MHD Nanofluid Flow Past a Vertical Plate Embedded in a Rotating Porous Medium with Chemical Reaction

In this paper, the problem of an unsteady free convection of MHD nanofluid flow past a semi-infinite vertical rotating plate embedded in a porous medium subject to a first-order chemical reaction has been studied numerically. The fluid flow has been mathematically modelled under the assumption that the viscosity of the fluid is temperature dependent. The governing dimensional partial differential equations were transformed into their equivalence non-dimensional form using a suitable variable. The resulting nonlinear partial differential equations were solved numerically using the bivariate interpolated spectral relaxation method (BI-SRM). The impacts of our physical parameters on the velocity components, temperature, concentration, and axial components of the induced magnetic field, as well as skin friction, are investigated using graphical and tabular forms. This establishes that as the rotation parameter increases, the secondary velocity field is rising, however when the Reynolds magnetic number increases, the opposite pattern is demonstrated.

• Temperature-Dependent Viscosity
• Variable Magnetic Field
• Rotating Frame
• Permeability
• Bivariate Spectral Relaxation Method

1. Introduction

Nanofluids are a relatively new technology consisting of nanoparticles mixed into a base fluid. Particles used in such a type of fluid are so small that each one is between 1 and 100 nanometres in size, which is made up of metal oxides, carbon nanotubes or other materials, while the base fluid is mainly a liquid such as water, oil or ethanol. This mixture forms a fluid with properties that are much more efficient than those of traditional fluids [1] . It was Choi [2] who first used the expression “nanofluid”. Nanofluids are found to have a higher thermal conductivity compared to the pure fluid [3] . In particular, the higher thermal conductivity allows nanofluids to be used as efficient coolants in electronics and other heat-generating devices. Nanofluids can also be engineered to have specific electrical conductivity and energy storage properties. In the medical field, nanofluids are also being used as drug delivery systems that can target specific areas of the body, and this type of delivery system can significantly improve the effectiveness of treatments and reduce the side effects of drugs. Earliest studies in the field of nanofluid by [4] - [10] .

Many researchers investigate the field of nanofluid, among them Salem [11] who studied the effects of variable viscosity, viscous dissipation and chemical reaction on heat and mass transfer. Hammed and Pop [12] studied the unsteady free convection flow of a nanofluid past a permeable semi-infinite plate in the presence of an applied magnetic field, theoretically. They have succeeded in approximating an analytical solution to the boundary layer equations by means of the perturbation method. Loganathan and Sangeetha [13] used the implicit finite difference method to analyse the augmentation of thermal conductivity and mass transfer rate across a semi-infinite vertical plate due to nanoparticles suspended in Williamson fluid. Ullah et al. [14] numerically studied the two-dimensional Oldroyd-B flow of a nanofluid past a stretching sheet using the finite element method, under the effects of magnetic, electrical and thermal radiation. In an important study, Krishna and Chamkha [15] used the perturbation approximation approach to study the unsteady free convective flow of nanofluids at room temperature over a moving vertical plate. The plate is immersed in a homogeneous porous medium under thermal buoyancy with a constant heat source and convective boundary conditions. Furthermore, Krishna and Chamkha [16] studied unsteady MHD convective rotating flow of a nanofluid past a permeable moving vertical plate with a constant heat source using the perturbation method. Riasat et al. [17] used nanofluid with multi-walled carbon nanotubes between two rotating discs in a Darcy-Forchheimer permeable medium subject to an induced magnetic field to improve thermal conductivity in squeezing 3D magnetohydrodynamic thin film flow. Maneengam et al. [18] carried out numerical study of the flow of a hybrid nanofluid within a corrugated porous cavity in the presence of a magnetic field using the generalised finite element technique. Abumandour et al. [19] investigated the effects of slip, temperature conditions, and porosity on nanoparticle fluid suspension as well as the thermal nanofluid behavior as it flowed within a stenotic artery. Utilizing a catheter and vertical orientation, they explored the role of magnetic fields both with and without their presence. Using the homotopy analysis method, Ur Rasheed et al. [20] , successfully evaluated the mixed convective MHD Jeffrey nanofluid flow. The study was conducted in the presence of a vertical stretchable cylinder with heat and radiation absorption effects. Hossain et al. [21] used the finite volume method to study the bio-nanofluid flow regimes in a real bifurcated artery. The study showed that the influence of a magnetic source induced gradual changes in the pressure and vorticity profiles. Several investigations on nanofluid properties and MHD flow are thoroughly detailed in the literature [22] - [29] .

