impact of slippage and permeability on the motion of two solid spheres in an infinite-stress fluid can be addressed through numerical approaches. The collocation method is particularly beneficial for analyzing the interaction of spheres in pair stress fluids, especially under complex boundary conditions or when the accurate identification of specific solution attributes is required. The research provides significant insights into the effects of pair stresses on flow patterns and interaction forces, elements that are generally difficult to depict using traditional numerical methods. The collocation approach, while beneficial for its flexibility and potential precision, requires the careful selection of basis functions and collocation points, along with effective solution strategies for the resulting system of equations. In El-Sapa [12], the numerical solution was obtained by combining the foundational solutions in two spherical coordinate systems at the poles of both the solid sphere and the spherical shell. This was achieved using a collocation method for a stationary micropolar fluid situated between the solid sphere and a non-concentric spherical shell with a sliding mechanism.

This study examines the interrelationship between two solid spheres moving through an infinite porous material where slippage occurs. The innovative aspect of this research is the introduction of two solid spheres navigating a porous material filled with a couple stress fluid. We employed semi-analytical solutions to derive the field equations and numerical methods to calculate the drag force for varying couple stress viscosity parameters, separation distances, size ratios, and velocity ratios. Furthermore, the discussion of flow patterns emphasizes the impact of the permeability parameter on the flow surrounding the two spheres in couple stress fluids, highlighting unique streamlines and hydrodynamic interactions.

2. Field Equations of Couple Stress Fluid

The Brinkman–couple stress fluid, which is uniformly incompressible, is supplied when there are no body and body couple forces:

$$\begin{array}{c}\hfill \nabla ·\overrightarrow{q}=0,\end{array}$$

(1)

$$\begin{array}{c}\hfill \eta {\nabla }^4\overrightarrow{q}-\mu {\nabla }^2\overrightarrow{q}+{\displaystyle \frac{\mu }{\kappa }}\overrightarrow{q}+\nabla p=0,\end{array}$$

(2)

where $\overrightarrow{q}$ and p represent the velocity field and fluid pressure, respectively, and $\rho$ and $\eta$ denote the fluid density and the couple stress viscosity coefficient, respectively. By assigning $\eta =0$, Equation (2) Formulate an appropriate equation for Newtonian fluids as articulated by Stokes and Navier [3]. The effective viscosity in the permeable medium during fluid flow is represented as ${\mu }_{eff}$. The correlation between viscosity and effective viscosity, as proposed by Ochoa-Tapia and Whitaker [4,5], is articulated as ${\mu }_{eff}={\displaystyle \frac{\mu }{\phi }}$. Conversely, Breugem [6] assessed the calculated value of ${\mu }_{eff}$ by both computational and empirical methods, enhancing the precision of the Brinkman solution over a broad spectrum of wave numbers and porosities. The primary hypothesis of this investigation is $\mu ={\mu }_{eff}$.

Moreover, the following equations delineate the essential correlations for the couple stress tensor and the stress tensor, respectively.

$$\begin{array}{cc}\hfill {\tau }_{ij}& =-p{\delta }_{ij}+2\lambda E_{ij}-{\displaystyle \frac{1}{2}}{ϵ}_{ijk}{m}_{sk,s},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill m_{ij}& =m{\delta }_{ij}+4\eta {\omega }_{i,j}+4{\eta }^{\prime }{\omega }_{j,i},\hfill \end{array}$$

(4)

where the deformed tensor is $E_{ij}={\displaystyle \frac{1}{2}}(q_{i,j}+q_{j,i})$, the alternative tensor is ${ϵ}_{ijk}$, the Kronecker delta is ${\delta }_{ij}$, $\mu$ is the fluid’s classical viscosity, and the material constant is $\lambda$. The viscosity coefficient ${\eta }^{\prime }$ signifies the secondary viscosity coefficient. This supplementary component is absent from the initial equation of the stress tensor and only emerges in the subsequent constitutive equation of the couple stress tensor [1]. The viscosity coefficients satisfy the following inequality requirements, as per the couple stress fluid equations:

$$\mu \ge 0,\eta \ge 0,\eta -{\eta }^{\prime }\ge 0,3\lambda +2\mu \ge 0.$$

(5)

3. Axisymmetric Slow Motion of Couple Stress Fluid Past an Object

This section examines the gradual, uniform translational movement of an axisymmetric object through an incompressible couple stress fluid. The stream function formulation is particularly advantageous for analyzing incompressible and axisymmetric flows because it inherently satisfies the continuity equation. In such flows, where the velocity field has no dependence on the azimuthal angle and no azimuthal component, the use of a stream function $\psi (r,\theta )$ ensures mass conservation via construction. Specifically, defining the radial and polar components of velocity in terms of derivatives of $\psi$ guarantees that the divergence of the velocity field is zero. This reduces the complexity of the problem by eliminating the need to explicitly enforce the continuity constraint, thereby simplifying the mathematical formulation and aiding in the derivation of analytical or semi-analytical solutions. The spherical coordinates $(r,\theta ,\varphi )$ are utilized in coupling with the unit vectors $({\overrightarrow{e}}_r,{\overrightarrow{e}}_{\theta },{\overrightarrow{e}}_{\varphi })$, with the origin located at the centroid of the axisymmetric body. Considering the axial symmetric of the boundary and the consequent motion, the subsequent formulae pertain to the velocity and vorticity vectors:

$$\left.\begin{array}{cc}\hfill \overrightarrow{q}& =q_r(r,\theta ){\overrightarrow{e}}_r+q_{\theta }(r,\theta ),\hfill \\ \hfill \overrightarrow{\omega }& ={\omega }_{\varphi }{\overrightarrow{e}}_{\varphi }.\hfill \end{array}\right\}$$

(6)

The stream function $\psi$ can be utilized to represent the components of velocity.

$$\left.\begin{array}{cc}\hfill q_r(r,\theta )& ={\displaystyle \frac{-1}{r^{2}sin\theta }}{\displaystyle \frac{\partial \psi }{\partial \theta }},\hfill \\ \hfill q_{\theta }(r,\theta )& ={\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\partial \psi }{\partial r}}.\hfill \end{array}\right\}$$

(7)

Also, the vorticity vector is defined as follows:

$$\overrightarrow{\omega }=\frac{1}{2}\nabla \wedge \overrightarrow{q}=\frac{E^2\psi }{2rsin\theta }{\overrightarrow{e}}_{\varphi }$$

(8)

where $E^2={\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1-{\zeta }^2}{r^2}}{\displaystyle \frac{{\partial }^2}{\partial {\xi }^2}},\zeta =cos\theta$ is the axisymmetric Stokesian differential operator. This reduces the field equations to the following:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mu }{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}\left(E^2\psi \right)-{\displaystyle \frac{\eta }{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}(E^2\left(E^2\psi \right))-{\displaystyle \frac{\mu }{\kappa }}{\displaystyle \frac{1}{r^{2}sin\theta }}{\displaystyle \frac{\partial \psi }{\partial \theta }}+{\displaystyle \frac{\partial p}{\partial r}}& =& 0,\hfill \\ \hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{\mu }{rsin\theta }}{\displaystyle \frac{\partial }{\partial r}}\left(E^2\psi \right)+{\displaystyle \frac{\eta }{rsin\theta }}{\displaystyle \frac{\partial }{\partial r}}(E^2\left(E^2\psi \right))+{\displaystyle \frac{\mu }{\kappa }}{\displaystyle \frac{1}{rsin\theta }}{\displaystyle \frac{\partial \psi }{\partial r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial p}{\partial \theta }}& =& 0.\hfill \end{array}$$

