Magnetohydrodynamic Analysis and Fast Calculation for Fractional Maxwell Fluid with Adjusted Dynamic Viscosity

Abstract

From the perspective of magnetohydrodynamics (MHD), the heat transfer properties of Maxwell fluids under MHD conditions with modified dynamic viscosity present complex challenges in numerical simulations. In this paper, we develop a time-fractional coupled model to characterize the heat transfer and MHD flow of Maxwell fluid with consideration of the Hall effect and Joule heating effect and incorporating a modified dynamic viscosity. The fractional coupled model is numerically solved based on the $L1$-algorithm and the spectral collocation method. We introduce a novel approach that integrates advanced algorithms with a fully discrete scheme, focusing particularly on the computational cost. Leveraging this approach, we aim to significantly enhance computational efficiency while ensuring accurate representation of the underlying physics. Through comprehensive numerical experiments, we explain the thermodynamic behavior in the MHD flow process and extensively examine the impact of various critical parameters on both MHD flow and heat transfer. We establish an analytical framework for the MHD flow and heat transfer processes, further investigate the influence of magnetic fields on heat transfer processes, and elucidate the mechanical behavior of fractional Maxwell fluids.

1. Introduction

Magnetohydrodynamics (MHD) studies the phenomenon of interaction between electromagnetic fields and conductive fluids, including plasmas, liquid metals, and electrolyte solutions. Heat transfer plays a crucial role in the MHD flow process and has important significance and application value in many aspects, including nuclear fusion reactors, liquid metal cooling systems, electromagnetic casting, and welding. In the process of MHD flow, heat transfer not only directly affects the dynamic behavior and electromagnetic effects of the fluid but also plays a key role in energy conversion, interface phenomena, and practical applications. A deep understanding of the heat transfer process is of great significance for optimizing and improving the performance and efficiency of MHD systems [1,2,3]. Fractional derivatives offer a means of describing the intricate interplay between magnetic fields and fluid motion in the context of MHD flow and heat transfer, particularly when dealing with nonlinear and nonlocal effects within magnetic fluids. With the help of fractional derivatives, it is possible to gain a deeper understanding of nonlinear and nonlocal effects in magnetic fluid systems. Numerous specialists and academics have devoted substantial effort to studying the complexities associated with fractional MHD flow and heat transfer phenomena [4,5,6]. In [7], Zheng et al. analyzed the effects of various parameters on the MHD flow and heat transfer of Oldroyd-B fluid. The effects of an accelerating plate on the flow patterns and shear stress characteristics were researched by Liu et al. in [8], where the fractional calculus approach was utilized to establish a constitutive relationship for viscoelastic fluids. In [9], Tassaddiq et al. considered MHD and porous effects for generalized Casson fluids.

Due to the inherent memory and inheritability of fractional operators and the complexity of coupled models, conventional analytical methods often struggle to effectively address fractional equations. Analytical solutions frequently involve special functions that are challenging to apply in practical scenarios. Therefore, developing numerical methods capable of efficiently handling fractional coupled models is crucial. Recently, a wealth of experts and scholars have delved into comprehensive research on numerical methods aimed at solving fractional coupled models [10,11,12,13]. Liu et al. [14] introduced an effective implicit numerical method for a class of fractional advection–dispersion models. A new fractional mathematical model incorporating a nonsingular derivative operator was explored by Jajarmi et al. in [15] in the context of the clinical implications of concurrent diabetes and tuberculosis. In [16], Yavuz et al. investigated the fractional damped generalized regularized long-wave equation, demonstrating the existence and uniqueness of its solution.

Traditional numerical methods consume a great deal of time and memory when tackling fractional coupled models, which limits the applications of existing numerical methods to large-scale problems. Consequently, developing a fast method which can effectively reduce computational time and memory demands has emerged as a paramount concern in contemporary society [17,18]. Wang et al. [19] developed a rapid characteristic finite difference method for efficiently solving space-fractional transient advection–diffusion equations. In [17], Wang et al. developed a rapid yet precise solution approach for the implicit finite difference discretization for space-fractional diffusion equations in two spatial dimensions. Drawing from an effective sum-of-exponentials approximation for the kernel, Jiang et al. [20] presented an efficient method for solving the time-fractional derivative.

In this paper, we examine the magnetohydrodynamic (MHD) flow and heat transfer characteristics of fractional Maxwell fluids by developing a coupled model. We take into account the effect of temperature on fluid viscosity and subsequently analyze its implications for MHD flow and heat transfer. For the numerical calculations, we introduce an efficient method designed to significantly reduce computation time and enhance computational efficiency. Additionally, we conduct a comprehensive analysis of how key parameters influence MHD flow and heat transfer. The remainder of this paper is structured as follows. In Section 2, we establish a fractional coupled model to characterize the MHD flow of the Maxwell fluid as well as heat transfer incorporating the Hall effect and Joule heating, while considering modified dynamic viscosity. In Section 3, we construct a numerical method combining the $L1$-algorithm and the spectral collocation method to solve the model. For improved computational efficiency, we suggest a quick method that relies on a fully discrete scheme. To demonstrate the effectiveness and stability of both the numerical method and the proposed rapid approach, we provide a specific numerical example; moreover, we conduct a thorough investigation of the impact of critical parameters on MHD flow and heat transfer, backed by numerical demonstrations in Section 4. Finally, Section 5 summarizes the pertinent conclusions.