The coupling of heat and mass transfer through porous media has a wide range of applications in thermal engineering, physics, mechanical engineering and chemistry, such as the design and optimization of heating, cooling, ventilation and air conditioning systems in various buildings and industries. The rate of heat and mass transfer through porous materials is influenced by the physical properties of these materials, including thermal conductivity, density, volume, temperature, humidity, and others. Porous media refers to materials that possess interconnected pores, channels, or voids. Porous materials are characterized by their porosity, which is a measure of the interconnected voids in the material. The intricate dynamics of nanofluid flow in a rotating porous media is a challenging subject to comprehend. It often requires the application of advanced mathematical models and simulations to accurately predict the transport phenomena. Moreover, the interplay between the nanoparticles and the pore surfaces plays a crucial role in determining the flow behaviour of nanofluids in porous media. The use of porous media is very encouraging, as it provides new insights into optimising the flow of nanofluids, thereby improving the efficiency of heat transfer and facilitating the sustainable use of energy. Reddy et al. [30] studied an unsteady natural convective flow of a nanofluid through a porous medium in a rotating system with convective and diffusive boundary conditions. They have solved the governing equations analytically using the perturbation approach. Kumar & Sood [31] investigated the three-dimensional MHD nanofluid flow over a vertical surface embedded in a porous rotating system. The principles of homogeneous-heterogeneous chemical reaction are introduced simultaneously. The governing equations are converted to dimensionless form and solved explicitly using a finite difference approach. Reddy et al. [32] demonstrated MHD flow of Cu-water and Ag-water nanofluids with heat and mass transfer in the presence of chemical reaction, thermal radiation and partial slip through a porous medium. They used similarity variables to convert the governing equations into ordinary differential equations, which were then solved numerically using the finite element method. Rashid et al. [33] investigated the MHD hybrid nanofluid flow between two parallel spinning discs in present of joule heating and chemical reactions. They utilised the Matlab programme “bvp4c” to numerically solve the governing equations. Recently, Abd-Alla et al. [34] introduced heat and mass transfer analysis on MHD Jeffery fluid flow through a porous medium utilising a rotating frame with a chemical reaction. Recently, nanofluids flow through rotating porous media with or without chemical reaction investigated widely by [35] - [41] .

Magnetohydrodynamics convective flows with simultaneous heat and mass transfer influenced by chemical reactions occur in many natural and artificial transport systems for instance in scientific and technological areas. Such applications of this phenomenon can be found in chemical vapour deposition on surfaces, nuclear reactor cooling, the petroleum industries, power and cooling sectors. In addition, convective flows occur naturally in temperature variations, concentration differences or a mixture of the two; for example, in air currents, concentration differences in the water where flow is affected by these concentration differences. Elsewhere this type of flow, which takes place through a porous plate, is of crucial importance in various scientific and mechanical applications. This prompted several scholars [42] - [48] to do extensive study on heat and mass transport of MHD free convective flow of fluid past a rotating plate with chemical reaction under different conditions. Prabhakar Reddy & Makinde [49] investigated a heat and mass transfer on MHD free convection flow under the influence of a chemical reaction. Also, Krishna et al. [50] studied numerically the problem of unsteady MHD natural convective rotating flow of an incompressible electrically conducting viscous fluid past an infinite vertical moving porous surface using cubic Bsplines collocation method. Very recently, Ayub et al. [51] investigated the problem of free convection on rotational flow of viscous fluid over an extended plate with heat and mass transfer in the presence of a constant-magnitude magnetic field through a porous medium.