(10)

Consequently, we express the non-dimensional values in relation to the radius of a rigid sphere a as follows:

$$\begin{array}{cc}\hfill & \widehat{\overrightarrow{q}}={\displaystyle \frac{\overrightarrow{q}}{U}},\widehat{\overrightarrow{\omega }}={\displaystyle \frac{a\overrightarrow{\omega }}{U}},\widehat{p}={\displaystyle \frac{ap}{\mu U}},\widehat{\nabla }=a\nabla ,\widehat{{\tau }_{rr}}={\displaystyle \frac{a{\tau }_{rr}}{\mu U}},{\alpha }^2={\displaystyle \frac{a^2}{\kappa }},\hfill \\ \hfill & {\xi }^2={\displaystyle \frac{a^2\mu }{\eta }},{\xi }^{\prime 2}={\displaystyle \frac{a^2\mu }{{\eta }^{\prime }}}.\hfill \end{array}$$

(11)

Inserting Equation (11) into Equations (9) and (10) and dropping that, we have the following:

$$\begin{array}{c}\hfill {\displaystyle \frac{{\xi }^{-2}}{r^{2}sin\theta }}{\displaystyle \frac{\partial }{\partial \theta }}{E}^2\left(E^2-{\xi }^2\right)\psi +{\displaystyle \frac{{\alpha }^2}{r^{2}sin\theta }}{\displaystyle \frac{\partial \psi }{\partial \theta }}-{\displaystyle \frac{\partial p}{\partial r}}=0,\end{array}$$

(12)

$$\begin{array}{c}\hfill {\displaystyle \frac{-{\xi }^{-2}}{rsin\theta }}{\displaystyle \frac{\partial }{\partial r}}{E}^2\left(E^2-{\xi }^2\right)\psi -{\displaystyle \frac{{\alpha }^2}{rsin\theta }}{\displaystyle \frac{\partial \psi }{\partial r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial p}{\partial \theta }}=0.\end{array}$$

(13)

Upon eliminating the pressure in Equations (12) and (13), we obtain the subsequent partial differential equation:

$$\begin{array}{c}\hfill E^2\left(E^4-{\xi }^2{E}^2+{\xi }^2{\alpha }^2\right)\psi =0,\end{array}$$

(14)

where the roots of Equation (14) are $k_i^2={\displaystyle \frac{{\xi }^2\pm \xi \sqrt{{\xi }^2-4{\alpha }^2}}{2}},i=1,2$. Furthermore, the modified Stokes equation for a non-polar fluid is expressed by (14) as $\xi \to \infty$ and $\alpha \to 0$ (absence of porous media).

Calculating the couple stress component involves the following:

$$\begin{array}{ccc}\hfill m_{r\varphi }& =& {\displaystyle \frac{4}{{\xi }^2}}{\displaystyle \frac{\partial {\omega }_{\varphi }}{\partial r}}-{\displaystyle \frac{4}{{\xi }^{\prime 2}}}{\displaystyle \frac{{\omega }_{\varphi }}{r}},\hfill \end{array}$$

(15)

The esteemed differential Equation (14) and the velocity field functions from (7) and vorticity (8) are expressed as follows:

$$\psi (r,\theta )=\sum _{n=2}^{\infty }\left[A_n{r}^{-n+1}+B_n{r}^{\frac{1}{2}}{K}_{n-\frac{1}{2}}\left(k_{1}r\right)+C_n{r}^{\frac{1}{2}}{K}_{n-\frac{1}{2}}\left(k_{2}r\right)\right]G_n\left(\zeta \right),$$

(16)

$$\begin{array}{cc}\hfill q_r(r,\theta )=& -\sum _{n=2}^{\infty }\left[A_n{r}^{-n-1}+B_n{r}^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_{1}r\right)+C_n{r}^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_{2}r\right)\right]P_{n-1}\left(\zeta \right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill q_{\theta }(r,\theta )=& \sum _{n=2}^{\infty }[(1-n)A_n{r}^{-n-1}+B_n{r}^{\frac{-3}{2}}(n{K}_{n-\frac{1}{2}}\left(k_{1}r\right)-k_{1}r{K}_{n+\frac{1}{2}}\left(k_{1}r\right))\hfill \\ \hfill & +C_n{r}^{\frac{-3}{2}}(n{K}_{n-\frac{1}{2}}\left(k_{2}r\right)-k_{2}r{K}_{n+\frac{1}{2}}\left(k_{2}r\right))]{\displaystyle \frac{G_n\left(\zeta \right)}{\sqrt{1-{\zeta }^2}}}.\hfill \end{array}$$

(18)

Based on Equation (8), the vorticity component is as follows:

$$\begin{array}{cc}\hfill {\omega }_{\varphi }(r,\theta )=& \sum _{n=2}^{\infty }{r}^{\frac{-1}{2}}\left[B_n{k}_1^2{K}_{n-\frac{1}{2}}\left(k_{1}r\right)+C_n{k}_2^2{K}_{n-\frac{1}{2}}\left(k_{2}r\right)\right]{\displaystyle \frac{G_n\left(\zeta \right)}{\sqrt{1-{\zeta }^2}}}.\hfill \end{array}$$

(19)

To compute an estimation examination of the flow of a pair stress fluid around an axisymmetric particle traversing an infinite porous medium in its direction of motion, we might employ the formula [2] for the drag force experienced by an incompressible fluid:

$$F_z=\mu {\alpha }^2\left(UV+4\pi \underset{r\to \infty }{lim}{\displaystyle \frac{r^3\psi }{{\rho }^2}}\right),$$

(20)

where V is the volume of the body. As a result, we get the following direct formula by substituting expression (16) into the drag formula (20):

$$F_z=\mu {\alpha }^2\left(UV+2\pi A_2\right).$$

(21)

4. How Two Solid Spheres Interact with One Another via a Porous Medium

This study examines the flow of an incompressible pair stress fluid in a steady state as it traverses between two translating collinear rigid spheres. Two solid spherical objects with radii $a_j$ (where $j=1,2$) and uniform velocities $U_j$ (where $j=1,2$) translate along a shared axis connecting their centers, therefore generating fluid movement. As illustrated in Figure 1, the two spherical objects are positioned externally to each other. The fluid is presumed to be stationary at a considerable distance from the two spheres. Two spherical coordinate systems, $(r_1,{\theta }_1,\varphi )$ and $(r_2,{\theta }_2,\varphi )$, originating from the centers of spheres $a_1$ and $a_2$, respectively, are analyzed for convenience. The axisymmetric bounds indicate that the angle $\varphi$ does not influence any of the flow field functions. Moreover, the subsequent relations connect the two coordinate systems $(r_1,{\theta }_1)$ and $(r_2,{\theta }_2)$.