2. Mathematical Model

This paper explores the flow dynamics and thermal transfer in MHD of fractional Maxwell fluids confined between parallel plates. Figure 1 depicts the physical model. Here, the length $L_w$ of the plate significantly exceeds the distance $2h$, i.e., $L_w\gg 2h$. We establish a Cartesian coordinate system, as shown in Figure 1, with the origin located on the centerline of the channel. The channel contains the fractional Maxwell fluids, and the flow experiences the influence of a steady strong magnetic field $\mathit{B}$, disregarding the impact of the electric field and incorporating the Hall effect. In the meantime, we take the flow to be a plane and parallel (independent of x). In addition, the flow is incompressible/solenoidal, meaning that that the only surviving spatial dependence is on the y coordinate. We assume a constant temperature $T_w$ on the cavity walls along with non-slip boundary condition within the system. Typically, the viscosity of a fluid is not a static constant, instead fluctuating with the fluid’s velocity and temperature, which in turn affects the flow and temperature distributions. In this paper, we modify the dynamic viscosity for the Maxwell fluid ${\mu }_v$, expressed as ${\mu }_v={\displaystyle \frac{{\mu }_0}{1+{ϵ}_v(T-T_w)}}$, where ${ϵ}_v$ is the viscosity coefficient of the Maxwell fluid, T is the temperatures of the fluid, and ${\mu }_0$ is the standard dynamic viscosity coefficient. Usually, the dynamic viscosity of temperature ${\mu }_T$ is a temperature dependent function and can be defined as ${\mu }_T=k_T(1-{ϵ}_T(T-T_w))$, where $k_T$ is the standard thermal conductivity coefficient and ${ϵ}_T$ is the viscosity coefficient of temperature. The dynamic viscosity of temperature indicates the dynamic effect of temperature changes on fluid viscosity, and describes the changes in fluid fluidity under different temperature conditions. By adding a temperature correction term ${ϵ}_T(T-T_w)$ to reflect the sensitivity or dependence of temperature changes on viscosity, this formula describes how the dynamic viscosity of a fluid adjusts with temperature changes in a nonlinear fashion.

Given these assumptions, the velocity of the model is expressed as $\mathbf{u}=(0,0,v(y,t\left)\right)$. Consequently, the momentum equation governing the MHD flow can be expressed as [21]

$$\nabla ·\mathbf{u}=0,$$

(1)

$$\rho \left({\displaystyle \frac{\partial \mathbf{u}}{\partial t}}+(\mathbf{u}·\nabla )\mathbf{u}\right)=\nabla ·\mathbf{S}+\mathbf{J}\times \mathbf{B}+\rho \mathbf{g}{\beta }_T(T-T_w),$$

(2)

where $\rho$ stands for the density of the fluid, $\mathbf{J}$ is the current density, $\mathbf{B}$ indicates the total magnetic field, $\mathbf{S}$ means the stress tensor, $\mathbf{g}$ is the gravitational acceleration vector, and ${\beta }_T$ reveals the thermal expansion coefficient. This equation is an extension of the classical momentum conservation equation in fluid dynamics, which combines the action of electromagnetic force with the flow of fluid.The dynamic diffusion behavior of complex MHD molecules between parallel plates challenges the applicability of classical laws, leading to a mechanical evolution process characterized by notable historical dependence and long-range correlation. Fractional derivatives have gained widespread adoption in fluid models and have proven effective in describing flow behavior. Utilizing the unique characteristics of fractional Maxwell fluids, we derive a modified constitutive relation by incorporating fractional derivatives into the classical equation [22]:

$$(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })\mathbf{S}={\mu }_v{\displaystyle \frac{\partial \mathbf{u}}{\partial \mathbf{x}}}$$

(3)

where $\lambda$ is the relaxation time of the velocity, $\mathbf{x}$ is the spatial variable vector, and ${\mu }_v$ is the dynamic viscosity of the flow. The operator ${}^{C}D_t^{\alpha }$ represents the Caputo fractional derivative of order $\alpha$ with respect to t, which can be defined as [23]

$${}^{C}D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{\displaystyle \frac{f^{\prime }\left(s\right)}{{(t-s)}^{\alpha }}}ds,$$

where $\mathrm{\Gamma }(·)$ is the Gamma function. The fractional constitutive relation is used to describe the mechanical behavior of complex materials, such as non-Newtonian fluids and viscoelastic materials. Compared with classical integer order models, fractional models can more accurately describe the mechanical response of materials with memory effects and nonlocality characteristics. Because of the significant impact of a strong magnetic field on the fluid and in accordance with Ohm’s law, the current density $\mathbf{J}$ and magnetic field $\mathbf{B}$ adhere to the following relationship [24]:

$$\mathbf{J}+\xi (\mathbf{J}\times \mathbf{B})=\sigma (\mathbf{E}+\mathbf{u}\times \mathbf{B})$$