Due to its importance in heat and mass transfer in some types of fluid flow, variable viscosity has been widely studied, recently. In a study by Ali et al. [52] , temperature-dependent viscosity, convective boundary conditions, thermo-diffusion, and radiation effects were all taken into consideration to analysed the problem of unsteady MHD nanofluid flow across a radially nonlinear stretching sheet. They used the variational finite element method to solve the governing equations. A numerical study was carried out by Khan et al. [53] to analysed the impact of the variable viscosity on the flow of a Casson nanofluid over non-linear stretching sheet with slip and convective boundary conditions. With the motivation from all of the above efforts on a fact, specialised literature investigation tolerates that the effects of temperature-dependent viscosity, an induced magnetic field, nanofluid fluid, natural convection flow with heat, and mass transfer in the presence of a first-order chemical reaction through a semi-infinite vertical rotating plate embedded in a porous medium have not yet been discussed. Therefore, the current study claims to extend the Prabhaker Reddy and Makinde [49] for it is limitation of considering unsteady MHD free convective flow past through vertical plate in presence of chemical reaction such that the current study offered advancements of adding more specific consideration of variable viscosity, induced magnetic field with viscous dissipation and Joule heating, rotating plate, porous media saturated in fluidity with Copper(Cu) nanoparticles. Then to obtain the problem solution the novel BI-SRM numerical approach [54] [55] was chosen because it convincingly demonstrated the best in terms of accuracy and convergence speed, and computational cost is lacking as a direct comparison with other established numerical methods (e.g., finite element, finite difference).

2. Mathematical Formulation

Let us consider an unsteady MHD boundary layer flow of an incompressible nanofluid driven by a semi-infinite vertical rotating plate embedded in a porous medium subject to chemical reaction. A coordinate frame is selected in which x-axis runs vertically along the plate, the y-axis is normal to the plate and directed into the fluid zone, in which the z-axis is taken to be normal to the xy-plane. Assumed that the nanofluid is formed with non-magnetic nanoparticles and base liquid with the velocity $\left(u\mathrm{,0,}w\right)$. Let ${\mu }_e\left(H_x\mathrm{,}H\mathrm{,}{H}_z\right)$ be the induced magnetic vector at the given point in the fluid. A homogeneous transverse magnetic field of intensity H is applied parallel to the y-axis and perpendicular to the plate. Together, the fluid and plate rotate about y-aixs with a constant angular velocity Ω in the anticlockwise direction. Initially ( $t=0$), both the fluid and plate are kept at a constant temperature $T_w$ where $T_w>T_{\infty }$ is for a heated plate and $T_w<T_{\infty }$ corresponding to a cooled plate. At a certain time $0<t<t_c$, the temperature of the plate is growing up to $T_{\infty }+\left(T_w-T_{\infty }\right)\frac{t}{t_0}$ here $t_0$ is the characteristic time. Moreover, the distribution of species concentration over the plate is boosted to $C_{\infty }+\left(C_w-C_{\infty }\right)\frac{t}{t_0}$. It is assumed that a homogeneous chemical reaction of first order with uniform rate $K_r$ between the diffusing species and the ambient fluid exists. Figure 1 shows the geometry of the flow problem. The boundary layer equations controlling the flow include the continuity, motion, energy, concentration, and induction under the using Boussinesq’s standard approximation respectively:

Figure 1

Figure 1. Physical model.

$\frac{\partial u}{\partial x}+\frac{\partial w}{\partial z}=\mathrm{0,}$(1)

$\begin{array}{c}{\rho }_{nf}\left(\frac{\partial u}{\partial t}+2\Omega w\right)=\frac{{\mu }_{nf}}{k}u+\frac{\partial }{\partial y}\left({\mu }_{nf}\frac{\partial u}{\partial y}\right)+{\sigma }_{nf}{\mu }_e^2\left(\left(w{H}_x-u{H}_z\right)H_z-u{H}^2\right)\\ +g{\left(\rho \beta \right)}_{nf}\left(T-T_{\infty }\right)+g{\left(\rho {\beta }^{*}\right)}_{nf}\left(C-C_{\infty }\right),\end{array}$(2)