$$\left.\begin{array}{cc}\hfill r_1^2& =r_2^2+h^2-2r_{2}hcos{\theta }_2,\hfill \\ \hfill r_2^2& =r_1^2+h^2+2r_{1}hcos{\theta }_1.\hfill \end{array}\right\}$$

(22)

The hypothesis of Stokesian flow will be utilized within the condition of minimal velocities. Under these conditions, the flow demonstrates axial symmetry, and all hydrodynamic parameters remain unaffected by $\varphi$. Let ${\overrightarrow{q}}^j$ denote the velocity vector of the couple stress fluid influenced by the presence of the spherical item $a_j$ in isolation from other objects. We select ${\overrightarrow{q}}^{\left(j\right)}$, ${\overrightarrow{\omega }}^{\left(j\right)}$, ${\tau }_{r\theta }^{\left(j\right)}$, and $m_{r\varphi }^{\left(j\right)}$ in the following manner:

$$\begin{array}{c}\hfill \left.\begin{array}{cc}\hfill {\overrightarrow{q}}^{\left(j\right)}(r_j,{\theta }_j)& =q_r^{\left(1\right)}(r_1,{\theta }_1)+q_r^{\left(2\right)}(r_2,{\theta }_2),\hfill \\ \hfill {\omega }_{\varphi }^{\left(j\right)}(r_j,{\theta }_j)& ={\omega }_{\varphi }^{\left(1\right)}(r_1,{\theta }_1)+{\omega }_{\varphi }^{\left(2\right)}(r_2,{\theta }_2),\hfill \\ \hfill t_{r_j{\theta }_j}^{\left(j\right)}(r_j,{\theta }_j)& =m_{r_1{\theta }_1}^{\left(1\right)}(r_1,{\theta }_1)+m_{r_2{\theta }_2}^{\left(2\right)}(r_2,{\theta }_2),\hfill \\ \hfill m_{r_j\varphi }^{\left(j\right)}(r_j,{\theta }_j)& =m_{r_1\varphi }^{\left(1\right)}(r_1,{\theta }_1)+m_{r_2\varphi }^{\left(2\right)}(r_2,{\theta }_2).\hfill \end{array}\right\}\end{array}$$

(23)

The typical approach to solving a sixth-order simple partial differential equation is the separation of variables strategy (14):

$$\begin{array}{cc}\hfill \psi (r,\theta )=& \sum _{j=1}^2\sum _{n=2}^{\infty }\left[A_n^{\left(j\right)}{r}_j^{-n+1}+B_n^{\left(j\right)}{r}_j^{\frac{1}{2}}{K}_{n-\frac{1}{2}}\left(k_1{r}_j\right)+C_n^{\left(j\right)}{r}_j^{\frac{1}{2}}{K}_{n-\frac{1}{2}}\left(k_2{r}_j\right)\right]G_n\left({\zeta }_j\right),\hfill \end{array}$$

(24)

The modified Bessel functions of the first kind of order, n, are denoted by the function $K_n(.)$, and the constants $A_n^{\left(j\right)},B_n^{\left(j\right)},C_n^{\left(j\right)},D_n^{\left(j\right)},E_n^{\left(j\right)},F_n^{\left(j\right)},j=1,2$ will be ascertained from the constraints. The elements of velocity are as follows:

$$\begin{array}{cc}\hfill q_r=& -\sum _{j=1}^2\sum _{n=2}^{\infty }\left[A_n^{\left(j\right)}{r}_j^{-n-1}+B_n^{\left(j\right)}{r}_j^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_1{r}_j\right)+C_n^{\left(j\right)}{r}_j^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_2{r}_j\right)\right]P_{n-1}\left({\zeta }_j\right).\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill q_{\theta }=& \sum _{j=1}^2\sum _{n=2}^{\infty }[(1-n)A_n^{\left(j\right)}{r}_j^{-n-1}+B_n^{\left(j\right)}{r}_j^{\frac{-3}{2}}(n{K}_{n-\frac{1}{2}}\left(k_1{r}_j\right)-k_1{r}_j{K}_{n+\frac{1}{2}}\left(k_1{r}_j\right))\hfill \\ \hfill & +C_n^{\left(j\right)}{r}_j^{\frac{-3}{2}}(n{K}_{n-\frac{1}{2}}\left(k_2{r}_j\right)-k_2{r}_j{K}_{n+\frac{1}{2}}\left(k_2{r}_j\right))]{\displaystyle \frac{G_n\left({\zeta }_j\right)}{\sqrt{1-{\zeta }_j^2}}}.\hfill \end{array}$$

(26)

From Equation (8), the vorticity component is as follows:

$$\begin{array}{cc}\hfill {\omega }_{\varphi }=& {\displaystyle \frac{1}{2}}\sum _{j=1}^2\sum _{n=2}^{\infty }{r}_j^{\frac{-1}{2}}\left[B_n^{\left(j\right)}{k}_1^2{K}_{n-\frac{1}{2}}\left(k_1{r}_j\right)+C_n^{\left(j\right)}{k}_2^2{K}_{n-\frac{1}{2}}\left(k_2{r}_j\right)\right]{\displaystyle \frac{G_n\left({\zeta }_j\right)}{\sqrt{1-{\zeta }_j^2}}}.\hfill \end{array}$$

(27)

The tangential stresses and couple stress are obtained as follows:

$$\begin{array}{cc}\hfill t_{r\theta }& (r,\theta )=\sum _{n=2}^{\infty }[2(n^2-1)A_n{r}^{-n-1}\hfill \\ \hfill & +B_n{r}^{\frac{-5}{2}}\left((k_1^2{r}^2+2n^2-4n-{\xi }^{-2}{k}_1^4{r}^2)K_{n-\frac{1}{2}}\left(k_{1}r\right)+2k_{1}r{K}_{n+\frac{1}{2}}\left(k_{1}r\right)\right)\hfill \\ \hfill & +D_n{r}^{\frac{-5}{2}}\left((k_2^2{r}^2+2n^2-4n-{\xi }^{-2}{k}_2^4{r}^2)K_{n-\frac{1}{2}}\left(k_{2}r\right)+2k_{2}r{K}_{n+\frac{1}{2}}\left(k_{2}r\right)\right)]{\displaystyle \frac{{\Im }_n\left(\zeta \right)}{\sqrt{1-{\zeta }^2}}},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill m_{r\varphi }(r,\theta )& =\sum _{j=1}^2\sum _{n=2}^{\infty }{r}_j^{\frac{-3}{2}}[B_n^{\left(j\right)}{k}_1^2\left(((n-1){\xi }^{-2}-{\xi }^{\prime -2})K_{n-\frac{1}{2}}-{\xi }^{-2}{k}_1{r}_j{K}_{n+\frac{1}{2}}\left(k_1{r}_j\right)\right)\hfill \\ \hfill & +C_n^{\left(j\right)}{k}_2^2\left(((n-1){\xi }^{-2}-{\xi }^{\prime -2})K_{n-\frac{1}{2}}-{\xi }^{-2}{k}_2{r}_j{K}_{n+\frac{1}{2}}\left(k_2{r}_j\right)\right)]{\displaystyle \frac{G_n\left({\zeta }_j\right)}{\sqrt{1-{\zeta }_j^2}}}.\hfill \end{array}$$