(4)

where $\xi$ is the Hall coefficient, $\mathbf{E}$ means the electric field, and $\sigma$ represents the electric conductivity. In Figure 1, we represent the uniform magnetic field as $\mathbf{B}=(0,B_0,0)$, where $B_0$ is a constant. The most critical part of the MHD flow is $\mathbf{J}\times \mathbf{B}$, known as the Lorentz force, which represents the force exerted by a magnetic field on charged fluid particles. Due to the interaction between the magnetic field $\mathbf{B}$ and the current $\mathbf{J}$, the fluid is subjected to a force in the magnetic field, which changes the motion of the fluid. By disregarding the impact of the electric field and incorporating the Hall effect, we can reformulate the governing equation of momentum into the following form:

$$\begin{array}{cc}\hfill & (1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha }){\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial }{\partial y}}\left({\mu }_v{\displaystyle \frac{\partial v}{\partial y}}\right)-{\displaystyle \frac{\sigma B_0^2}{\rho (1+B_0^2{\xi }^2)}}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })v-g{\beta }_T(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })(T-T_w).\hfill \end{array}$$

(5)

Specifically, the discrepancy in the sign of the thermal buoyancy contributions depends on the orientation of gravity.

As is well known, heat transfer is universal in the process of MHD flow, as reflected in various application fields, basic physical mechanisms, multi-physics field coupling, and both numerical simulation and experimental verification. Understanding and optimizing the heat transfer process is of great theoretical and practical significance for improving the performance and efficiency of MHD systems. Next, we study the heat transfer of fractional Maxwell fluid by formulating the following heat equation incorporating both viscous dissipation and the Joule heat effect [21,25]:

$$\rho C_p\left({\displaystyle \frac{\partial T}{\partial t}}+(\mathbf{u}·\nabla )T\right)=-\nabla ·\mathbf{q}+{\mu }_v{\left({\displaystyle \frac{\partial v}{\partial y}}\right)}^2+{\displaystyle \frac{\sigma B_0^2}{1+B_0^2{\xi }^2}}{v}^2$$

(6)

where $C_p$ represents the specific heat capacity and $\mathbf{q}$ represents the heat flux. In Equation (6), the second and third terms on the right respectively account for the viscous dissipation and Joule heat in the flow. Viscous dissipation refers to the process in which some mechanical energy (usually kinetic energy) inside a fluid is converted into internal energy (thermal energy) due to the action of viscous forces. This energy loss increases the temperature of the fluid. The Joule heat refers to the process in which electrical energy is converted into thermal energy due to the resistance effect when current passes through a conductor. This phenomenon is based on Joule’s law, and is an important manifestation of energy loss during current flows. Through application of the Taylor formula, we can obtain $\mathbf{q}=-{\mu }_T\nabla T$. Hence, the energy equation is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{1}{\rho C_p}}{\displaystyle \frac{\partial }{\partial y}}({\mu }_T{\displaystyle \frac{\partial T}{\partial y}})+{\displaystyle \frac{{\mu }_v}{\rho C_p}}({\displaystyle \frac{\partial v}{\partial y}}{)}^2+{\displaystyle \frac{\sigma B_0^2}{\rho C_p(1+B_0^2{\xi }^2)}}{v}^2.\end{array}$$

(7)

Due to the form of the velocity vector being $\mathbf{u}=(0,0,v(y,t\left)\right)$, we have $\mathbf{u}·\nabla =0$ in the three-dimensional coordinate system, resulting in the disappearance of the convective term. With the no-slip boundary condition prevailing within the channel and a constant wall temperature $T=T_w$, we can formulate the initial and boundary conditions as follows:

$$v(y,0)=v_0\left(y\right),T(y,0)=T_w,{\partial }_{t}T(y,0)=0,-h\le y\le h$$

(8)

$$v(y,t)=0,T(y,t)=T_w,y=\pm h,t>0$$

(9)

where $v_0$ is a smooth function.

For ease of analysis, the following dimensionless quantities are introduced:

$$y^{\ast }={\displaystyle \frac{y}{h}},v^{\ast }={\displaystyle \frac{\rho hv}{{\mu }_0}},v_0^{\ast }={\displaystyle \frac{\rho h{v}_0}{{\mu }_0}},t^{\ast }={\displaystyle \frac{{\mu }_{0}t}{\rho h^2}},{\lambda }^{\ast }={\displaystyle \frac{{\mu }_0\lambda }{\rho h^2}},\theta ={\displaystyle \frac{T-T_w}{T_w}},{ϵ}_v^{\ast }={ϵ}_v{T}_w,{ϵ}_T^{\ast }={ϵ}_T{T}_w$$

$$H{a}^2={\displaystyle \frac{\sigma B_0^2{h}^2}{{\mu }_0}},m=\xi B_0,Gr={\displaystyle \frac{{\rho }^{2}g{h}^3{\beta }_T{T}_w}{{\mu }_0^2}},Pr={\displaystyle \frac{C_p{\mu }_0}{k_T}},Br={\displaystyle \frac{{\mu }_0{\left(v_0^{\ast }\right)}^2}{k_T{T}_w}}$$

where $Ha$ is the Hartmann number, m is the Hall number, $Gr$ is the Grashof number, $Pr$ is the Prandtl number, and $Br$ is the Brinkmann number. Then, the dimensionless form (dropping *) of Equations (5) and (7) can be obtained as follows:

$$\begin{array}{cc}\hfill & (1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha }){\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}({\displaystyle \frac{1}{1+{ϵ}_v\theta }}{\displaystyle \frac{\partial v}{\partial y}})-{\displaystyle \frac{H{a}^2}{1+m^2}}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })v-Gr(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })\theta \hfill \end{array}$$