${\rho }_{nf}\left(\frac{\partial w}{\partial t}-2\Omega u\right)=\frac{{\mu }_{nf}}{k}w+\frac{\partial }{\partial y}\left({\mu }_{nf}\frac{\partial w}{\partial y}\right)+{\sigma }_{nf}{\mu }_e^2\left(\left(u{H}_z-w{H}_x\right)H_x-w{H}^2\right)\mathrm{,}$(3)

$\begin{array}{c}\frac{\partial T}{\partial t}={\alpha }_{nf}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\mu }_{nf}}{{\left(\rho C_p\right)}_{nf}}{\left(\frac{\partial u}{\partial y}\right)}^2+\frac{{\mu }_{nf}}{{\left(\rho C_p\right)}_{nf}}{\left(\frac{\partial w}{\partial y}\right)}^2\\ +\frac{{\sigma }_{nf}{\mu }_e^2}{{\left(\rho C_p\right)}_{nf}}\left[H^2\left(w^2+u^2\right)+{\left(w{H}_x-u{H}_z\right)}^2\right]\mathrm{,}\end{array}$(4)

$\frac{\partial C}{\partial t}=D_m\frac{{\partial }^{2}C}{\partial y^2}-k_r\left(C-C_{\infty }\right)\mathrm{,}$(5)

$\frac{\partial H_x}{\partial t}=H\frac{\partial u}{\partial y}+\frac{1}{{\sigma }_{nf}{\mu }_e}\frac{{\partial }^2{H}_x}{\partial y^2}\mathrm{,}$(6)

$\frac{\partial H_z}{\partial t}=-H\frac{\partial w}{\partial y}+\frac{1}{{\sigma }_{nf}{\mu }_e}\frac{{\partial }^2{H}_z}{\partial y^2}\mathrm{,}$(7)

where u and w represents the velocity components in the x and z directions, respectively, t is the time, $T_w$ and $C_w$ are the wall-dimensional temperatures and concentrations, respectively, $T_{\infty }$ and $C_{\infty }$, respectively are the free stream’s starting temperature and concentration, ${\rho }_{nf}$ is the nanofluid density, ${\mu }_{nf}$ is the nanofluid dynamic viscosity, ${\sigma }_{nf}$ is the nanofluid electrical conductivity, ${\alpha }_{nf}$ is the nanofluid thermal diffusivity, T is the dimensional temperature, C is the dimensional mass concentration, $C_p{}_{nf}$ is the specific heat of the nanofluid at constant pressure, g is the gravitational acceleration, ${\beta }_{nf}$ and ${\beta }^{\mathrm{<mo>\; *\; </mo>}}$ are the nanofluid thermal and concentration expansion coefficients respectively, ${\mu }_e$ and k are the magnetic and porous plate permeabilities respectively, $K_r$ stands for the chemical reaction parameter.

The problem’s initial boundary conditions are similar to those of Prabhaker Reddy and Makinde [49] , and are presented as follows

$\begin{array}{l}u=0,w=0,T=T_{\infty }\text{for}\text{all}L\ge y\ge 0\text{and}t=0,\\ C=C_{\infty },H_x=0,H_z=0\text{for}\text{all}L\ge y\ge 0\text{and}t=0,\\ u=u_0,w=0,H_x=H,H_z=H\text{at}y=0\text{for}t>0,\\ T=T_{\infty }+\left(T_w-T_{\infty }\right)\frac{t}{t_0}\text{when}0<t\le t_0\text{at}y=0\text{for}t>0,\\ T=T_w\text{when}t>t_0\text{at}y=0\text{for}t>0,\\ C=C_{\infty }+\left(C_w-C_{\infty }\right)\frac{t}{t_0}\text{when}0<t\le t_0\text{at}y=0\text{for}t>0,\\ C=C_w\text{when}t>t_0\text{at}y=0\text{for}t>0,\\ u\to 0,w\to 0,H_x\to 0,H_z\to 0,T\to T_{\infty },C\to C_{\infty }\text{as}y\to \infty \text{for}t>0.\end{array}$(8)

where Initial velocity and time are $u_0$ and $t_0$, respectively. As a function of the nanoparticle, the following definitions for effective density, effective dynamic viscosity, thermal expansion coefficient, thermal diffusivity, heat capacitance, thermal conductivity, and electrical conductivity (see Minea [56] , and Hamad & Pop [12] ) are given, respectively:

$\begin{array}{l}{\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s,\\ {\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}},\\ {\left(\rho \beta \right)}_{nf}=\left(1-\varphi \right){\left(\rho \beta \right)}_f+\varphi {\left(\rho \beta \right)}_s,\\ {\alpha }_{nf}=\frac{{\kappa }_{nf}}{{\left(\rho C_p\right)}_{nf}},\\ {\left(\rho C_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho C_p\right)}_f+\varphi {\left(\rho C_p\right)}_s,\\ \frac{{\kappa }_{nf}}{{\kappa }_f}=\frac{\left({\kappa }_s+2{\kappa }_f\right)-2\varphi \left({\kappa }_f-{\kappa }_s\right)}{\left({\kappa }_s+2{\kappa }_f\right)+2\varphi \left({\kappa }_f-{\kappa }_s\right)}\\ {\sigma }_{nf}={\sigma }_f\left[1+\frac{3\left(\epsilon -1\right)\varphi }{\left(\epsilon +2\right)-\left(\epsilon -1\right)\varphi }\right],\text{where}\epsilon =\frac{{\sigma }_s}{{\sigma }_f},\end{array}$(9)

where $\varphi$ denotes the volume of the nanoparticles that are made of solid material, ${\rho }_f$ and ${\rho }_s$ denote the densities of the fluid and solid particles, respectively, and ${\beta }_s$ and ${\beta }_f$ denote the coefficients of thermal expansion of the solid and fluid, respectively. The thermal conductivity of the base fluid and the solid are ${\kappa }_f$ and ${\kappa }_s$, respectively, whereas the electrical conductivity of the fluid and the solid are ${\sigma }_f$ and ${\sigma }_s$. Table 1 lists the thermo-physical properties of the base fluid (water) and copper that are utilized for code validation.

Table 1

Table 1. Base fluid and nanoparticle thermophysical properties.

Thermo-physical properties	Water	Copper (Cu)
Heat capacity $C_p\left(\text{J}/\text{kg}\cdot \text{K}\right)$	4179	385
Density $\rho \left(\text{kg}/{\text{m}}^{\text{3}}\right)$	997.1	8933
Thermal conductivity $\kappa \left(\text{W}/\text{m}\cdot \text{K}\right)$	0.613	400
Thermal expansion $\beta \times {10}^{-5}\left(\text{1}/\text{K}\right)$	21	1.67
Electrical conductivity $\sigma {\left(\Omega \cdot \text{m}\right)}^{-1}$	0.05	5.96 × 107

The fluid dynamic viscosity ${\mu }_f$ is considered to vary as an inverse function of temperature as given by Lai & Kulacki [57] :

$\frac{1}{{\mu }_f}=\frac{1}{{\mu }_{\infty }}\left[1+\gamma \left(T-T_{\infty }\right)\right]\mathrm{.}$(10)

The transformation variables in dimensionless form are introduced as follows:

$\begin{array}{l}{H}_x^{*}=\frac{H_x}{H},H_z^{*}=\frac{H_z}{H};u^{*}=\frac{u}{u_0},w^{*}=\frac{w}{u_0}\\ t^{*}=\frac{u_{0}t}{L},y^{*}=\frac{y}{L};C^{*}=\frac{C-C_{\infty }}{C_w-C_{\infty }};T^{*}=\frac{T-T_{\infty }}{T_w-T_{\infty }}.\end{array}$(11)

Such that, Equations (2)-(7) can be written in the following dimensionless form (dropping asterisks):