(29)

5. Boundary Conditions

Six boundary conditions are required for Equation (14) to comprehensively define the problem. Stokes [1] posited assumptions regarding two probable scenarios: The boundary conditions (A) specify that there are no shear stresses at the region’s edge, and the second condition states that the boundary’s rotational velocity is equivalent to the vorticity along the boundary. The six criteria involve two spheres moving at different velocities, $U_j,j=1,2$, with the surfaces of the solid objects described by $r_j=a_j,j=1,2$ conforming to particular kinematic and dynamic boundary conditions.

$$\left.\begin{array}{cc}\hfill q_r(r_j,{\theta }_j)& =U_{j}cos{\theta }_j,\hfill \\ \hfill q_{\theta }(r_j,{\theta }_j)& ={\beta }_1{t}_{r\theta }(r_j,{\theta }_j)-U_{j}sin{\theta }_j,\hfill \\ \hfill {\omega }_{\varphi }(r_j,{\theta }_j)& ={\beta }_2{m}_{r_j\varphi }(r_j,{\theta }_j).\hfill \end{array}\right\}$$

(30)

where ${\beta }_j,j=1,2$, is a constant referred to as frictional resistance. The outcome is contingent upon the composition of the surfaces submerged in the fluid and the nature of the fluid itself. In the limiting scenario where ${\beta }_1\to 0$, a no-slip condition prevails, and the solid sphere behaves akin to a spherical gas bubble as ${\beta }_1\to \infty$. Furthermore, as ${\beta }_2$ approaches infinity, the couple stress at the boundary becomes null, indicating that the mechanical interactions at the surface are equivalent solely to a force distribution. Conversely, as ${\beta }_2\to 0$, this corresponds to inhibiting the relative rotation of the fluid element near the aerosol’s surface.

The formulations for normal and tangential velocities, as well as coupling stress, derived from Equations (25)–(29) are inserted into the boundary conditions (30) as follows:

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }\left[A_n^{\left(1\right)}{a}_1^{-n-1}+B_n^{\left(1\right)}{a}_1^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_1{a}_1\right)+C_n^{\left(1\right)}{a}_1^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_2{a}_1\right)\right]P_{n-1}\left({\zeta }_1\right)\hfill \\ \hfill & +\sum _{n=2}^{\infty }\left[A_n^{\left(2\right)}{r}_2^{-n-1}+B_n^{\left(2\right)}{r}_2^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_1{r}_2\right)+C_n^{\left(j\right)}{r}_2^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_2{r}_2\right)\right]{|}_{r_1=a_1}{P}_{n-1}\left({\zeta }_2\right)\hfill \\ \hfill & =-U_{1}cos{\theta }_1,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }\left[A_n^{\left(1\right)}{r}_1^{-n-1}+B_n^{\left(1\right)}{r}_1^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_1{r}_1\right)+C_n^{\left(1\right)}{r}_1^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_2{r}_1\right)\right]{|}_{r_2=a_2}{P}_{n-1}\left({\zeta }_1\right)\hfill \\ \hfill & +\sum _{n=2}^{\infty }\left[A_n^{\left(2\right)}{a}_2^{-n-1}+B_n^{\left(2\right)}{a}_2^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_1{a}_2\right)+C_n^{\left(j\right)}{a}_2^{\frac{-3}{2}}{K}_{n-\frac{1}{2}}\left(k_2{a}_2\right)\right]P_{n-1}\left({\zeta }_2\right)\hfill \\ \hfill & =-U_{2}cos{\theta }_2,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }[\left((1-n)-2{\eta }_1(n^2-1)\right)A_n^{\left(1\right)}{a}_1^{-n-1}+B_n^{\left(1\right)}{a}_1^{\frac{-3}{2}}((n-{\eta }_1(k_1^2{a}_1^2+2n^2-4n\hfill \\ \hfill & -{\xi }^{-2}{k}_1^4{a}_1^2\left)\right)K_{n-\frac{1}{2}}\left(k_1{a}_1\right)-(1+2{\eta }_1)k_1{a}_1{K}_{n+\frac{1}{2}}\left(k_1{a}_1\right))+C_n^{\left(1\right)}{a}_1^{\frac{-3}{2}}((n-{\eta }_1(k_2^2{a}_1^2\hfill \\ \hfill & +2n^2-4n-{\xi }^{-2}{k}_2^4{a}_1^2\left)\right)K_{n-\frac{1}{2}}\left(k_2{a}_1\right)-(1+2{\eta }_1)k_2{a}_1{K}_{n+\frac{1}{2}}\left(k_2{a}_1\right)\left)\right]{\displaystyle \frac{{\Im }_n\left({\zeta }_1\right)}{\sqrt{1-{\zeta }_1^2}}}\hfill \\ \hfill & +\sum _{n=2}^{\infty }[\left((1-n)-2{\eta }_1{\displaystyle \frac{a_1}{r_2}}(n^2-1)\right)A_n^{\left(2\right)}{r}_2^{-n-1}+B_n^{\left(2\right)}{r}_2^{\frac{-3}{2}}((n-{\eta }_1{\displaystyle \frac{a_1}{r_2}}(k_1^2{r}_2^2+2n^2-4n\hfill \\ \hfill & -{\xi }^{-2}{k}_1^4{r}_2^2\left)\right)K_{n-\frac{1}{2}}\left(k_1{r}_2\right)-(1+2{\eta }_1{\displaystyle \frac{a_1}{r_2}})k_1{r}_2{K}_{n+\frac{1}{2}}\left(k_1{r}_2\right))+C_n^{\left(2\right)}{r}_2^{\frac{-3}{2}}((n-{\eta }_1{\displaystyle \frac{a_1}{r_2}}(k_2^2{r}_2^2\hfill \\ \hfill & +2n^2-4n-{\xi }^{-2}{k}_2^4{r}_2^2\left)\right)K_{n-\frac{1}{2}}\left(k_2{r}_2\right)-(1+2{\eta }_1{\displaystyle \frac{a_1}{r_2}})k_2{r}_2{K}_{n+\frac{1}{2}}\left(k_2{r}_2\right){\left)\right]|}_{r_1=a_1}{\displaystyle \frac{{\Im }_n\left({\zeta }_2\right)}{\sqrt{1-{\zeta }_2^2}}}\hfill \\ \hfill & =-U_{1}sin{\theta }_1,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }[\left((1-n)-2{\eta }_1{\displaystyle \frac{a_1}{r_1}}(n^2-1)\right)A_n^{\left(1\right)}{r}_1^{-n-1}+B_n^{\left(1\right)}{r}_1^{\frac{-3}{2}}((n-{\eta }_1{\displaystyle \frac{a_1}{r_1}}(k_1^2{r}_1^2+2n^2-4n\hfill \\ \hfill & -{\xi }^{-2}{k}_1^4{r}_1^2\left)\right)K_{n-\frac{1}{2}}\left(k_1{r}_1\right)-(1+2{\eta }_1{\displaystyle \frac{a_1}{r_1}})k_1{r}_1{K}_{n+\frac{1}{2}}\left(k_1{r}_1\right))+C_n^{\left(1\right)}{r}_1^{\frac{-3}{2}}((n-{\eta }_1{\displaystyle \frac{a_1}{r_1}}(k_2^2{r}_1^2\hfill \\ \hfill & +2n^2-4n-{\xi }^{-2}{k}_2^4{r}_1^2\left)\right)K_{n-\frac{1}{2}}\left(k_2{r}_1\right)-(1+2{\eta }_1{\displaystyle \frac{a_1}{r_1}})k_2{r}_1{K}_{n+\frac{1}{2}}\left(k_2{r}_1\right)\left)\right]{|}_{r_2=a_2}{\displaystyle \frac{{\Im }_n\left({\zeta }_1\right)}{\sqrt{1-{\zeta }_1^2}}}\hfill \\ \hfill & +\sum _{n=2}^{\infty }[\left((1-n)-2{\eta }_1{\displaystyle \frac{a_1}{a_2}}(n^2-1)\right)A_n^{\left(2\right)}{a}_2^{-n-1}+B_n^{\left(2\right)}{a}_2^{\frac{-3}{2}}((n-{\eta }_1{\displaystyle \frac{a_1}{a_2}}(k_1^2{a}_2^2+2n^2-4n\hfill \\ \hfill & -{\xi }^{-2}{k}_1^4{r}_2^2\left)\right)K_{n-\frac{1}{2}}\left(k_1{a}_2\right)-(1+2{\eta }_1{\displaystyle \frac{a_1}{a_2}})k_1{a}_2{K}_{n+\frac{1}{2}}\left(k_1{a}_2\right))+C_n^{\left(2\right)}{a}_2^{\frac{-3}{2}}((n-{\eta }_1{\displaystyle \frac{a_1}{a_2}}(k_2^2{a}_2^2\hfill \\ \hfill & +2n^2-4n-{\xi }^{-2}{k}_2^4{a}_2^2\left)\right)K_{n-\frac{1}{2}}\left(k_2{a}_2\right)-(1+2{\eta }_1{\displaystyle \frac{a_1}{a_2}})k_2{a}_2{K}_{n+\frac{1}{2}}\left(k_2{a}_2\right)\left)\right]{\displaystyle \frac{{\Im }_n\left({\zeta }_2\right)}{\sqrt{1-{\zeta }_2^2}}}\hfill \\ \hfill & =-U_{2}sin{\theta }_2,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }{a}_1^{\frac{-1}{2}}\left[B_n^{\left(1\right)}{k}_1^2\right((\frac{1}{2}-2{\eta }_2\frac{a_2}{a_1}((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_1{a}_1\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_1{a}_1{\eta }_2\frac{a_2}{a_1}{K}_{n+\frac{1}{2}}\left(k_1{a}_1\right))+C_n^{\left(1\right)}{k}_2^2((\frac{1}{2}-2{\eta }_2\frac{a_2}{a_1}((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_2{a}_1\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_2{a}_1{\eta }_2\frac{a_2}{a_1}{K}_{n+\frac{1}{2}}\left(k_2{a}_1\right)\left]\right){\displaystyle \frac{{\Im }_n\left({\zeta }_1\right)}{\sqrt{1-{\zeta }_1^2}}}\hfill \\ \hfill & +\sum _{n=2}^{\infty }{r}_2^{\frac{-1}{2}}\left[B_n^{\left(2\right)}{k}_1^2\right((\frac{1}{2}-2{\eta }_2\frac{a_2}{r_2}((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_1{r}_2\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_1{r}_2{\eta }_2\frac{a_2}{r_2}{K}_{n+\frac{1}{2}}\left(k_1{r}_2\right))+C_n^{\left(2\right)}{k}_2^2((\frac{1}{2}-2{\eta }_2\frac{a_2}{r_2}((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_2{r}_2\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_2{r}_2{\eta }_2\frac{a_2}{r_2}{K}_{n+\frac{1}{2}}\left(k_2{r}_2\right)\left)\right]{|}_{r_1=a_1}{\displaystyle \frac{{\Im }_n\left({\zeta }_2\right)}{\sqrt{1-{\zeta }_2^2}}}=0,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \sum _{n=2}^{\infty }{r}_1^{\frac{-1}{2}}\left[B_n^{\left(1\right)}{k}_1^2\right((\frac{1}{2}-2{\eta }_2\frac{a_2}{r_1}((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_1{r}_1\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_1{r}_1{\eta }_2\frac{a_2}{r_1}{K}_{n+\frac{1}{2}}\left(k_1{r}_1\right))+C_n^{\left(1\right)}{k}_2^2((\frac{1}{2}-2{\eta }_2\frac{a_2}{r_1}((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_2{r}_1\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_2{r}_1{\eta }_2\frac{a_2}{r_1}{K}_{n+\frac{1}{2}}\left(k_2{r}_1\right)\left)\right]{|}_{r_2=a_2}{\displaystyle \frac{{\Im }_n\left({\zeta }_1\right)}{\sqrt{1-{\zeta }_1^2}}}\hfill \\ \hfill & +\sum _{n=2}^{\infty }{a}_2^{\frac{-1}{2}}\left[B_n^{\left(2\right)}{k}_1^2\right((\frac{1}{2}-2{\eta }_2((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_1{a}_2\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_1{a}_2{\eta }_2{K}_{n+\frac{1}{2}}\left(k_1{a}_2\right))+C_n^{\left(2\right)}{k}_2^2((\frac{1}{2}-2{\eta }_2((n-1){\xi }^{-2}-{\xi }^{\prime -2}))K_{n-\frac{1}{2}}\left(k_2{a}_2\right)\hfill \\ \hfill & +2{\xi }^{-2}{k}_2{a}_2{\eta }_2{K}_{n+\frac{1}{2}}\left(k_2{a}_2\right)\left)\right]{\displaystyle \frac{{\Im }_n\left({\zeta }_2\right)}{\sqrt{1-{\zeta }_2^2}}}=0.\hfill \end{array}$$

(36)

This issue may be resolved with the boundary collocation approach developed by Gluckman et al. [18]. Nevertheless, resolving the entire infinite system of equations containing the unknown constants is unfeasible. Consequently, the hydrodynamic, dimensionless drag force from (21) on the two rigid spheres $a_j,j=1,2$ relative to the drag in the unbounded viscous region $F_{\infty }^{\left(j\right)}=2\mu \pi U_j{a}_j$ is expressed as follows:

$$F_z^{\left(j\right)}=2\mu \pi {\alpha }^2{U}_j{a}_j\left({\displaystyle \frac{2}{3}}+A_2^{\left(j\right)}\right).$$

(37)

The procedure for obtaining a numerical solution to the aforementioned system can be delineated as follows: Initially, each infinite series is terminated after a designated number, N, of terms, resulting in a finite quantity of unknown constants. A suitable collection of colocation points on each spherical particle is subsequently chosen to implement boundary conditions, yielding a system of linear equations comprising $6N$ equations that correspond to an equal number of unknown constants. This system is resolved to ascertain the unknown constants, hence defining the flow field. To improve precision, particularly when particles are in close proximity, additional colocation sites are required. The hydrodynamic drag force $F_j$ exerted on particle $a_j$ is determined using Formula (28). The initial boundary condition points are ${\theta }_j=0$, ${\theta }_j=\pi$, and ${\theta }_j=2\pi$; however, employing these points may result in a singular coefficient matrix. To mitigate this problem, alternative points ${\theta }_j=ϵ,2\pi -ϵ,2\pi +ϵ,\pi -ϵ$ are utilized, along with supplementary mirror-image points to enhance precision.