(10)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{1}{Pr}}{\displaystyle \frac{\partial }{\partial y}}\left((1-{ϵ}_T\theta ){\displaystyle \frac{\partial \theta }{\partial y}}\right)+{\displaystyle \frac{Br}{1+{ϵ}_v\theta }}({\displaystyle \frac{\partial v}{\partial y}}{)}^2+{\displaystyle \frac{H{a}^{2}Br}{1+m^2}}{v}^2\end{array}$$

(11)

with the following initial conditions and the boundary conditions:

$$v(y,0)=v_0\left(y\right),\theta (y,0)=0,{\partial }_t\theta (y,0)=0,-1\le y\le 1$$

(12)

$$v(y,t)=0,\theta (y,t)=0,y=\pm 1,t>0.$$

(13)

3. Numerical Method

In this section, we derive the fully discrete form for the coupled model presented in Equations (10)–(13). We employ the $L1$-algorithm for time derivative discretization and the spectral collocation method for spatial discretization. To explore a broader scope, we introduce source terms on the right-hand side of Equations (10) and (11), respectively, and consider the following time-fractional coupled model:

$$\begin{array}{cc}\hfill & (1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha }){\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}({\displaystyle \frac{1}{1+{ϵ}_v\theta }}{\displaystyle \frac{\partial v}{\partial y}})-{\displaystyle \frac{H{a}^2}{1+m^2}}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })v-Gr(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })\theta +f_1(y,t)\hfill \end{array}$$

(14)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{1}{Pr}}{\displaystyle \frac{\partial }{\partial y}}\left((1-{ϵ}_T\theta ){\displaystyle \frac{\partial \theta }{\partial y}}\right)+{\displaystyle \frac{Br}{1+{ϵ}_v\theta }}({\displaystyle \frac{\partial v}{\partial y}}{)}^2+{\displaystyle \frac{H{a}^{2}Br}{1+m^2}}{v}^2+f_2(y,t)\end{array}$$

(15)

subject to the initial condition

$$v(y,0)=v_0\left(y\right),\theta (y,0)={\theta }_0\left(y\right),{\partial }_t\theta (y,0)={\theta }_{t,0}\left(y\right),y\in \mathrm{\Omega }$$

(16)

and boundary condition

$$v(y,t)=\overline{v}(y,t),\theta (y,t)=\overline{\theta }(y,t),y\in \partial \mathrm{\Omega },t>0,$$

(17)

where $(y,t)\in \mathrm{\Omega }\times (0,\widehat{T}]$, $\mathrm{\Omega }=[-1,1]$, $v_0$, ${\theta }_0\left(y\right)$, ${\theta }_{t,0}\left(y\right)$, $\overline{v}(y,t)$, $\overline{\theta }(y,t)$ and $f_1(y,t)$, $f_2(y,t)$ are the known smooth functions.

3.1. Direct Method

To discretize time, the time interval, $[0,\overline{T}]$ can be partitioned into Q equally-sized subintervals, yielding a time step size of $\tau =\overline{T}/Q$. We define $t_n=n\tau$, $n=0,1,\cdots ,Q$, and $v^n\left(y\right)=v(y,t_n)$, ${\theta }^n\left(y\right)=\theta (y,t_n)$. Incorporating the $L1$-algorithm, we obtain the discrete representation of the Caputo fractional derivative ${}^{C}D_t^{\alpha }$ [26,27]:

$$D_{\tau }^{\alpha }f\left(t_n\right)={\displaystyle \frac{{\tau }^{-\alpha }}{\mathrm{\Gamma }(2-\alpha )}}\left[f\left(t_n\right)-{\alpha }_{n-1}f\left(t_0\right)-\sum _{s=1}^{n-1}({\alpha }_{s-1}-{\alpha }_s)f\left(t_{n-s}\right)\right]+O\left({\tau }^{2-\alpha }\right)$$

where ${\alpha }_s={(s+1)}^{1-\alpha }-s^{1-\alpha }$, $s=0,1,\cdots ,Q$. We use the backward difference formulas to discretize the first-order differential operator

$${\displaystyle \frac{\partial v}{\partial t}}{|}_{t=t_n}={\displaystyle \frac{v^n-v^{n-1}}{\tau }},{\displaystyle \frac{\partial \theta }{\partial t}}{|}_{t=t_n}={\displaystyle \frac{{\theta }^n-{\theta }^{n-1}}{\tau }}.$$

Next, we propose the spectral collocation method to discretize the spatial direction [28]. Let ${\left\{y_i\right\}}_{i=1}^N\subseteq \mathrm{\Omega }$ be the Legendre–Gauss–Lobatto points, where $\mathrm{\Omega }=[-1,1]$, $y_1=-1$, $y_N=1$. Here, ${\mathbb{P}}_N$ represents a space composed of all polynomials with the degree less than or equal to N. Let ${\left\{{\psi }_p\right\}}_{p=1}^N$ represent the Lagrange interpolation polynomials at the points ${\left\{y_p\right\}}_{p=1}^N$. Denoting $v_i^n=v_N\left(y_i\right)$, ${\theta }_i^n={\theta }_N\left(y_i\right)$, we can then express $v^n\left(y\right)$ and ${\theta }^n\left(y\right)$ as

$$v^n\left(y\right)=\sum _{i=1}^N{v}_i^n{\psi }_i\left(y\right),{\theta }^n\left(y\right)=\sum _{i=1}^N{\theta }_i^n{\psi }_i\left(y\right).$$

