Theoretical Analysis of Blood Rheology as a Non-Integer Order Nanofluid Flow with Shape-Dependent Nanoparticles and Thermal Effects

Abstract

This study theoretically investigates blood rheology in arteries by modeling blood as an Oldroyd-B nanofluid with uniformly suspended $Au$, $Cu$, and $A{l}_2{O}_3$ nanoparticles. A fractional order framework is employed to capture memory and hereditary effects while preserving the invariance of governing equations. The influence of nanoparticle geometry is examined by considering spherical (isotropic), cylindrical (axial), and platelet-like (planar) shapes. Using integral transform techniques, a comparative analysis highlights how particle symmetry and system parameters affect flow behavior and heat transfer. Thermal effects are further analyzed as both a contributor to flow resistance and a source of symmetry breaking in conduction, with implications for optimizing nanofluid-based blood rheology in biomedical applications such as cryosurgery.

1. Introduction

Hemodynamics refers to the study of blood rheology and its flow behavior within the circulatory system. It involves analyzing the physical and biological properties of blood, the forces governing circulation (pressure, velocity, resistance), and their interactions with vascular structures. As a multidisciplinary field, it integrates principles of physics, biology, and engineering, and has critical relevance in cardiology, vascular surgery, intensive care, and cryosurgery.

Nanofluids (NFs), introduced by Choi [1], are suspensions of nanoparticles (NPs, 1−100 nm) in a base fluid such as water, oils, or polymer solutions. Owing to their enhanced thermal and rheological properties, NFs have attracted significant attention in energy systems [2], manufacturing [3], and particularly biomedical applications [4]. The size and shape of NPs are key determinants of the thermal, rheological, and stability characteristics of NFs [5]. Smaller NPs, due to their higher surface area-to-volume ratio, enhance interaction with the base fluid, thereby improving thermal conductivity, heat transfer efficiency, and Brownian motion-induced microconvection [6]. However, particles below $10$ nm exhibit high surface energy, leading to agglomeration and sedimentation, which degrade thermal performance. At equivalent volume fractions, smaller particles also cause a lower increase in viscosity compared to larger ones, favoring fluid flow and pumping efficiency.

In terms of morphology, spherical NPs are easily dispersible, exhibit good suspension stability, moderately enhance thermal conductivity, and induce minimal viscosity rise [7,8]. Conversely, rod, tube, and platelet-like NPs possess higher aspect ratios that facilitate conductive network formation, significantly improving heat transfer, particularly anisotropic conductivity, but often at the cost of increased viscosity and reduced suspension stability if not properly stabilized [9,10]. Hence, properties such as thermal conductivity, lubrication and targeted bio-functionality are strongly influenced by NP morphology, which affects aggregation, flow dynamics, and interfacial interactions. Spherical, rod-like, platelet, and fiber-shaped NPs are commonly employed, and tailoring NP shape can optimize drug delivery, imaging, and hyperthermia treatments.

Blood is inherently a non-Newtonian fluid, exhibiting nonlinear shear stress–shear rate relations and viscoelastic properties. In large arteries, blood behaves nearly Newtonian [11], and at low shear rates in micro-vessels, it demonstrates non-Newtonian features [12]. The Casson model is often applied due to its yield stress, which realistically captures blood flow in narrow vessels [13]. Thurston [14] first reported blood’s viscoelasticity and introduced extended Maxwell-type models, later refined into three-parameter Oldroyd-B formulations [15,16]. These models account for both viscous and elastic components, reflecting the dual fluid- and solid-like behavior of red blood cells.

Research on blood flow has expanded to include effects of magnetohydrodynamics (MHD), oscillatory flows, and NP suspensions. Shah et al. [17] studied MHD flow in cylinders, while Khan, Ahmed, and Javaid [18,19,20] analyzed Newtonian and second-grade fluids in oscillating domains. More recently, metallic and non-metallic NPs dispersed in blood have been examined for improving thermal and biomedical processes. For example, Sarwar et al. [21] modeled copper-based NFs in arteries, highlighting their impact on flow patterns. Biomedical applications include hyperthermia, MRI, and nano-cryosurgery, where highly conductive, biocompatible NPs (e.g., $Au$, $A{l}_2{O}_3$, $F{e}_3{O}_4$) enhance freeze efficiency and minimize collateral damage [22,23,24]. Beyond heat transfer, NP morphology influences circulation time, clot prevention, targeted drug release and imaging, making morphology-sensitive models essential for biomedical optimization.

Boundary effects also play a crucial role in hemodynamics. Slip velocity, introduced by Navier [25], describes finite velocity differentials at the fluid–wall interface and is linked to boundary shear stress. Although widely applied in engineering fluids, its relevance to blood and complex bio-fluids has only recently gained attention [26,27,28,29,30,31,32].

Fractional calculus (FC), which generalizes differentiation and integration to non-integer orders, has emerged as a powerful framework for modeling systems with memory and hereditary effects [33,34]. FC has been successfully applied to biological systems and hemodynamics, including fractional order liver models [35] and magnetized blood flow. Awrejcewicz et al. [36] recently applied FC to study coronary and femoral artery flow under body acceleration. Moreover, Azmi et al. [37] presents the blood Cason nanofluid fractional order model for cryosurgery.

In most existing studies, blood rheology is modeled by treating blood either as a Newtonian or Casson fluid. However, these models fail to adequately capture the simultaneous viscous and elastic components that are fundamental under oscillatory arterial flow. The Oldroyd-B fluid [38] framework more accurately represents this viscoelastic nature of blood [39]. From the perspective of cryosurgical applications, fractional order models have predominantly employed Caputo or Caputo–Fabrizio operators. The Caputo operator, with its singular power-law kernel, characterizes long-term memory and hereditary behavior, while the Caputo–Fabrizio operator, defined through a non-singular exponential kernel, highlights fading memory with rapid relaxation and numerical stability. In contrast, the Atangana–Baleanu operator, with its non-local Mittag–Leffler kernel, provides a unified description of both short- and long-term memory effects, enabling more realistic modeling of anomalous diffusion and coupled transport mechanisms. Despite its advantages, the use of the Atangana–Baleanu operator in fractional order hemodynamical models remains limited. Additionally, nanoparticle-based thermal control has been explored using various materials (e.g., iron oxide, Au), but comparative analyses addressing the efficiency of different NP types and morphologies remain scarce. Since particle shape strongly affects thermal regulation, systematic investigation is essential. Motivated by these gaps, the present work develops a fractional order hemodynamical model in which blood is represented as an Oldroyd-B NF containing uniformly dispersed $Au$, $Cu$, and $A{l}_2{O}_3$ NPs of varying morphologies. The Atangana–Baleanu–Caputo (ABC) derivative is employed to capture memory effects, while integral transform methods yield analytical solutions. Comparative analyses are conducted to evaluate the influence of NP type, shape, and thermophysical parameters on blood rheology. The results are interpreted with reference to enhancing NF-assisted biomedical applications, particularly cryosurgery.

2. Research Design

Our study begins with the formulation of the physical problem, expressed through dimensional governing equations in the form of PDEs. These equations are subjected to appropriate initial and boundary conditions that describe the physical constraints of the system. Moreover, to simplify the analysis and reduce the number of parameters, the governing PDEs, along with the associated initial and boundary conditions, are converted into a dimensionless form using suitable scaling variables. Further, we have obtained the non-integer order analogous model by replacing time derivative order terms with the fractional order derivative operator. Then, to simulate the model, the dimensionless governing equations are then transformed with respect to the temporal variable using the Laplace transform. This step yields algebraic equations in the Laplace domain, along with transformed initial and boundary conditions. Subsequently, the finite Hankel transform is applied to the governing equations with respect to the radial coordinate. This procedure reduces the PDEs to a simpler form, facilitating further analytical treatment. Additionally, the Laplace inversion is performed using the Stehfest numerical algorithm, which provides efficient and accurate recovery of time-domain solutions from their Laplace-transformed counterparts. The inverse finite Hankel transform is employed to retrieve the solutions in the original spatial domain, thereby completing the transformation process. Finally, the obtained analytical solutions are represented graphically to visualize the effects of various governing parameters on the system behavior, and the findings are compared with the existing literature where applicable.