6. Results and Discussion

This study introduces the interaction among two solid spheres moving into an unbounded porous medium filled with an incompressible couple stress fluid with different values of size parameter $a_1/a_2$, distance separation $\delta =h/(a_1+a_2)$, slip and spin slip ${\eta }_1,{\eta }_2$, the velocity ratio $U_2/U_1$, and couple stress fluid parameters $\overline{\eta },{\overline{\eta }}^{\prime }$. The normalized drag force is shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6, and

Table 1. Normalized drag force for two solid spheres translating to the same velocity in the same direction throughan unbounded porous medium filled with couple stress fluid at ${\eta }_1={\eta }_2=\eta , {\overline{\eta }}^{\prime }=0.1, \alpha =1.0$.

$\mathit{\delta }$	${\mathit{a}}_1/{\mathit{a}}_2$	$\overline{\mathit{\eta }}=0.01$			$\overline{\mathit{\eta }}=0.1$
		$\mathit{\eta }=\mathbf{0.0}$	$\mathit{\eta }=\mathbf{1.0}$	$\mathit{\eta }=\infty$	$\mathit{\eta }=\mathbf{0.0}$	$\mathit{\eta }=\mathbf{1.0}$	$\mathit{\eta }=\infty$
	0.1	2.536719	1.235271	0.978776	4.597639	1.927397	1.533600
	0.2	2.553545	1.315319	1.065318	4.603861	2.043600	1.654458
	0.3	2.550512	1.367235	1.115399	4.570197	2.122442	1.723979
	0.4	2.545211	1.403428	1.146771	4.538908	2.181066	1.767469
	0.5	2.541381	1.429929	1.167603	4.516444	2.227392	1.796264
$1.5$	0.6	2.539217	1.449999	1.182029	4.501257	2.265501	1.816166
	0.7	2.538301	1.465603	1.192395	4.491201	2.297803	1.830495
	0.8	2.538227	1.478013	1.231000	4.484694	2.325835	1.841071
	0.9	2.538697	1.488087	1.205805	4.480680	2.350629	1.849281
	0.99	2.539416	1.495656	1.209707	4.478621	2.370758	1.855021
	0.999	2.539499	1.496348	1.210346	4.478480	2.372673	1.855552
	0.1	2.576117	1.469819	1.230211	4.607312	2.275053	1.906473
	0.2	2.577045	1.482036	1.233443	4.608850	2.301485	1.910487
	0.3	2.577842	1.491859	1.235318	4.610379	2.325251	1.912552
	0.4	2.578526	1.499983	1.236529	4.611842	2.346961	1.913733
	0.5	2.579102	1.506840	1.237385	4.613185	2.366982	1.914478
$4.0$	0.6	2.579577	1.512720	1.238031	4.614375	2.385564	1.914993
	0.7	2.579963	1.517824	1.238550	4.615396	2.402886	1.915372
	0.8	2.580271	1.522299	1.238981	4.616257	2.419089	1.915692
	0.9	2.580515	1.526258	1.239347	4.616971	2.434287	1.915963
	0.99	2.580689	1.529451	1.239640	4.617506	2.447187	1.916152
	0.999	2.580705	1.529754	1.239667	4.617554	2.448439	1.916209
	0.1	2.578898	1.480754	1.241764	4.609750	2.291391	1.923706
	0.2	2.579074	1.491252	1.242637	4.610008	2.315955	1.924750
	0.3	2.579230	1.499883	1.243132	4.610271	2.338502	1.925242
	0.4	2.579368	1.507121	1.243442	4.610531	2.359336	1.925480
	0.5	2.579490	1.513286	1.243655	4.610784	2.378680	1.925597
$6.0$	0.6	2.579597	1.518606	1.243812	4.611025	2.396707	1.925648
	0.7	2.579692	1.523246	1.243935	4.611251	2.413557	1.925675
	0.8	2.579776	1.527331	1.244039	4.611462	2.429349	1.925681
	0.9	2.579849	1.530956	1.244130	4.611657	2.444182	1.925686
	0.99	2.579908	1.533886	1.244200	2.579908	1.533886	1.244200
	0.999	2.579914	1.534164	1.244208	2.579914	1.534164	1.244208
	0.1	2.580240	1.485579	1.246841	4.611078	2.298611	1.931287
	0.2	2.580241	1.495501	1.246841	4.611091	2.322649	1.931287
	0.3	2.580241	1.503740	1.246841	4.611094	2.344908	1.931287
	0.4	2.580241	1.510690	1.246841	4.611097	2.365575	1.931287
	0.5	2.580242	1.516631	1.246841	4.611098	2.384818	1.931287
∞	0.6	2.580242	1.521769	1.246841	4.611099	2.402778	1.931287
	0.7	2.580242	1.526256	1.246841	4.611099	2.419579	1.931287
	0.8	2.580242	1.530208	1.246841	4.611100	2.435330	1.931287
	0.9	2.580242	1.533716	1.246841	4.611100	2.450126	1.931287
	0.99	2.580242	1.536552	1.246841	4.611101	2.462697	1.931287
	0.999	2.580242	1.536820	1.246841	4.611101	2.463917	1.931287

and

Table 2. Comparison for the normalized drag force for two solid spheres translating to the same velocity in the same direction through an unbounded couple stress fluid at ${\overline{\eta }}^{\prime }=0.01,$ in the presence and absence of permeability $\alpha$ as per [15].