Subsequently, we can obtain the fully discrete formulation as follows: find $v_N^n\in {\mathbb{P}}_N$, ${\theta }_N^n\in {\mathbb{P}}_N$ such that, for $1\le n\le Q$ and $2\le i\le N-1$,

$$\begin{array}{cc}\hfill & (1+{\displaystyle \frac{\tau H{a}^2}{1+m^2}})(1+{\lambda }^{\alpha }{D}_{\tau }^{\alpha })v_N^n\left(y_i\right)-\tau {\displaystyle \frac{\partial }{\partial y}}({\displaystyle \frac{1}{1+{ϵ}_v{\theta }_N^{n-1}\left(y_i\right)}}{\displaystyle \frac{\partial v_N^n\left(y_i\right)}{\partial y}})\hfill \\ \hfill =& (1+{\lambda }^{\alpha }{D}_{\tau }^{\alpha })v_N^{n-1}\left(y_i\right)-\tau Gr(1+{\lambda }^{\alpha }{D}_{\tau }^{\alpha }){\theta }_N^{n-1}\left(y_i\right)+f_1^n\left(y_i\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & {\theta }_N^n\left(y_i\right)-{\displaystyle \frac{\tau }{Pr}}{\displaystyle \frac{\partial }{\partial y}}\left((1-{ϵ}_T{\theta }_N^{n-1}\left(y_i\right)){\displaystyle \frac{\partial {\theta }_N^n\left(y_i\right)}{\partial y}}\right)\hfill \\ \hfill =& {\theta }_N^{n-1}\left(y_i\right)+{\displaystyle \frac{\tau Br}{1+{ϵ}_v{\theta }_N^{n-1}\left(y_i\right)}}({\displaystyle \frac{\partial v_N^n\left(y_i\right)}{\partial y}}{)}^2+{\displaystyle \frac{\tau H{a}^{2}Br}{1+m^2}}{v}_N^n{\left(y_i\right)}^2+f_2^n\left(y_i\right).\hfill \end{array}$$

(19)

$$\begin{array}{c}\hfill v_N^0\left(y_i\right)=v_0\left(y_i\right),{\theta }_N^0\left(y_i\right)={\theta }_0\left(y_i\right),1\le i\le N,\end{array}$$

(20)

$$\begin{array}{c}\hfill v_N^n\left(y_i\right)=\overline{v}(y_i,t_n),{\theta }_N^n\left(y_i\right)=\overline{\theta }(y_i,t_n),i=1,N.\end{array}$$

(21)

3.2. Fast Method

To minimize computational expense, we explore the sum-of-exponentials (SOE) method [20] and introduce an efficient approach to significantly reduce computational time. Initially, we decompose the Caputo fractional derivative into two components: the local part $C_l\left(t_n\right)$, and the historical part $C_h\left(t_n\right)$. This decomposition applies to any $\alpha$ and any function $u\left(t\right)$, yielding

$$\begin{array}{cc}\hfill D_{\tau }^{\alpha }{u}^n& ={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^{t_n}{\displaystyle \frac{u^{\prime }\left(s\right)}{{(t_n-s)}^{\alpha }}}ds,\hfill \\ & ={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_{t_{n-1}}^{t_n}{\displaystyle \frac{u^{\prime }\left(s\right)}{{(t_n-s)}^{\alpha }}}ds+{\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^{t_{n-1}}{\displaystyle \frac{u^{\prime }\left(s\right)}{{(t_n-s)}^{\alpha }}}ds,\hfill \\ & =:C_l\left(t_n\right)+C_h\left(t_n\right).\hfill \end{array}$$

For the local part $C_l\left(t_n\right)$, direct computation can be achieved through the $L1$-algorithm and spectral collocation method; thus, $C_l\left(t_n\right)$ can be approximated as follows:

$$C_l\left(t_n\right)\approx {\displaystyle \frac{u^n-u^{n-1}}{\tau \mathrm{\Gamma }(1-\alpha )}}{\int }_{t_{n-1}}^{t_n}{\displaystyle \frac{1}{{(t_n-s)}^{\alpha }}}ds={\displaystyle \frac{u^n-u^{n-1}}{{\tau }^{\alpha }\mathrm{\Gamma }(2-\alpha )}}.$$

The historical component $C_h\left(t_n\right)$ can be obtained by

$$\begin{array}{cc}\hfill C_h\left(t_n\right)& ={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^{t_{n-1}}{\displaystyle \frac{u^{\prime }\left(s\right)}{{(t_n-s)}^{\alpha }}}ds\hfill \\ & ={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}\left[{\displaystyle \frac{u^{n-1}}{{\tau }^{\alpha }}}-{\displaystyle \frac{u^0}{{\tau }^{\alpha }}}-\alpha {\int }_0^{t_{n-1}}{\displaystyle \frac{u\left(s\right)}{{(t_n-s)}^{1+\alpha }}}ds\right]\hfill \\ & \approx {\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}\left[{\displaystyle \frac{u^{n-1}}{{\tau }^{\alpha }}}-{\displaystyle \frac{u^0}{{\tau }^{\alpha }}}-\alpha {\int }_0^{t_{n-1}}u\left(s\right)\sum _{i=1}^{N_e}{\overline{\omega }}_i{e}^{-s_i(t_n-s)}ds\right].\hfill \end{array}$$

In this context, the positive real numbers ${\overline{\omega }}_i$ and $s_i$ respectively denote the weights and nodes, while $N_e$ represents the number of exponentials. The definitions of ${\overline{\omega }}_i$, $s_i$, and $N_e$ can be found in [20,29].