3. Description and Mathematical Modeling of the Problem

Assuming blood as an Oldroyd-B fluid, its flow is modeled through an artery represented as a cylindrical glass tube with a radius $r_0$, as shown in Figure 1. This tube contains uniformly distributed NPs of various shapes. The flow is directed along the z-axis and is influenced by a magnetic field, periodic body acceleration, and an axial pulsatile pressure gradient.

The blood and magnetic NPs are both at rest for $t=0$. The fluid with suspended particles begins to flow shortly after $t=0$, driven by an oscillating pressure gradient and convective heat transfer (see Figure 2), while the Navier–Stokes equation governs fluid motion.

The nomenclature is shown in

Table 1. Nomenclature.

Symbol	Unit	Quantity
$V_f$	${\mathrm{ms}}^{-1}$	Velocity of the fluid
$T_f$	K	Temperature of the fluid
$T_w$	K	Temperature at wall
$T_{f\infty }$	K	Temperature of atmosphere
$B_{app}$	${\mathrm{kgs}}^{-2}{A}^{-1}$	Magnetic field
$K_f$	${\mathrm{Wm}}^{-1}$${\mathrm{K}}^{-1}$	Thermal conductivity of base fluid
$k_{nf}$	${\mathrm{Wm}}^{-1}$${\mathrm{K}}^{-1}$	Thermal conductivity of Nanofluid
$K_s$	${\mathrm{Wm}}^{-1}$${\mathrm{K}}^{-1}$	Thermal conductivity of nanoparticles
$P_0$	${\mathrm{Nm}}^{-2}$	Constant amplitude
$P_1$	${\mathrm{Nm}}^{-1}$	Pulsatile component’s amplitude
${\rho }_f$	${\mathrm{kgm}}^{-3}$	Density of fluid
${\rho }_{nf}$	${\mathrm{kgm}}^{-3}$	Density of nanofluid
${\mu }_{nf}$	${\mathrm{kgm}}^{-1}{s}^{-1}$	Viscosity of nanofluid
${\mu }_f$	${\mathrm{kgm}}^{-1}{s}^{-1}$	Viscosity of fluid
${\nu }_{nf}$	${\mathrm{m}}^2{s}^{-1}$	Kinematic viscosity of the nanofluid
${\beta }_{T_f}$	${\mathrm{K}}^{-1}$	Thermal expansion coefficient of base fluid
${\beta }_{T_{nf}}$	${\mathrm{K}}^{-1}$	Thermal expansion coefficient of nanofluid
${\beta }_{T_s}$	${\mathrm{K}}^{-1}$	Thermal expansion coefficient of nanoparticles
${\lambda }_1$	s	Relaxation time parameter
${\lambda }_2$	s	Retardation time parameter
$\omega$	${\mathrm{s}}^{-1}$	Frequency of body acceleration
${\sigma }_f$	${\mathrm{m}}^2$${\mathrm{s}}^{-1}$	Electrical conductivity of base fluid
${\sigma }_{nf}$	${\mathrm{m}}^2$${\mathrm{s}}^{-1}$	Electrical conductivity of nanofluid
${\sigma }_s$	${\mathrm{m}}^2$${\mathrm{s}}^{-1}$	Electrical conductivity of nanoparticles
$\varphi$	rad	Lead angle
$A_g$	${\mathrm{ms}}^{-2}$	Amplitude of body acceleration
$\alpha$	no units	Fractional parameter
$\beta$	no units	Fractional parameter
$\gamma$	no units	Fractional parameter
$Ha$	no units	Hartman number
$P_r$	no units	Prandtl number
$\phi$	no units	Solid volume fraction of nanoparticles
m	no units	Shapes parameter of nanoparticles

.

Governing Equations

The Cauchy Stress Tensor T in incompressible Oldroyd-B fluid [38,39] is given by

$$\mathbf{T}=-p\mathbf{I}+\mathbf{S},\mathbf{S}+{\lambda }_1\left(\dot{\mathbf{S}}-\mathbf{LS}-{\mathbf{SL}}^T\right)=\mu \left[A+{\lambda }_2\left(\dot{\mathbf{A}}-\mathbf{LA}-{\mathbf{AL}}^T\right)\right].$$

(1)

where $p\mathbf{I}$ denotes the indeterminate spherical stress and $\mathbf{S}$ represents the extra stress tensor. The velocity gradient tensor is expressed as $\mathbf{L}=\nabla \mathbf{V}$, and the first Rivilin–Erickson tensor is given by $\mathbf{A}=\mathbf{L}+{\mathbf{L}}^T$. Dynamic viscosity is denoted by $\mu$, and relaxation and retardation time are represented by ${\lambda }_1$ and ${\lambda }_2$ respectively. The superscript T signifies the transpose operation, and the superposed dot indicates the material time derivative.

The model is characterized by constitutive Equation (1), which contains, as a special case, the Maxwell model when ${\lambda }_2\to 0$, and the Newtonian model when ${\lambda }_2\to 0$ and ${\lambda }_1\to 0$. For the problem under consideration, we shall assume a velocity field and extra stress tensor of the form

$$\mathbf{V}=\mathbf{V}(r,t)=V_f(r,t)e_z,\mathbf{S}=\mathbf{S}(r,t),$$

(2)

where $e_z$ is the unit vector in the z-direction of the system of cylindrical coordinates $(r,\theta ,z)$. If fluid is at rest up to the moment $t=0$, then

$$\mathbf{v}(r,0)=\mathbf{0}, \mathbf{S}(r,0)=\mathbf{0}$$

(3)

and Equations (1)–(3) imply $S_{rr}=S_{r\theta }=S_{\theta z}=S_{\theta \theta }.$

For axisymmetry flows, the governing equations correspond to a motion of Oldroyd-B fluid as

$$\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)\tau (r,t)=\mu \left(1+{\lambda }_2{\displaystyle \frac{\partial }{\partial t}}\right){\displaystyle \frac{\partial V_f(r,t)}{\partial r}}$$

(4)

$${\rho }_{nf}({\displaystyle \frac{\partial V_f(r,t)}{\partial r}})=-{\displaystyle \frac{\partial p}{\partial z}}+{\rho }_{nf}G+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r\tau (r,t)\right)+{\rho }_{nf}g{\left({\beta }_T\right)}_{nf}(T_f(r,t)-T_{f\infty }(r,t))-{\sigma }_{nf}{B}_{app}^2{V}_f(r,t)$$

(5)

where $\tau (r,t)=S_{rz}(r,t)$ is the non-trivial shear stress. Eliminate $\tau (r,t)$ from (4) and (5), and we get

$$\begin{array}{c}\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right){\displaystyle \frac{\partial V_f(r,t)}{\partial t}}=-\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right){\displaystyle \frac{1}{{\rho }_{nf}}}{\displaystyle \frac{\partial p}{\partial z}}+\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)G+\hfill \\ +{\nu }_{nf}\left(1+{\lambda }_2{\displaystyle \frac{\partial }{\partial t}}\right)\left({\displaystyle \frac{{\partial }^2{V}_f(r,t)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial V_f(r,t)}{\partial r}}\right)-\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right){\displaystyle \frac{{\sigma }_{nf}}{{\rho }_{nf}}}{B}_{app}^2{V}_f(r,t)+\hfill \\ +\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)g{\left({\beta }_T\right)}_{nf}(T_f(r,t)-T_{f\infty }(r,t)).\hfill \end{array}$$

(6)

In the above equation, G is the body acceleration, $-{\sigma }_{nf}{B}_{app}^2{V}_f(r,t)$ is the electromagnetic force, and ${\rho }_{nf}g{\left({\beta }_T\right)}_{nf}(T_f(r,t)-T_{f\infty }(r,t))$ is the body force representing the Buoyancy term due to the temperature difference.