$\mathit{\delta }$	${\mathit{a}}_1/{\mathit{a}}_2$	$\mathit{\alpha }=0.0=0.0$			$\mathit{\alpha }=1.0$
		$\overline{\mathit{\eta }}=\mathbf{0.05}$	$\overline{\mathit{\eta }}=\mathbf{0.1}$	$\overline{\mathit{\eta }}=\mathbf{0.15}$	$\overline{\mathit{\eta }}=\mathbf{0.05}$	$\overline{\mathit{\eta }}=\mathbf{0.1}$	$\overline{\mathit{\eta }}=\mathbf{0.15}$
	0.1	0.428236	0.428645	0.429572	2.489293	3.353901	3.345714
	0.2	0.508549	0.509332	0.511094	2.505031	3.363621	3.355504
	0.3	0.575068	0.576156	0.578593	2.500761	3.344673	3.336725
	0.4	0.629770	0.631086	0.634029	2.494275	3.325142	3.317346
$1.5$	0.5	0.674783	0.676262	0.679564	2.489391	3.310325	3.302651
	0.6	0.712008	0.713598	0.717147	2.486322	3.299905	3.292307
	0.7	0.743013	0.744676	0.748388	2.484630	3.292696	3.285196
	0.8	0.769049	0.770756	0.774572	2.483889	3.287797	3.280240
	0.9	0.791095	0.792828	0.796703	2.483752	3.284376	3.277143
	0.1	0.780131	0.780221	0.780429	2.528909	3.381715	3.373650
	0.2	0.804235	0.804391	0.804748	2.529839	3.382847	3.374779
	0.3	0.823871	0.824074	0.824538	2.530619	3.383844	3.375775
	0.4	0.840125	0.840361	0.840903	2.531277	3.384726	3.376654
$4.0$	0.5	0.853769	0.854029	0.854625	2.531825	3.385493	3.377419
	0.6	0.865363	0.865641	0.866275	2.532275	3.386144	3.378067
	0.7	0.875325	0.875614	0.876274	2.532639	3.386684	3.378607
	0.8	0.883965	0.884261	0.884939	2.532928	3.387126	3.379047
	0.9	0.891524	0.891825	0.892513	2.533157	3.387480	3.379399
	0.1	0.851932	0.851974	0.852071	2.531661	3.384312	3.376249
	0.2	0.866928	0.867001	0.867166	2.531842	3.384517	3.376454
	0.3	0.879253	0.879346	0.879560	2.531998	3.384701	3.376637
	0.4	0.889545	0.889653	0.889901	2.532132	3.384865	3.376801
$6.0$	0.5	0.898261	0.898379	0.898651	2.532248	3.385015	3.376950
	0.6	0.905730	0.905855	0.906144	2.532350	3.385150	3.377085
	0.7	0.912197	0.912328	0.912627	2.532439	3.385274	3.377209
	0.8	0.917850	0.917984	0.918290	2.532517	3.385386	3.377321
	0.9	0.922830	0.922966	0.923276	2.532586	3.385489	3.377424
	0.1	1.000000	1.000000	1.000000	2.532986	3.385621	3.377558
	0.2	1.000000	1.000000	1.000000	2.532986	3.385621	3.377558
	0.3	1.000000	1.000000	1.000000	2.532986	3.385621	3.377559
	0.4	1.000000	1.000000	1.000000	2.532986	3.385621	3.377559
∞	0.5	1.000000	1.000000	1.000000	2.532986	3.385621	3.377559
	0.6	1.000000	1.000000	1.000000	2.532986	3.385621	3.377559
	0.7	1.000000	1.000000	1.000000	2.532986	3.385621	3.377559
	0.8	1.000000	1.000000	1.000000	2.532986	3.385621	3.377559
	0.9	1.000000	1.000000	1.000000	2.532987	3.385622	3.377559

, for comparison, depict the normalized drag force for two solid spheres translating with the same velocity in the same direction through an unbounded porous medium filled with couple stress fluid at ${\overline{\eta }}^{\prime }=0.1,\alpha =1.0$. The collocation method was implemented with a sufficiently large and carefully chosen number of basis functions and collocation points to ensure that accuracy is reached, $N=40$.

Also, Figure 2 illustrates the normalized drag force on two spherical particles as a function of their size ratio ($a_1/a_2$) and the first couple stress parameter ($\overline{\eta }$), revealing that drag force increases with both the size ratio and $\overline{\eta }$, indicating stronger particle interactions and greater fluid resistance at comparable particle sizes and higher couple stress values; comparing cases (a) and (b), where (a) approaches Newtonian behavior with minimal couple stress, and (b) exhibits significant couple stress effects, highlights that the drag force’s sensitivity to size ratio is amplified by couple stresses, resulting in a steeper increase in (b) and underscoring the necessity of accounting for couple stress in non-Newtonian fluids to accurately predict particle dynamics, particularly in applications where particle interactions in complex fluids are critical.

Figure 3 presents the normalized drag force on two spherical particles as a function of their size ratio ($a_1/a_2$) and the slip parameters (${\eta }_1={\eta }_2$). The analysis was conducted under specific conditions: $\overline{\eta }=0.001$, ${\eta }^{\prime }=0.1$, $U_2/U_1=5.0$, $\delta =2.0$. The figure is divided into two cases, (a) with $\alpha =1.0$ and (b) with $\alpha =3.0$, to illustrate the influence of $\alpha$ on the relationship between drag force and slip parameters. In both cases, it is observed that the normalized drag force increases with the size ratio ($a_1/a_2$), indicating stronger particle interactions as the particles become more similar in size. However, the key finding is the significant impact of the slip parameters (${\eta }_1={\eta }_2$) on the drag force. As the slip parameters increase from 0.0 to ∞, the drag force decreases. This demonstrates that higher levels of surface slippage reduce the resistance to motion, likely due to a diminished interaction between the fluid and the particle surfaces. When comparing cases (a) and (b), it is evident that the parameter $\alpha$ amplifies the effect of the slip parameters. In one case (a), where $\alpha =1.0$, the drag force curves are relatively close, suggesting a moderate influence of slippage. In contrast, in the other case (b), with $\alpha =3.0$, a more pronounced spread among the curves is shown, indicating a stronger dependence of drag force on the slip parameters. This suggests that $\alpha$ plays a crucial role in modulating hydrodynamic interactions.

The asymptotic behavior of the curves as $a_1/a_2$ approaches 1.0 indicates that, beyond a certain size ratio, further increases have diminishing effects on the drag force. This observation is consistent across both cases. In summary, Figure 3 highlights the critical role of surface slippage in reducing drag force in fluids with couple stress properties. The parameter $\alpha$ significantly influences the sensitivity of drag force to slip parameters.

Figure 4 illustrates the normalized drag force on two spherical particles as a function of their size ratio ($a_1/a_2$) and the slip parameter ${\eta }^{\prime }$. The analysis is conducted under specific conditions: $\eta =0.1$, ${\eta }^{\prime }=0.1$, $U_2/U_1=0.5$, $\delta =1.01$, $\alpha =1.0$. The figure is divided into two cases: (a) with ${\eta }_1={\eta }_2=3.0$ and (b) with ${\eta }_1={\eta }_2\to \infty$, to investigate the influence of ${\eta }_1={\eta }_2$ on the relationship between drag force and ${\eta }^{\prime }$. In both cases, it is observed that the normalized drag force generally increases with the size ratio ($a_1/a_2$), indicating stronger particle interactions as the particles become more similar in size. However, the primary focus is on the impact of the slip parameter ${\eta }^{\prime }$ on the drag force. As ${\eta }^{\prime }$ increases from 0.0 to 0.9, the drag force also increases. This indicates that higher values of ${\eta }^{\prime }$ lead to greater resistance to motion, likely due to a reduction in surface slippage. Comparing cases (a) and (b), it is evident that the parameter ${\eta }_1={\eta }_2$ amplifies the effect of ${\eta }^{\prime }$. In one case (a), where ${\eta }_1={\eta }_2=3.0$, the drag force curves are relatively close, suggesting a moderate influence of ${\eta }^{\prime }$. In contrast, in the other case (b), with ${\eta }_1={\eta }_2\to \infty$, a more pronounced spread among the curves is shown, indicating a stronger dependence of drag force on ${\eta }^{\prime }$. This suggests that ${\eta }_1={\eta }_2$ plays a crucial role in modulating hydrodynamic interactions. Additionally, both cases show a peak in the drag force at a size ratio between 0.2 and 0.3, followed by a slight decline as $a_1/a_2$ approaches 1.0. This suggests a complex interaction between particle size and slip effects. In summary, Figure 4 highlights the critical role of the slip parameter, ${\eta }^{\prime }$, in influencing drag force in fluids with couple stress properties. The parameter ${\eta }_1={\eta }_2$ significantly influences the sensitivity of drag force to ${\eta }^{\prime }$.