4. Numerical Examples

In the upcoming section, we provide numerical examples to validate the efficiency and stability of both the numerical and fast methods. Additionally, we examine the impact of various parameters on the velocity and temperature fields. These numerical examples are defined within the domain $\mathrm{\Omega }=[0,1]$ and solved using the $L1$-algorithm method along with the spectral collocation method. Initially, the convergence rates in time and space, measured in the $L^2$-error sense, are defined as follows:

$$\mathrm{rate}=\left\{\begin{array}{cc}{\displaystyle \frac{log(\parallel e({\tau }_1,N)\parallel /\parallel e({\tau }_2,N)\parallel )}{log({\tau }_1/{\tau }_2)}},\hfill & \mathrm{time},\hfill \\ \\ {\displaystyle \frac{log(\parallel e(\tau ,N_1)\parallel /\parallel e(\tau ,N_2)\parallel )}{log(N_1/N_2)}},\hfill & \mathrm{space},\hfill \end{array}\right.$$

where $e({\tau }^{\ast },h^{\ast })$ is the $L^2$-error of v or $\theta$ under ${\tau }^{\ast }$ and $h^{\ast }$, ${\tau }_1\ne {\tau }_2$, $N_1\ne N_2$.

4.1. Example 1

To validate both the convergence rate of the variables along with the efficiency of the numerical and fast methods, we examine the coupled model within the domain $\mathrm{\Omega }=[-1,1]$:

$$\begin{array}{c}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha }){\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}({\displaystyle \frac{1}{1+{\mu }_v\theta }}{\displaystyle \frac{\partial v}{\partial y}})-{\displaystyle \frac{H{a}^2}{1+m^2}}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })v-Gr(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })\theta +f_1(y,t),\hfill \\ {\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{1}{Pr}}{\displaystyle \frac{\partial }{\partial y}}\left((1-{ϵ}_T\theta ){\displaystyle \frac{\partial \theta }{\partial y}}\right)+{\displaystyle \frac{Br}{1+{ϵ}_v\theta }}({\displaystyle \frac{\partial v}{\partial y}}{)}^2+{\displaystyle \frac{H{a}^{2}Br}{1+m^2}}{v}^2+f_2(y,t),\hfill \\ v(y,0)=0,\theta (y,0)=0,{\partial }_t\theta (y,0)={\displaystyle \frac{1}{2}}(1+cos\left(\pi y\right)),y\in \mathrm{\Omega },\hfill \\ v(-1,t)=v(1,t)=0,\theta (-1,t)=\theta (1,t)=0,t>0,\hfill \end{array}$$

where $t\in (0,1]$. Let the source items be

$$\begin{array}{cc}\hfill f_1(y,t)=& (y-1)(y+1)\left(2+2{\lambda }^{\alpha }{\displaystyle \frac{t^{1-\alpha }}{\gamma (2-\alpha )}}\right)+{\displaystyle \frac{H{a}^2}{1+m^2}}(y-1)(y+1)\left(t^2+2{\lambda }^{\alpha }{\displaystyle \frac{t^{2-\alpha }}{\gamma (3-\alpha )}}\right)\hfill \\ & +{\displaystyle \frac{{ϵ}_v\frac{\pi t}{2}sin\left(\pi y\right)2t^{2}y-2t^2}{{(1+{ϵ}_v\frac{t}{2}(1+cos\left(\pi y\right)))}^2}}+Gr(1+cos\left(\pi y\right))\left({\displaystyle \frac{t}{2}}+{\lambda }^{\alpha }{\displaystyle \frac{t^{1-\alpha }}{\gamma (2-\alpha )}}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill f_2(y,t)=& {\displaystyle \frac{1}{2}}(1+cos\left(\pi y\right))+{\displaystyle \frac{{\pi }^2{t}^2{ϵ}_T{sin}^2\left(\pi y\right)}{4Pr}}-{\displaystyle \frac{{\pi }^{2}tcos\left(\pi y\right)}{2Pr}}(1-{\displaystyle \frac{{ϵ}_{T}t}{2}}(1+cos\left(\pi y\right)))\hfill \\ & -{\displaystyle \frac{Br{\left(2t^{2}y\right)}^2}{1+\frac{{ϵ}_{v}t}{2}(1+cos\left(\pi y\right))}}-{\displaystyle \frac{BrH{a}^2}{1+m^2}}{t}^4{(y-1)}^2{(y+1)}^2;\hfill \end{array}$$

with these assumptions in place, the exact solutions for the model are

$$v(y,t)=t^2(y^2-1),\theta (y,t)={\displaystyle \frac{t}{2}}(1+cos\left(\pi y\right)).$$

We first illustrate the values of these related parameters. The typical values of these parameters are listed as follows: $Ha=1$, $m=1$, $\lambda =0.4$, $Pr=0.2$, $Br=1$, ${ϵ}_v=1$, ${ϵ}_T=0.8$.