Moreover, the energy equation is given as

$$\begin{array}{cc}\hfill {\left(\rho c_p\right)}_{nf}{\displaystyle \frac{\partial T_f(r,t)}{\partial t}}& =k_{nf}\left({\displaystyle \frac{{\partial }^2{T}_f(r,t)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_f(r,t)}{\partial r}}\right).\hfill \end{array}$$

(7)

Similarly, in [41], ${\displaystyle \frac{\partial p}{\partial z}}$ is considered as

$$\begin{array}{cc}\hfill -{\displaystyle \frac{\partial p}{\partial z}}& =P_0-P_{1}cos\left(\omega t\right),\hfill \end{array}$$

(8)

where $P_0$ represents the constant amplitude, and $P_1$ denotes the pulsatile component’s amplitude that indicates systolic or diastolic pressure.The body acceleration along the axial direction is determined by the following expression:

$$\begin{array}{cc}\hfill G& =A_{g}cos(\omega t+\varphi ),\hfill \end{array}$$

(9)

where $A_g$ represents amplitude, $\varphi$ is the lead angle regarding heart activity, and $\omega$ is the frequency of body acceleration.

The associated constraints for the problem are

$$\begin{array}{cc}\hfill V_f(r,0)& =0,T_f(r,0)=T_{f\infty },forr>0,\hfill \end{array}$$

(10)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial V_f(0,t)}{\partial r}}=0,{\displaystyle \frac{\partial T_f(0,t)}{\partial r}}=0fort>0,\end{array}$$

(11)

$$\begin{array}{c}\hfill V_f(r,t)=U,T_f(r,t)=T_{fw},forr=r_0.\end{array}$$

(12)

The NF expression for NPs were taken from [42]. The mathematical expression associated to NFs are given below:

${\rho }_{nf}=(1-\phi ){\rho }_f+\phi {\rho }_s$, ${\mu }_{nf}={\displaystyle \frac{{\mu }_f}{{(1-\phi )}^{2.5}}}$, $k_{nf}=K_f\left[{\displaystyle \frac{K_s+(m+1)K_f-(m+1)\phi (K_f-K_s)}{K_s+(m+1)K_f+\phi (K_f-K_s)}}\right]$, ${\left(\rho {\beta }_{Tf}\right)}_{nf}=(1-\phi ){\left(\rho {\beta }_T\right)}_f+\phi {\left(\rho {\beta }_T\right)}_s$, ${\left(\rho c_p\right)}_{nf}=(1-\phi ){\left(\rho c_P\right)}_f+\phi {\left(\rho c_P\right)}_s$, ${\sigma }_{nf}={\sigma }_f\left[1+{\displaystyle \frac{3(\sigma -1)\phi }{(\sigma +2)-(\sigma -1)\phi }}\right]$, $\sigma ={\displaystyle \frac{{\sigma }_s}{{\sigma }_f}}$.

Moreover, the shape factor of different types of NPs is shown in Figure 3 [43].

To make the problem under consideration geometry-free, the following dimensionless quantities are introduced:

$$\begin{array}{cc}\hfill & r^{\ast }={\displaystyle \frac{r}{r_0}},t^{\ast }={\displaystyle \frac{{\nu }_f}{r_0^2}}t,V_f^{\ast }={\displaystyle \frac{V_f}{V_0}},U^{\ast }={\displaystyle \frac{U}{V_0}},T_f^{\ast }={\displaystyle \frac{T_f-T_{f\infty }}{T_{fw}-T_{f\infty }}},P_0^{\ast }={\displaystyle \frac{r_0^2{P}_0}{{\mu }_f{V}_0}},P_1^{\ast }={\displaystyle \frac{r_0^2{P}_1}{{\mu }_f{V}_0}},\hfill \\ & {\lambda }_1^{\ast }={\displaystyle \frac{{\nu }_f{\lambda }_1}{r_0^2}},{\lambda }_2^{\ast }={\displaystyle \frac{{\nu }_f{\lambda }_2}{r_0^2}}.\hfill \end{array}$$

(13)

The dimensionless system of coupled partial DEs after dropping ‘*’ becomes

$$\begin{array}{cc}& \left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right){\displaystyle \frac{\partial V_f(r,t)}{\partial t}}=\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)X_1(P_0+P_{1}cos\left(\omega t\right))\hfill \\ \hfill & +\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)A_{g}cos(\omega t+\varphi )+\left(1+{\lambda }_2{\displaystyle \frac{\partial }{\partial t}}\right)X_2\left({\displaystyle \frac{{\partial }^2{V}_f(r,t)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial V_f(r,t)}{\partial r}}\right)\hfill \\ & -\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)X_{5}H{a}^2{V}_f(r,t)+\left(1+{\lambda }_1{\displaystyle \frac{\partial }{\partial t}}\right)X_4{G}_r{T}_f(r,t),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial T_f(r,t)}{\partial t}}& =X_8\left({\displaystyle \frac{{\partial }^2{T}_f(r,t)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_f(r,t)}{\partial r}}\right).\hfill \end{array}$$

(15)

Moreover, the initial and boundary conditions associated with the fluid flow model in dimensionless relations are

$$\begin{array}{cc}\hfill V_f(r,0)& =0,T_f(r,0)=0,rϵ[0,1],\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial V_f(0,t)}{\partial r}}=0,{\displaystyle \frac{\partial T_f(0,t)}{\partial r}}=0,t>0,\end{array}$$

(17)

$$\begin{array}{cc}\hfill V_f(r,t)& =U,T_f(r,t)=1,\mathrm{at}r=1,t>0,\hfill \end{array}$$

(18)

Into the above expressions,

$$\begin{array}{c}{X}_1=(1-\phi )+{\displaystyle \frac{\phi {\rho }_s}{{\rho }_f}},X_2={\displaystyle \frac{1}{{(1-\phi )}^{2.5}[(1-\phi )+\frac{\phi {\rho }_s}{{\rho }_f}]}},X_3=1+{\displaystyle \frac{3(\sigma -1)\phi }{(\sigma +2)-(\sigma -1)\phi }},X_4={\displaystyle \frac{(1-\phi ){\rho }_f+\phi \frac{{\left(\rho {\beta }_T\right)}_s}{{\left({\beta }_T\right)}_f}}{(1-\phi ){\rho }_f+\phi {\rho }_s}},\hfill \\ X_5={\displaystyle \frac{X_3}{X_1}},X_6={\displaystyle \frac{K_s+(m+1)K_f-(m+1)\phi (K_f-K_s)}{K_s+(m+1)K_f+\phi (K_f-K_s)}},X_7={\displaystyle \frac{1}{(1-\phi )+\frac{\phi {\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}}},X_8={\displaystyle \frac{X_6}{X_7{P}_r}},H{a}^2={\displaystyle \frac{r_0^2{\sigma }_f{B}_{app}^2}{{\mu }_f}},\hfill \\ G_r={\displaystyle \frac{{\left({\beta }_T\right)}_f{r}_0^2(T_{fw}-T_{f\infty })}{{\nu }_f{V}_0}},P_r={\displaystyle \frac{{\left(\rho c_p\right)}_f{\nu }_f}{K_f}}.\hfill \end{array}$$