Figure 5 illustrates the normalized drag force on two spherical particles as a function of their size ratio ($a_1/a_2$) and the velocity ratio ($U_2/U_1$). The analysis was conducted under specific conditions: $\overline{\eta }=0.1$, ${\eta }^{\prime }=0.1$, $\alpha =1.0$, $\delta =1.01$. The figure is divided into two cases, (a) with ${\eta }_1={\eta }_2=3.0$ and (b) with ${\eta }_1={\eta }_2\to \infty$, to investigate the influence of ${\eta }_1={\eta }_2$ on the relationship between drag force and $U_2/U_1$. In both cases, it is observed that the normalized drag force generally decreases with the size ratio ($a_1/a_2$), indicating reduced drag as the particles become more similar in size. However, the primary focus is on the impact of the velocity ratio ($U_2/U_1$) on the drag force. As $U_2/U_1$ increases from −2.0 to 2.0, the drag force decreases. This demonstrates that the relative velocities of the particles significantly influence the resistance to motion. When comparing cases (a) and (b), it is evident that the parameter ${\eta }_1={\eta }_2$ amplifies the effect of $U_2/U_1$. In one case (a), where ${\eta }_1={\eta }_2=3.0$, the drag force curves are relatively close, suggesting a moderate influence of $U_2/U_1$. In contrast, in the other case (b), with ${\eta }_1={\eta }_2\to \infty$, a more pronounced spread among the curves is shown, indicating a stronger dependence of drag force on $U_2/U_1$. This suggests that ${\eta }_1={\eta }_2$ plays a crucial role in modulating the hydrodynamic interactions. Additionally, negative velocity ratios ($U_2/U_1=-2.0$ and $-1.0$) result in higher drag forces compared to positive velocity ratios. This indicates that the direction of relative motion significantly affects the drag force. In summary, Figure 5 highlights the critical role of the velocity ratio ($U_2/U_1$) in influencing drag force in fluids with couple stress properties. The parameter ${\eta }_1={\eta }_2$ significantly influences the sensitivity of drag force to $U_2/U_1$.

Figure 6 illustrates the normalized drag force on two spherical particles as a function of their size ratio ($a_1/a_2$) and the parameter $\alpha$. The analysis was conducted under specific conditions: $\overline{\eta }=0.1$, ${\eta }^{\prime }=0.1$, ${\eta }_1={\eta }_2=1.0$. The figure is divided into two cases, (a) with $\delta =1.0$ and (b) with $\delta =1.01$, to investigate the influence of $\delta$ on the relationship between drag force and $\alpha$. In both cases, it was observed that the normalized drag force generally increases with the size ratio ($a_1/a_2$), indicating stronger particle interactions as the particles become more similar in size. However, the primary focus is on the impact of the parameter $\alpha$ on the drag force. As $\alpha$ increases from 1.0 to 2.5, the drag force also increases. This demonstrates that higher values of $\alpha$ lead to greater resistance to motion. When comparing cases (a) and (b), it is evident that the parameter $\delta$ moderates the effect of $\alpha$. In one case (a), where $\delta =1.0$, the drag force curves show a wider range of values, indicating a more significant influence of $\alpha$. In contrast, the other case (b), with $\delta =1.01$, shows a narrower range of values, indicating a less pronounced dependence of drag force on $\alpha$. This suggests that $\delta$ plays a crucial role in modulating hydrodynamic interactions. Additionally, one case (a) exhibits a negative drag force at low size ratios for $\alpha =1.0$. This indicates a complex interaction where the smaller particle might be experiencing a “pulling” force, rather than a drag force. In summary, Figure 6 highlights the critical role of the parameter $\alpha$ in influencing drag force in fluids with couple stress properties. The parameter $\delta$ significantly influences the sensitivity of drag force to $\alpha$. These findings have important implications for applications in microfluidics, biomedical engineering, and material science, where controlling particle interactions is essential.

In our revised analysis, we provide quantitative results for the drag force behavior under varying conditions. Specifically, when the parameter $\alpha$ increases from 1.0 to 2.5, the drag force increases by approximately 40%. This change can be attributed to the increased viscosity and enhanced frictional interactions as $\alpha$ increases. As $\alpha$ reflects the ratio of surface roughness to fluid viscosity, a higher value indicates more resistance to motion, which results in a significant increase in drag force. These findings align with theoretical predictions for viscous flow in the presence of surface roughness, where higher roughness or greater fluid viscosity leads to increased frictional resistance. The updated results emphasize the sensitivity of drag force to changes in $\alpha$, providing a more comprehensive understanding of the flow dynamics in our system.

7. Conclusions

This investigation clarifies the interaction between two solid spheres moving through a porous medium filled with a couple stress fluid, emphasizing the critical influences of permeability, couple stress coefficients, and surface slip coefficients on their hydrodynamic behavior. The findings indicate that increased permeability enhances fluid transport, leading to higher drag coefficients and stronger inter-particle interaction forces. Additionally, higher couple stress coefficients significantly increase the sensitivity of drag coefficients to particle size ratios, highlighting the important role of non-Newtonian fluid rheology in altering particle interactions. Conversely, elevated surface slip coefficients result in lower drag coefficients due to reduced fluid-particle interfacial resistance. These insights demonstrate the necessity of incorporating these parameters when analyzing particle dynamics in porous media, with significant implications for various applications, including microfluidic device design, biomedical transport phenomena, and advanced materials processing. Consequently, continued and rigorous research is essential to further refine our understanding of fluid-particle interactions in complex, multi-phase environments. It is important to acknowledge the limitations of the present study. First, the analysis is based on the Stokes flow assumption (low Reynolds number), which is valid for microscale or highly viscous environments but does not account for inertial effects present in moderate or high Reynolds number regimes (Happel & Brenner, 1983 [3]). Second, the use of idealized rigid spherical particles simplifies the geometry and neglects surface roughness, deformation, or non-sphericity that can influence hydrodynamic interactions (Kim & Karrila, 1991 [19]). Finally, the study focuses on steady-state conditions and does not incorporate time-dependent or transient effects, which may be significant in dynamic or evolving systems. These assumptions, while standard in theoretical microhydrodynamics, may limit the direct applicability of the results to more complex real-world scenarios.