First, we fix the polynomial order in each direction at 16, that is, $N=16$, to showcase the efficacy of the numerical method in temporal discretization. With this setup, the spatial discretization error becomes negligible compared to the temporal error. To accurately depict the convergence rate of this method,

Table 1. Errors and convergence orders of $\tau$ at $\overline{T}=1$.

	$\mathit{\tau }$	$\mathit{\alpha }=0.2$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.8$
		Error	Rate	Error	Rate	Error	Rate
v	1/32	$1.4956\times {10}^{-3}$	-	$1.4014\times {10}^{-3}$	-	$1.3277\times {10}^{-3}$	-
	1/64	$7.6617\times {10}^{-4}$	0.9650	$7.2277\times {10}^{-4}$	0.9553	$6.8276\times {10}^{-4}$	0.9595
	1/128	$3.8784\times {10}^{-4}$	0.9822	$3.6774\times {10}^{-4}$	0.9748	$3.4684\times {10}^{-4}$	0.9771
	1/256	$1.9517\times {10}^{-4}$	0.9907	$1.8579\times {10}^{-4}$	0.9850	$1.7518\times {10}^{-4}$	0.9854
	1/512	$9.7915\times {10}^{-5}$	0.9951	$9.3492\times {10}^{-5}$	0.9907	$8.8220\times {10}^{-5}$	0.9897
$\theta$	1/32	$9.8802\times {10}^{-4}$	-	$9.6060\times {10}^{-4}$	-	$9.1978\times {10}^{-4}$	-
	1/64	$4.9348\times {10}^{-4}$	1.0015	$4.7734\times {10}^{-4}$	1.0089	$4.5806\times {10}^{-4}$	1.0057
	1/128	$2.4695\times {10}^{-4}$	0.9988	$2.3814\times {10}^{-4}$	1.0032	$2.2862\times {10}^{-4}$	1.0026
	1/256	$1.2356\times {10}^{-4}$	0.9989	$1.1892\times {10}^{-4}$	1.0018	$1.1414\times {10}^{-4}$	1.0021
	1/512	$6.1807\times {10}^{-5}$	0.9994	$5.9407\times {10}^{-5}$	1.0013	$5.6989\times {10}^{-5}$	1.0021

presents the $L^2$-error and convergence rate for both velocity v and temperature $\theta$ at $\alpha =0.2,0.5,0.8$. The numerical procedure described in the preceding section demonstrates first-order convergence for both velocity and temperature across all values of $\alpha$, affirming its accuracy. Based on these data, it is evident that this numerical method exhibits first-order convergence accuracy and spectral accuracy for both velocity v and temperature $\theta$ in the temporal and spatial direction respectively, thereby fulfilling the required accuracy criteria.

To showcase the efficiency of the rapid approach, we set $N=32$ to solve the coupled model using both the direct and the fast methods. Figure 2 depicts the computational times associated with each approach when $\alpha =0.7$. The computational time for the fast method is notably shorter compared to the direct method. This underscores the significant reduction in computational time achieved by our proposed fast method. Additionally, Figure 3 presents the exact and numerical solutions for velocity v (first row) and temperature $\theta$ (second row) obtained via both the direct and the fast methods with $\tau =1/1000$ and $\alpha =0.3,0.6,0.9$. It is evident that the numerical solutions from both methods closely match the exact solutions. These findings underscore the stability and effectiveness of the fast method in resolving the time-fractional coupled model. Considering both error accuracy and computational times, we opt for a time step of $\tau =1/1000$ in our subsequent numerical experiments.

4.2. Example 2

In order to better study the influence of temperature on MHD flow and heat transfer and explain the thermodynamic behavior in magnetic fluid flow processes, we examine the following fractional coupled model to visually depict how various parameters affect the velocity and temperature fields:

$$\begin{array}{c}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha }){\displaystyle \frac{\partial v}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}({\displaystyle \frac{1}{1+{\mu }_v\theta }}{\displaystyle \frac{\partial v}{\partial y}})-{\displaystyle \frac{H{a}^2}{1+m^2}}(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })v-Gr(1+{\lambda }^{\alpha } {}^{C}D_t^{\alpha })\theta ,\hfill \\ {\displaystyle \frac{\partial \theta }{\partial t}}={\displaystyle \frac{1}{Pr}}{\displaystyle \frac{\partial }{\partial y}}\left((1-{ϵ}_T\theta ){\displaystyle \frac{\partial \theta }{\partial y}}\right)+{\displaystyle \frac{Br}{1+{ϵ}_v\theta }}({\displaystyle \frac{\partial v}{\partial y}}{)}^2+{\displaystyle \frac{H{a}^{2}Br}{1+m^2}}{v}^2,\hfill \\ v(y,0)=e^{-y^2}(1-y^2)\theta (y,0)=0,{\partial }_t\theta (y,0)=0,y\in [-1,1],\hfill \\ v(y,t)=0,\theta (y,t)=0,y=\pm 1,t>0,\hfill \end{array}$$

where $(y,t)\in [-1,1]\times (0,1]$. We set the parameters as follows: $\lambda =0.5$, $\alpha =0.3$, $Pr=0.8$, $Br=1.2$, $Gr=1.2$, $Ha=0.7$, $m=1$, ${ϵ}_v=0.9$, ${ϵ}_T=0.2$. The detailed velocity and temperature distributions are depicted in Figure 4, Figure 5 and Figure 6.