4. Non-Integer Order Analog of the Problem

The non-integer order analog of Equations (14) and (15) is obtained as follows:

$$\begin{array}{cc}\hfill & (1+{\lambda }_1{D}_t^{\alpha })V_f(r,t)=(1+{\lambda }_1{D}_t^{\alpha })X_1(P_0+P_{1}cos\left(\omega t\right))+(1+{\lambda }_1{D}_t^{\alpha })A_{g}cos(\omega t+\varphi )\hfill \\ & +(1+{\lambda }_2{D}_t^{\beta })X_2\left({\displaystyle \frac{{\partial }^2{V}_f(r,t)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial V_f(r,t)}{\partial r}}\right)\hfill \\ & -(1+{\lambda }_1{D}_t^{\alpha })X_{5}H{a}^2{V}_f(r,t)+(1+{\lambda }_1{D}_t^{\alpha })X_4{G}_r{T}_f(r,t),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill D_t^{\gamma }{T}_f(r,t)& =X_8\left({\displaystyle \frac{{\partial }^2{T}_f(r,t)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T_f(r,t)}{\partial r}}\right),\hfill \end{array}$$

(20)

where we have replaced the time derivative operator in Equations (14) and (15) with the ABC non-integer order derivative operator $D_t^{\alpha }(·)$, defined as [44]

$$\begin{array}{cc}\hfill D_t^{\alpha }h(r,t)& ={\displaystyle \frac{N\left(\alpha \right)}{(1-\alpha )}}{\int }_a^{t}h\prime (r,\tau )E_{\alpha }\left[{\displaystyle \frac{-\alpha {(t-\tau )}^{\alpha }}{1-\alpha }}\right]d\tau ,\alpha \in (0,1),\hfill \end{array}$$

(21)

in which $N\left(\alpha \right)$ is a normalization function satisfying the following conditions: $N\left(1\right)=N\left(0\right)=1$; $k(\alpha ,t)$ is the kernel of the derivative operator; and $k(\alpha ,t)={\displaystyle \frac{N\left(\alpha \right)}{1-\alpha }}{E}_{\alpha }\left[{\displaystyle \frac{-\alpha t^{\alpha }}{1-\alpha }}\right]$ and $E_{\alpha }(.)$ is the one-parameter Mittag–Leffler function. Furthermore, the Laplace transform of the ABC non-integer order derivative operator is

$$\begin{array}{cc}\hfill \mathcal{L}\left(D_t^{\alpha }h(r,t)\right)& ={\displaystyle \frac{N\left(\alpha \right)q^{\alpha }\mathcal{L}h(r,t)}{(1-\alpha )q^{\alpha }+\alpha }}-{\displaystyle \frac{N\left(\alpha \right)q^{\alpha -1}h(r,0)}{(1-\alpha )q^{\alpha }+\alpha }}.\hfill \end{array}$$

(22)

5. Computational Framework

In this section, we will compute the temperature and velocity profiles by employing integral transforms.

5.1. Computation for the Temperature Profiles

The equation derived by applying Laplace Transform (LT) to Equation $\left(20\right)$ is

$$\begin{array}{cc}\hfill {\displaystyle \frac{q^{\gamma }}{(1-\gamma )q^{\gamma }+\gamma }}{\tilde{T}}_f(r,q)& =X_8\left({\displaystyle \frac{{\partial }^2{\tilde{T}}_f(r,q)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial {\tilde{T}}_f(r,q)}{\partial r}}\right),\hfill \end{array}$$

(23)

along with

$$\begin{array}{cc}\hfill {\tilde{T}}_f(1,q)& ={\displaystyle \frac{1}{q}}.\hfill \end{array}$$

(24)

Here, ${\tilde{T}}_f(r,q)$ denotes the Laplace transform of $T_f(r,t)$, and q is the Laplace transform parameter. The equation derived by applying the finite Hankel transform (FHT) [45] to Equation $\left(23\right)$ is

$$\begin{array}{cc}\hfill {\tilde{T}}_{Hf}(r_n,q)& ={\displaystyle \frac{J_1\left(r_n\right)}{r_n}}{\displaystyle \frac{1}{q}}-{\displaystyle \frac{a_{5n}{J}_1\left(r_n\right)}{r_n{q}^{\gamma }+a_{4n}}}.\hfill \end{array}$$

(25)

In the above expression, ${\tilde{T}}_{Hf}(r_n,q)$ denotes the finite Hankel transform of ${\tilde{T}}_f(r,q)$. Employing inverse LT, we get

$$\begin{array}{cc}\hfill T_{Hf}(r_n,t)& ={\displaystyle \frac{J_1\left(r_n\right)}{r_n}}-{\displaystyle \frac{a_{5n}{J}_1\left(r_n\right)}{r_n}}{F}_{\gamma }(-a_{4n},t),\hfill \end{array}$$

(26)

where $L^{-1}\left\{{\displaystyle \frac{1}{(q^a-c)}}\right\}={\sum }_{n=0}^{\infty }{\displaystyle \frac{c^n{t}^{(n+1)c}}{\mathrm{\Gamma }(n+1)a}}:$ is the Robotnov and Hartley function [46].

Taking the inverse FHT of Equation $\left(26\right)$ yields the following equation:

$$\begin{array}{cc}\hfill T_f(r,t)& =1-2\sum _{n=0}^{\infty }{\displaystyle \frac{a_{5n}{J}_0\left(r{r}_n\right)}{r_n{J}_1\left(r_n\right)}}{F}_{\gamma }(-a_{4n},t)\hfill \end{array}$$

(27)

with $a_0={\displaystyle \frac{1}{1-\gamma }}$, $a_1=a_0\gamma$, $a_2={\displaystyle \frac{a_0}{X_8}}$, $a_{3n}=a_2+r_n^2$, $a_{4n}={\displaystyle \frac{a_1{r}_n^2}{a_{3n}}}$, $a_{5n}={\displaystyle \frac{a_2}{a_2+r_n^2}}$.

5.2. Computation for the Velocity Profiles

Applying Laplace transform to Equation $\left(19\right)$, we get

$$\begin{array}{cc}\hfill & \left(q+{\lambda }_1{\displaystyle \frac{q^{\alpha +1}}{(1-\alpha )q^{\alpha }+\alpha }}\right){\tilde{V}}_f(r,q)=\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_1\left({\displaystyle \frac{P_0}{q}}+P_1{\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}\right)\hfill \\ & +\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{\omega }{q^2+{\left(\omega \right)}^2}}\right)\hfill \\ & +\left(1+{\lambda }_2{\displaystyle \frac{q^{\beta }}{(1-\beta )q^{\beta }+\beta }}\right)X_2\left({\displaystyle \frac{{\partial }^2{\tilde{V}}_f(r,q)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial {\tilde{V}}_f(r,q)}{\partial r}}\right)\hfill \\ & -\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_{5}H{a}^2{\tilde{V}}_f(r,q)+\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_4{G}_r{\tilde{T}}_f(r,q),\hfill \end{array}$$

(28)

Applying FHT

$$\begin{array}{cc}\hfill & \left(q+{\lambda }_1{\displaystyle \frac{q^{\alpha +1}}{(1-\alpha )q^{\alpha }+\alpha }}\right){\tilde{V}}_{Hf}(r_n,q)={\displaystyle \frac{J_1\left(r_n\right)}{r_n}}\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)[X_1\left({\displaystyle \frac{P_0}{q}}+{\displaystyle \frac{P_1}{q^2+{\left(\omega \right)}^2}}\right)\hfill \\ & +A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{\omega }{q^2+{\left(\omega \right)}^2}}\right)]+\left(1+{\lambda }_2{\displaystyle \frac{q^{\beta }}{(1-\beta )q^{\beta }+\beta }}\right)X_2\left[(-r_n^2){\tilde{V}}_{Hf}(r_n,q)+{\displaystyle \frac{U{J}_1\left(r_n\right)r_n}{q}}\right]\hfill \\ & -\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_{5}H{a}^2{\tilde{V}}_{Hf}(r_n,q)+\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_4{G}_r{\tilde{T}}_{Hf}(r_n,q),\hfill \end{array}$$

(29)

where ${\tilde{V}}_{Hf}(r_n,q)=H\left({\tilde{V}}_f(r,q)\right)={\int }_0^{1}r{\tilde{V}}_f(r,q)J_2\left(r{r}_n\right)dr$ is the finite Hankel transform of the function ${\tilde{V}}_f(r,q)$ and $r_n; n=1,2\dots$ are positive roots of transcendental equation $J_2\left(x\right)=0$, with $J_{\nu }$ being Bessel function of first kind of order $\nu$. Into the above expression, we use $H\left({\displaystyle \frac{{\partial }^2{\tilde{V}}_f(r,q)}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial {\tilde{V}}_f(r,q)}{\partial r}}\right)=r_n{\tilde{V}}_f(1,q)J_1\left(r_n\right)-r_n^2{\tilde{V}}_{Hf}(r_n,q)$.

$$\begin{array}{cc}\hfill & \left(q+{\lambda }_1{\displaystyle \frac{q^{\alpha +1}}{(1-\alpha )q^{\alpha }+\alpha }}\right){\tilde{V}}_{Hf}(r_n,q)+\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_{5}H{a}^2{\tilde{V}}_{Hf}(r_n,q)\hfill \\ \hfill & +\left(1+{\lambda }_2{\displaystyle \frac{q^{\beta }}{(1-\beta )q^{\beta }+\beta }}\right)X_2\left(r_n^2\right){\tilde{V}}_{Hf}(r_n,q)={\displaystyle \frac{J_1\left(r_n\right)}{r_n}}\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)[X_1\left({\displaystyle \frac{P_0}{q}}+{\displaystyle \frac{P_1}{q^2+{\left(\omega \right)}^2}}\right)\hfill \\ \hfill & +A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{\omega }{q^2+{\left(\omega \right)}^2}}\right)]+\left(1+{\lambda }_2{\displaystyle \frac{q^{\beta }}{(1-\beta )q^{\beta }+\beta }}\right)X_2\left({\displaystyle \frac{U{J}_1\left(r_n\right)r_n}{q}}\right)\hfill \\ \hfill & +\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_4{G}_r{\tilde{T}}_{Hf}(r_n,q)\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \left[\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)(q+X_{5}H{a}^2)+\left(1+{\lambda }_2{\displaystyle \frac{q^{\beta }}{(1-\beta )q^{\beta }+\beta }}\right)X_2\left(r_n^2\right)\right]{\tilde{V}}_{Hf}(r_n,q)\hfill \\ & ={\displaystyle \frac{J_1\left(r_n\right)}{r_n}}\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)\left[X_1\left({\displaystyle \frac{P_0}{q}}+{\displaystyle \frac{P_1}{q^2+{\left(\omega \right)}^2}}\right)+A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{\omega }{q^2+{\left(\omega \right)}^2}}\right)\right]\hfill \\ & +\left(1+{\lambda }_2{\displaystyle \frac{q^{\beta }}{(1-\beta )q^{\beta }+\beta }}\right)X_2\left({\displaystyle \frac{U{J}_1\left(r_n\right)r_n}{q}}\right)+\left(1+{\lambda }_1{\displaystyle \frac{q^{\alpha }}{(1-\alpha )q^{\alpha }+\alpha }}\right)X_4{G}_r{\tilde{T}}_{Hf}(r_n,q),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{((1-\beta )q^{\beta }+\beta )\left((1-\alpha )q^{\alpha }+\alpha +{\lambda }_1{q}^{\alpha }\right)(q+X_{5}H{a}^2)+X_2\left(r_n^2\right)\left((1-\alpha )q^{\alpha }+\alpha \right)\left((1-\beta )q^{\beta }+\beta +{\lambda }_2{q}^{\beta }\right)}{\left((1-\alpha )q^{\alpha }+\alpha \right)\left((1-\beta )q^{\beta }+\beta \right)}}\right){\tilde{V}}_{Hf}(r_n,q)\hfill \\ \hfill & =\left({\displaystyle \frac{((1-\beta )q^{\beta }+\beta )\left((1-\alpha )q^{\alpha }+\alpha +{\lambda }_1{q}^{\alpha }\right)}{\left((1-\alpha )q^{\alpha }+\alpha \right)\left((1-\beta )q^{\beta }+\beta \right)}}\right)[{\displaystyle \frac{J_1\left(r_n\right)}{r_n}}\left(X_1\left({\displaystyle \frac{P_0}{q}}+{\displaystyle \frac{P_1}{q^2+{\left(\omega \right)}^2}}\right)+A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{\omega }{q^2+{\left(\omega \right)}^2}}\right)\right)\hfill \\ \hfill & +X_4{G}_r{\tilde{T}}_{Hf}(r_n,q)]+\left({\displaystyle \frac{\left((1-\alpha )q^{\alpha }+\alpha \right)\left((1-\beta )q^{\beta -1}+\beta q^{-1}+{\lambda }_2{q}^{\beta -1}\right)}{\left((1-\alpha )q^{\alpha }+\alpha \right)\left((1-\beta )q^{\beta }+\beta \right)}}\right)X_2\left(U{J}_1\left(r_n\right)r_n\right)\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\tilde{V}}_{Hf}(r_n,q)=\left[{\displaystyle \frac{((1-\beta )q^{\beta }+\beta )((1-\alpha )q^{\alpha }+\alpha +{\lambda }_1{q}^{\alpha })}{((1-\beta )q^{\beta }+\beta )((1-\alpha )q^{\alpha }+\alpha +{\lambda }_1{q}^{\alpha })(q+X_{5}H{a}^2)+X_2\left(r_n^2\right)((1-\alpha )q^{\alpha }+\alpha )((1-\beta )q^{\beta }+\beta +{\lambda }_2{q}^{\beta })}}\right]\hfill \\ & \left[{\displaystyle \frac{J_1\left(r_n\right)}{r_n}}\left(X_1\left({\displaystyle \frac{P_0}{q}}+{\displaystyle \frac{P_1}{q^2+{\left(\omega \right)}^2}}\right)+A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{\omega }{q^2+{\left(\omega \right)}^2}}\right)\right)+X_4{G}_r{\tilde{T}}_{Hf}(r_n,q)\right]+\hfill \\ & \left({\displaystyle \frac{((1-\alpha )q^{\alpha }+\alpha )((1-\beta )q^{\beta -1}+\beta q^{-1}+{\lambda }_2{q}^{\beta -1})}{((1-\beta )q^{\beta }+\beta )((1-\alpha )q^{\alpha }+\alpha +{\lambda }_1{q}^{\alpha })(q+X_{5}H{a}^2)+X_2\left(r_n^2\right)((1-\alpha )q^{\alpha }+\alpha )((1-\beta )q^{\beta }+\beta +{\lambda }_2{q}^{\beta })}}\right)X_2\left(U{J}_1\left(r_n\right)r_n\right),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & {\tilde{V}}_{Hf}(r_n,q)={\displaystyle \frac{J_1\left(r_n\right)}{r_n}}[{\displaystyle \frac{b_1{q}^{\alpha +\beta }+b_2{q}^{\alpha }+b_3{q}^{\beta }+b_4}{b_1{q}^{\alpha +\beta +1}+b_2{q}^{\beta +1}+b_3{q}^{\alpha +1}+b_{4}q+b_{1n}{q}^{\alpha +\beta }+b_{2n}{q}^{\beta }+b_{3n}{q}^{\alpha }+b_{4n}}}\hfill \\ & \left(X_1\left({\displaystyle \frac{P_0}{q}}+P_1{\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}\right)+A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(2\pi \omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{2\pi \omega }{q^2+{\left(2\pi \omega \right)}^2}}\right)+X_4{G}_r\left({\displaystyle \frac{1}{q}}-{\displaystyle \frac{a_{5n}}{q^{\gamma }+a_{4n}}}\right)\right)\hfill \\ & +{\displaystyle \frac{(b_{5n}{q}^{\alpha +\beta -1}+b_{6n}{q}^{\beta -1}+b_{7n}{q}^{\alpha -1}+b_{8n}{q}^{-1})U}{b_1{q}^{\alpha +\beta +1}+b_2{q}^{\beta +1}+b_3{q}^{\alpha +1}+b_{4}q+b_{1n}{q}^{\alpha +\beta }+b_{2n}{q}^{\beta }+b_{3n}{q}^{\alpha }+b_{4n}}}],\hfill \end{array}$$