In this paper, we investigate the influence of modified dynamic viscosity on the velocity v and temperature $\theta$ distributions. By considering the effects of dynamic viscosity on flow and temperature, we can more accurately and comprehensively capture the impact of temperature changes on MHD flow. Figure 4 and Figure 5 illustrate the impact of dynamic viscosity of flow ${ϵ}_v$ and dynamic viscosity of temperature ${ϵ}_T$ on the velocity v and temperature $\theta$ distributions, respectively. From examining Figure 4, it is evident that the velocity and temperature gradually rise with increasing ${ϵ}_v$, albeit at a decreasing change rate. Additionally, we analyzed the velocity and temperature distributions under a constant dynamic viscosity, i.e., ${ϵ}_v=0$; under this condition, both the velocity and temperature are lower compared to scenarios where ${ϵ}_v\ne 0$. Figure 5 depicts the impact of ${ϵ}_T$ on the distributions of velocity v and temperature $\theta$. Notably, as ${ϵ}_T$ increases, the velocity gradually declines, whereas the temperature exhibits an upward trend; simultaneously, the rate of change in both velocity and temperature progressively accelerates with the increasing value of ${ϵ}_T$. This observation suggests that alterations in temperature can exert significant influence on fluid dynamics and heat transfer by modulating the dynamic viscosity, which governs the internal friction of the fluid. Additionally, these temperature-induced variations not only amplify their impact on temperature distribution but also contribute to intricate changes in fluid flow patterns, such as boundary layer behavior and turbulence intensity, leading to complex interactions between thermal gradients and fluid movement.

The influence of the Hall effect on the MHD flow and thermal energy transfer of the Maxwell fluid between parallel plates is illustrated in Figure 6. It is important to note that we kept all of the other parameters constant during our study. In Figure 6a, the velocity distribution v is plotted against increasing Hall numbers $(m=0,1,2,5)$. It is evident from the graph that the peak velocity rises progressively with increasing m at a constant Hartmann number $Ha$, while the rate of change decreases. This phenomenon occurs because the Hall number m alters the axial velocity magnitude by diminishing the effective conductivity $(1/(1+m^2))$ of the Maxwell fluid. Consequently, the Lorentz force in the momentum equation and the Joule heat term in the energy equation gradually diminish, leading to a reduction in the fluid’s damping effect and an acceleration in the flow of the Maxwell fluid. Figure 6b presents the temperature distribution $\theta$ with increasing Hall number m. From observing the graph, it is apparent that the amplitude of temperature variation gradually diminishes as m escalates, while the peak temperature progressively rises. This observation indicates that the Hall number, which is a dimensionless parameter characterizing the relative significance of the Hall effect in magnetohydrodynamics (MHD), influences the rate of temperature fluctuation by modifying the behavior of charged particles in a conducting fluid under the influence of a magnetic field. The Hall effect alters the current distribution and generates secondary electric fields, which in turn affect the fluid’s motion and energy dissipation. This leads to changes in heat conduction and convection processes. Referring to Equation (11), it is evident that the Joule heat effect diminishes with the increase in the Hall number, leading to a gradual reduction in the kinetic energy loss attributed to the Joule heat effect. Concurrently, the Hall effect enhances fluid flow, expedites temperature alterations, and accelerates energy transfer. This accelerated fluid motion not only increases the rate of energy transfer within the fluid but also impacts the thermal boundary layer, leading to a more uniform temperature distribution and quicker response to thermal gradients.

5. Conclusions

This study has addressed numerical computation and rapid methods for the MHD flow and thermal exchange of a Maxwell fluid with the modified constitutive relation and studied the thermodynamic behavior of fractional Maxwell fluids through numerical simulation results. Through the development of a fractional coupled model incorporating the Hall effect and the Joule heating and considering modified dynamic viscosity, we provide a comprehensive framework for characterizing the complex dynamics of Maxwell fluid in MHD scenarios. By employing a numerical method combining the $L1$-algorithm and spectral collocation method along with the introduction of a fast method based on the fully discrete scheme, we have demonstrated significant advancements in computational efficiency without compromising accuracy. Through a numerical example, we validate both the numerical approach and the rapid technique in terms of their practical applicability, efficacy, and stability. The numerical experiment shows that the convergence accuracy of our proposed numerical scheme is $O(\tau +N^{-r})$. Meanwhile, the fast method can effectively reduce computational time and cost without introducing additional errors. Our investigation into the effects of significant parameters on MHD flow and the thermal exchange mechanism provides valuable insights into the underlying physical phenomena. It can be seen that both velocity and temperature gradually rise with increasing ${ϵ}_v$. However, as ${ϵ}_T$ increases, the velocity gradually declines, whereas the temperature exhibits an upward trend. In addition, the Hall number reduces the damping effect of the fluid, accelerates the flow of Maxwell fluid, influences the rate of temperature change, and intensifies temperature variations. This research contributes to the advancement of understanding and modeling of Maxwell fluids in MHD systems, paving the way for further developments in this interdisciplinary field.