(34)

where

$b_1=((1-\alpha )+{\lambda }_1)(1-\beta )$, $b_2=(1-\beta )\alpha$, $b_3=((1-\alpha )+{\lambda }_2)\left(\beta \right)$, $b_4=\beta \alpha$, $b_{1n}=X_2{r}_n^2((1-\beta )+{\lambda }_2)(1-\alpha )+X_{5}H{a}^2(1-\beta )((1-\alpha )+{\lambda }_1)$, $b_{2n}=X_2{r}_n^2(1-\beta +{\lambda }_2)\alpha +X_5\alpha (1-\beta )H{a}^2$, $b_{3n}=X_2{r}_n^2\beta (1-\alpha )+X_5\beta (1-\alpha +\lambda )H{a}^2$, $b_{4n}=\alpha \beta (X_{5}H{a}^2+X_2{r}_n^2)$, $b_{5n}=X_2{r}_n^2(1-\beta +{\lambda }_2)(1-\alpha )$, $b_{6n}=X_2{r}_n^2(1-\beta +{\lambda }_2)\alpha$, $b_{7n}=(1-\alpha )\beta X_2{r}_n^2$, $b_{8n}=\alpha \beta X_2{r}_n^2$.

Applying inverse FHT, we get

$$\begin{array}{cc}\hfill & {\tilde{V}}_f(r,q)=2\sum _{n=0}^{\infty }{\displaystyle \frac{J_0\left(r{r}_n\right)}{r_n{J}_1\left(r_n\right)}}[{\displaystyle \frac{b_1{q}^{\alpha +\beta }+b_2{q}^{\alpha }+b_3{q}^{\beta }+b_4}{b_1{q}^{\alpha +\beta +1}+b_2{q}^{\beta +1}+b_3{q}^{\alpha +1}+b_{4}q+b_{1n}{q}^{\alpha +\beta }+b_{2n}{q}^{\beta }+b_{3n}{q}^{\alpha }+b_{4n}}}\hfill \\ & \left(X_1\left({\displaystyle \frac{P_0}{q}}+P_1{\displaystyle \frac{q}{q^2+{\left(\omega \right)}^2}}\right)+A_g\left(cos\left(\varphi \right){\displaystyle \frac{q}{q^2+{\left(2\pi \omega \right)}^2}}+sin\left(\varphi \right){\displaystyle \frac{2\pi \omega }{q^2+{\left(2\pi \omega \right)}^2}}\right)+X_4{G}_r\left({\displaystyle \frac{1}{q}}-{\displaystyle \frac{a_{5n}}{q^{\gamma }+a_{4n}}}\right)\right)\hfill \\ & +{\displaystyle \frac{(b_{5n}{q}^{\alpha +\beta -1}+b_{6n}{q}^{\beta -1}+b_{7n}{q}^{\alpha -1}+b_{8n}{q}^{-1})U}{b_1{q}^{\alpha +\beta +1}+b_2{q}^{\beta +1}+b_3{q}^{\alpha +1}+b_{4}q+b_{1n}{q}^{\alpha +\beta }+b_{2n}{q}^{\beta }+b_{3n}{q}^{\alpha }+b_{4n}}}].\hfill \end{array}$$

(35)

The required results in the t-domain can be obtained by taking the inverse LT of these expressions, which is indeed a tedious job. So, we apply Stehfest’s algorithm for Laplace inversion to these expressions. Stehfest’s algorithm for Laplace inversion is defined as [47]

$$L^{-1}\left\{{\tilde{V}}_f\left(q\right)\right\}=\psi \left(t\right)={\displaystyle \frac{ln2}{t}}\sum _{j=1}^{2l}{(-1)}^{j+l}\sum _{w=\frac{j+1}{2}}^{min(j,l)}\frac{w^l\left(2w\right)!}{(l-w)!w!(w-1)!(j-w)!(2w-j)!}{\tilde{V}}_f\left({\displaystyle \frac{ln2}{t}}\right).$$

(36)

It is worth mentioning that the main algorithmic parameter is the number l. A moderate choice (here, $l=5$) is generally recommended in the literature as it provides a good compromise between convergence and numerical stability, since larger l can amplify round-off errors due to alternating large coefficients. The convergence of the method is well established for smooth Laplace-domain functions, with errors decreasing as l increases until round-off dominates. To ensure robustness, we verified the stability of our results by comparing computations with ($l=4$) and ($l=5$), which showed negligible differences in the graphical outcomes reported.

6. Heat Transfer Rate

The dimensional expression of the rate of heat transfer is given by

$$Nu=-{\displaystyle \frac{r_0{k}_{nf}}{(T_w-T_{\infty })k_f}}{\left({\displaystyle \frac{\partial T_f(r,t)}{\partial r}}\right)}_{r=r_0}.$$

(37)

Using the dimensionless variables from Equation $\left(8\right)$, the non-dimensional form of the heat transfer rate by ignoring the * notation is given as

$$Nu=-{\displaystyle \frac{k_{nf}}{k_f}}{\left({\displaystyle \frac{\partial T_f(r,t)}{\partial r}}\right)}_{r=1}.$$

(38)

7. Result Validation

Our proposed model produces several well-known results as special cases. Specifically, by allowing the fractional order parameters to approach unity and setting ${\lambda }_1={\lambda }_2=H{a}^2=0$ in the absence of a pressure gradient and body acceleration for $Au$ NPs, our velocity profiles are in agreement with the velocity profiles obtained by Azmi et al. [23] (for ${\displaystyle \frac{1}{\beta }}=0$) (see Figure 4a). Unlike [23], which is restricted to $Au$ NPs, the present study provides a comparative analysis for multiple NPs with varying morphologies. The results of Awrejcewicz et al. [36] emerge as special cases of our general results under the parameter restrictions $\phi =0$ and $U=0$, and in the absence of thermal influences as depicted in Figure 4b. Furthermore, since most existing works on blood rheology with NFs assume Newtonian behavior, our general expressions (when the relaxation and retardation time parameters approach zero and only $Au$ NPs are utilized) are similar to the corresponding results of [24] for the case when ${\displaystyle \frac{1}{\beta }}=0$, as velocity of the two models overlap under these conditions (shown in Figure 4c). These limiting cases confirm both the validity and the broader applicability of our model.

8. Graphs and Discussion

This article presents the semi-analytical solution for the flow of blood (modeled as an Oldroyd-B fluid) mixed with various shapes of NPs in arteries. The study focuses on analyzing the impact of various parameters on the flow behavior. Different graphs are provided to illustrate these effects. The physical properties of blood and NPs are given in

Table 2. Physical properties [43].

Physical Properties	Blood	$\mathit{Au}$	$\mathit{Cu}$	Al2O3
$C_p$ (J/kgK)	3594	128.8	385	686.2
$\rho$ (kg/${\mathrm{m}}^3)$	1063	19,300	8933	4250
K (W/mK)	0.492	314.4	400	8.95
${\beta }_T$ (${\mathrm{K}}^{-1})$	0.18	1.4	1.67	0.9
$\sigma$ (s/m)	0.67	4.11	5.17	${10}^{-10}$

[43]. The fixed values chosen for the parameters are $\alpha =0.7$, $\beta =0.7$, $\gamma =0.7$, $w={\displaystyle \frac{\pi }{6}}$, amplitude $A_g=0.5$ (for ordinary walking [48]), $U=0$, lead angle $\varphi =0$ (for body acceleration and pressure oscillations that occur simultaneously [48]), and NP volume fraction $\phi =0.05$. It is important to mention that in theoretical and numerical NF studies, researchers often use volume fraction values up to 0.08 or even higher, but in physiology and biomedical applications, the situation is very different. For $Au$, $Cu$, or $A{l}_2{O}_3$ NPs in blood, volume fractions up to 0.08 are not physiologically realistic. Real biomedical applications typically work with lower fractions of orders of magnitude, mainly for toxicity and rheological reasons.

Other key parameters include the relaxation time ${\lambda }_1=0.7$, retardation time parameter ${\lambda }_2=0.5$, Prandtl number $P_r=2$, Hartman number $Ha=2$, and Grashof number $G_r=5$. Note that the choice of these parameters is adapted from [24,43,48,49]. The velocity profiles for gold NPs are depicted by black lines, while those for copper ($Cu$) NPs are shown in yellow and $A{l}_2{O}_3$ NPs are in blue. Additionally, the study explores the effects of NP shapes on the flow behavior.

(a)

Influence of shapes of NPs

Figure 5 illustrates the velocities of NPs with different shapes. It shows that the velocity of platelet-shaped NPs is higher than that of cylindrical and brick-shaped particles. Additionally, the figure highlights that gold particles exhibit higher velocities compared to copper and aluminum oxide particles. The enhancement of velocity profiles with platelet-shaped NPs in non-Newtonian fluids confined within cylindrical geometries can be attributed to their higher surface area-to-volume ratio and favorable alignment with the axial flow. The increased interfacial area facilitates stronger momentum and energy exchange with the surrounding fluid, which effectively reduces the apparent viscosity. In addition, platelet-like NPs tend to align with the streamlines, minimizing flow resistance compared to cylindrical or brick-shaped particles that generate higher form drag and disrupt the flow structure. The improved thermal conductivity and intensified Brownian/thermophoretic effects of platelet-shaped particles further thin the thermal boundary layer, reducing viscous damping and resulting in higher axial velocities. Further, gold NPs exhibit higher velocity profiles than copper or aluminum oxide counterparts due to their superior thermal conductivity, which enhances heat transport and reduces viscous damping, and their higher density, which promotes stronger momentum exchange with the base fluid. Additionally, their enhanced colloidal stability minimizes agglomeration, further facilitating efficient energy transfer and elevated fluid velocities.

(b)

Influence of radial parameter

Figure 6 depicts velocities for different NPs. It has been shown that the central artery has the highest fluid particle velocities and blood layers near the walls to travel more slowly than those along the axis. It is justified because in cylindrical flows, the no-slip boundary condition forces fluid particles adjacent to the wall to have zero velocity, while viscous drag progressively decreases toward the central axis. With minimal shear resistance at the centerline, fluid particles there attain the maximum velocity, whereas those near the walls are slowed by wall friction and strong shear stresses.

(c)

Influence of fractional parameter α and β

The effects of the non-integer order parameter $\alpha$ and $\beta$ on the velocity fields are seen in Figure 7a,b and Figure 8a,b. These curves show unique behaviors known as the memory effect that are not captured by classical derivatives and are produced at a fixed period. The fractional parameters $\alpha$ and $\beta$ behave differently at greater and smaller times. It is found that for smaller times velocity decreases for large values of $\alpha$ and $\beta$. Additionally, it is noted that the gold particles has a comparatively higher blood velocity than other NPs. The observed reversal in the influence of the fractional order parameters on velocity is a consequence of memory effects intrinsic to fractional calculus. At small times, higher fractional orders enhance relaxation and suppress velocity growth, while at large times, the reduced memory associated with higher fractional orders allows the system to relax more completely, leading to higher asymptotic velocities. It is worth mentioning that the curves that are obtained will help researchers in fitting the curve using data from their experiments.

(d)

Influence of fractional parameter $\gamma$

The effects of the non-integer order parameter $\gamma$ on the velocity fields and temperature are illustrated in Figure 9a,b and Figure 10a,b. The Figure 9a,b show that the velocity increases as the fractional parameter $\gamma$ is increased. Figure 10a,b depict temperature profiles versus r for different values of $\gamma$ at $t=0.4$ and $t=2$. It is observed that temperature profiles increase as the fractional parameter $\gamma$ is increased at $t=0.4$ and $t=2$. As the fractional order parameter $\gamma$ increases, the memory effect in the fractional heat conduction model weakens, allowing faster relaxation toward classical diffusion. This accelerates heat transport through the medium, leading to enhanced thermal diffusion and thus higher temperature and velocity profiles for larger values of $\gamma$ in the interval $(0,1)$.

(e)

Influence of amplitude $A_g$ and frequency of body acceleration ω

Figure 11a,b show velocity profiles against radial parameter for different NPs, respectively, with varying $A_g$ at $t=0.4$ and $t=2$. It is shown that raising $A_g$ increases the blood and magnetic particles to move faster. An increase in body acceleration introduces an additional inertial force term into the blood flow equations, which supplements the natural pressure gradient driving circulation. This external forcing enhances momentum transfer, reduces the relative influence of viscous resistance, and reinforces the pulsatile nature of arterial flow, thereby resulting in increased blood velocity. Figure 12a,b indicate that blood velocity decreases with increasing $\omega$. An increase in the frequency of body acceleration reduces blood velocity because the fluid cannot effectively follow rapid oscillations. At higher frequencies, phase lag, enhanced viscous dissipation, and stronger boundary layer effects dominate, so the inertial forcing is increasingly absorbed as internal resistance rather than forward flow, leading to diminished velocities.

(f)

Influence of lead angle ϕ

Figure 13a,b show velocity profiles over r for different lead angle $\varphi$ values, where blood velocities decrease as the lead angle increases. As the lead angle of the vessel increases, the blood is forced to travel along a longer and more tortuous path, which enhances wall friction, induces stronger centrifugal effects, and generates secondary vortices. These mechanisms redirect energy away from the axial direction, thereby reducing the net forward velocity of blood in the circulatory system.

(g)

Influence of Hartman number $Ha$, and Grashof Number $G_r$

Figure 14a,b show how the magnetic parameter impacts velocity profiles; increasing magnetic parameter values causes the fluid velocity to drop significantly. This is physically conceivable because the transverse magnetic field generates a Lorentz drag force, which opposes the bulk flow and reduces longitudinal velocity.

Figure 15a,b show velocity profiles vs. the radius for various values of $G_r$ at $t=0.4$ and $t=2$. It is observed that increasing the Grashof number $G_r$ results in higher velocities because larger $G_r$ implies stronger buoyancy forces relative to viscous resistance. This buoyancy-driven acceleration enhances natural convection currents, thereby increasing the fluid velocity within the system.

(h)

Influence of relaxation time parameter  ${\lambda }_1$ and retardation time parameter ${\lambda }_2$

Figure 16 and Figure 17 were prepared to investigate the influence of ${\lambda }_1$ and ${\lambda }_2$ (Oldroyd-B fluid parameters) on the velocity distribution. It is observed that as ${\lambda }_1$ increases, the velocity profile also increases (see Figure 16), while the opposite trend is observed for the case of retardation time parameter ${\lambda }_2$ (see Figure 17). Velocity increases with relaxation time because larger ${\lambda }_1$ prolongs stress memory, reducing viscous damping and allowing elastic energy to accelerate the flow. In contrast, velocity decreases with retardation time because larger ${\lambda }_2$ delays the strain–rate response, effectively enhancing viscous resistance and dissipating energy, which suppresses forward motion.

(i)

Influence of slip effect U

Figure 18 shows velocities for spherical $Au$, $Cu$, and $A{l}_2{O}_3$ NPs with different values of slip velocities. It is noted that the velocity of the fluid increases with larger slip velocity because the wall exerts less resistive drag under slip conditions. This reduction in shear resistance allows the driving pressure or body forces to accelerate the fluid more effectively, shifting the velocity profile upward and resulting in higher overall flow rates compared to the no-slip case.

(j)

Influence of NP volume fraction φ

Figure 19 shows velocity profiles for spherical $Au$, $Cu$ and $A{l}_2{O}_3$ vs. radius for various values of $\phi$ at $t=0.4$ and $t=2$. It is observed that the velocity of a NF increases with NP volume fraction because a higher concentration of NP enhances effective thermal conductivity and microconvection via Brownian motion, which reduces viscous resistance and strengthens buoyancy-driven forces. These mechanisms collectively improve momentum transport in the fluid, yielding higher velocity profiles despite the modest rise in viscosity.

The rate of heat transfer for various NPs is summarized in

Table 3. Variation in Nusselt number for various values of $\phi$.

$\mathit{\phi }$	$\mathit{Nu}$	%	$\mathit{Nu}$	%	$\mathit{Nu}$	%	$\mathit{Nu}$	%	$\mathit{Nu}$	%	$\mathit{Nu}$	%
	for  $\mathit{Au}$	Enhancement	for  $\mathit{Au}$	Enhancement	for  $\mathit{Cu}$	Enhancement	for  $\mathit{Cu}$	Enhancement	for Al2O3	Enhancement	for Al2O3	Enhancement
	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{2}$	$\mathit{t}=\mathbf{2}$	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{2}$	$\mathit{t}=\mathbf{2}$	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{2}$	$\mathit{t}=\mathbf{2}$
$0.00$	1.214	-	0.345	-	1.214	-	0.345	-	1.214	-	0.345	-
$0.01$	1.306	7.5%	0.377	9 %	1.303	7.3 %	0.376	8.7%	1.288	6%	0.372	7.7 %
$0.02$	1.398	15.1%	0.409	18.5 %	1.390	14.3 %	0.407	17.9 %	1.363	12 %	0.399	15.6 %
$0.03$	1.502	23.4 %	0.442	29.4%	1.483	23.2 %	0.444	28.3 %	1.452	19.6 %	0.430	26.4%
$0.04$	1.607	32.3%	0.486	40.8 %	1.596	31 %	0.481	39.4 %	1.542	27.1%	0.461	33.6%
$0.05$	1.725	42%	0.532	54.2 %	1.711	40.9 %	0.521	51 %	1.638	34.9%	0.497	44%

. It is observed that the highest rate of heat transfer is achieved with $Au$ NPs, followed by $Cu$ and $A{l}_2{O}_3$. When NPs are dispersed with a volume fraction of $\phi =0.05$, the heat transfer rate with $Au$ shows the greatest enhancement at $54.2\%$, followed by $Cu$ at $51\%$ and $A{l}_2{O}_3$ at $44\%$.

8.1. Pairwise Comparative Analysis of NP Performance

8.1.1. $Au$ vs. $Cu$

$Au$ NPs exhibit a heat transfer enhancement of 54.24 %, which is about 3.24% higher than that of $Cu$ NPs (51%). This indicates that $Au$-based NFs have superior thermal conductivity and stronger energy transport efficiency, likely due to gold’s higher intrinsic thermal conductivity and better interfacial compatibility with the base fluid.

8.1.2. $Au$ vs. $A{l}_2{O}_3$

Compared to $A{l}_2{O}_3$ NPs (44% enhancement), $Au$ shows a remarkable improvement of 10.24%. The difference reflects $Au$’s metallic nature and higher electron-driven heat transfer, while $A{l}_2{O}_3$, being a ceramic, mainly conducts heat through phonon mechanisms, resulting in lower enhancement.

8.1.3. $Cu$ vs. $A{l}_2{O}_3$

$Cu$ NPs outperform $Cu$ vs. $A{l}_2{O}_3$ by 7%, demonstrating better thermal coupling and fluid–particle interaction due to their metallic conductivity. However, $Cu$ performance remains below $Au$, suggesting that while $Cu$ offers a cost-effective enhancement, $Au$ achieves maximum energy transport per unit volume fraction.

The findings of this study indicate that $Au$ NPs provide the highest heat transfer rate, making them particularly effective for improving heat transfer, especially in applications like cancer treatment. Additionally, as the heat transfer time increases, the heat transfer rate also increases.

The influence of the system parameters other than rate of heat transfer on the blood rheology are summarized in

Table 4. Numerical trends of the effects of the parameters.

Parameters	${\mathit{V}}_{\mathit{f}}$	${\mathit{V}}_{\mathit{f}}$	${\mathit{T}}_{\mathit{f}}$	${\mathit{T}}_{\mathit{f}}$
(Increasing)	Velocity	Velocity	Temperature	Temperature
	of Fluid	of Fluid	of Fluid	of Fluid
	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{2}$	$\mathit{t}=\mathbf{0}.\mathbf{4}$	$\mathit{t}=\mathbf{2}$
r	decreases	decreases	-	-
$\alpha$	decreases	increases	-	-
$\beta$	decreases	increases	-	-
$A_g$	increases	increases	-	-
$\omega$	decreases	decreases	-	-
$\varphi$	decreases	decreases	-	-
$\phi$	increases	increases	-	-
${\lambda }_1$	increases	increases	-	-
${\lambda }_2$	decreases	decreases	-	-
${Ha}^2$	decreases	decreases	-	-
$G_r$	increases	increases	-	-
$\gamma$	increases	increases	increases	increases

.

9. Conclusions

This study demonstrates that platelet-shaped NPs produce superior velocity enhancements compared to cylindrical and brick-shaped geometries, highlighting the role of particle morphology in governing blood flow behavior. The memory-dependent parameter distinctly modulates short- and long-term velocity responses, while magnetic effects (Hartmann number) suppress circulation. Physiological drivers such as cardiac-induced body acceleration amplify velocity, whereas larger lead angles attenuate it. Among the Oldroyd-B parameters, relaxation time enhances velocity, while retardation time diminishes it; similarly, buoyancy forces (Grashof number) and higher nanoparticle volume fractions intensify flow dynamics.

From a thermal standpoint, gold nanoparticles deliver the highest heat transfer efficiency relative to copper and aluminum oxide, reinforcing their suitability for biomedical thermal applications, particularly cryosurgery. These findings deepen the mechanistic understanding of nanoparticle blood interactions and establish a foundation for optimizing fluid-thermal responses in diagnostic and therapeutic platforms.

Importantly, the results suggest that nanoparticle-assisted cryosurgery can accelerate ice formation, thereby enhancing tumor ablation efficiency. In parallel, the velocity augmentation induced by nanoparticle suspensions offers opportunities for improving the precision and efficacy of targeted anti-cancer drug delivery. Collectively, this work underscores the translational potential of nanoparticle-mediated bio-thermal transport processes in advancing minimally invasive treatment strategies for malignant cell eradication.

10. Limitations in the Current Study and Future Directions

This study is theoretical and subject to certain limitations. Blood is modeled as an Oldroyd-B viscoelastic fluid, which neglects yield–stress and shear–thinning effects observed under specific shear–rate regimes. NPs are assumed to be uniformly dispersed, without considering agglomeration, Brownian motion, or sedimentation. The arterial segment is simplified as a rigid cylinder, excluding curvature, wall elasticity, and branching effects.

While the present study is theoretical in nature and involves several simplifying assumptions, these limitations provide a clear roadmap for future research. The modeling of blood as an Oldroyd-B viscoelastic fluid neglects yield–stress and shear–thinning behaviors that could be explored by extending the framework to hybrid or generalized non-Newtonian models. Similarly, the assumption of uniformly dispersed NPs may be relaxed in future work by incorporating the effects of agglomeration, Brownian motion, and sedimentation to better capture realistic particle dynamics. Furthermore, the simplification of the arterial geometry as a rigid cylinder could be improved by considering wall elasticity, curvature, and branching to enhance physiological accuracy.