$\begin{array}{c}\frac{\partial u}{\partial t}+2Rw=a_1\left(K\frac{u}{1+\gamma \Delta TT}+\frac{1}{Re}\frac{\partial }{\partial y}\left(\frac{1}{1+\gamma \Delta TT}\frac{\partial u}{\partial y}\right)\right)\\ +a_{2}ReGrT+ReGmC+a_{3}M\left(\left[w{H}_x-u{H}_z\right]H_z-u\right)\mathrm{,}\end{array}$(12)

$\begin{array}{c}\frac{\partial w}{\partial t}-2Rw=a_1\left(K\frac{w}{1+\gamma \Delta TT}+\frac{1}{Re}\frac{\partial }{\partial y}\left(\frac{1}{1+\gamma \Delta TT}\frac{\partial w}{\partial y}\right)\right)\\ +a_{3}M\left(\left[u{H}_z-w{H}_x\right]H_x-w\right)\mathrm{,}\end{array}$(13)

$\begin{array}{c}\frac{\partial T}{\partial t}=a_4\left(\frac{1}{Re}\left[\frac{1}{Pr}\frac{{\partial }^{2}T}{\partial y^2}\right]\right)+a_5\left(\frac{Ec}{Re}\frac{1}{1+\gamma \Delta TT}\left[{\left(\frac{\partial u}{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial y}\right)}^2\right]\right)\\ +a_6\left(JeRe\left[\left(w^2+u^2\right)+{\left(w{H}_x-u{H}_z\right)}^2\right]\right)\mathrm{,}\end{array}$(14)

$\frac{\partial C}{\partial t}=\frac{a_7}{ScRe}\frac{{\partial }^{2}C}{\partial y^2}-a_{7}KcReC\mathrm{,}$(15)

$\begin{array}{l}\frac{\partial H_x}{\partial t}=\frac{\partial u}{\partial y}+\frac{a_8}{R{e}_m}\frac{{\partial }^2{H}_x}{\partial y^2}\mathrm{,}\hfill \end{array}$(16)

$\frac{\partial H_z}{\partial t}=-\frac{\partial w}{\partial y}+\frac{a_8}{R{e}_m}\frac{{\partial }^2{H}_z}{\partial y^2}\mathrm{.}$(17)

Here $R=\frac{\Omega L}{u_0}$ is the Rotation parameter, $K=\frac{{\mu }_{f}L}{k{\rho }_f{u}_0}$ is the Permeability parameter, $Re=\frac{{\rho }_f{u}_{0}L}{{\mu }_f}$ is the Reynolds number, $Gr=\frac{g{\beta }_f{\nu }_f\left(T_w-T_{\infty }\right)}{u_0^3}$ is the local temperature Grashof number, $Gm=\frac{g{\beta }^{\mathrm{<mo>\; *\; </mo>}}{\nu }_f\left(C_w-C_{\infty }\right)}{u_0^3}$ is the local mass Grashof number, $M=\frac{{\sigma }_f{\mu }_e^2{H}^{2}L}{{\rho }_f{u}_0}$ is the Magnetic field parameter, $Pr=\frac{{\mu }_{f}C{p}_f}{{\kappa }_f}$ is the Prandtl number, $Ec=\frac{u_0^2}{C{p}_f\Delta T}$ is the Eckert number, $Je=\frac{{\sigma }_f{\mu }_e^2{H}^2{\mu }_f}{{\rho }_f^{2}C{p}_f\Delta T}$ is the Joule heating parameter, $Sc=\frac{{\mu }_f}{{\rho }_f{D}_m}$ is the Schmidt number, $Kc=\frac{k_r{\mu }_f}{{\rho }_f{u}_0^2}$ is the Chemical reaction parameter, $R{e}_m={\sigma }_f{\mu }_e{u}_{0}L$ is the Magnetic Reynolds number, $\Delta T=T_w-T_{\infty }$ is the temperature difference and where,

$a_1=\frac{{\left(1-\varphi \right)}^{-2.5}}{\left(1-\varphi \right)+\varphi {\rho }_s/{\rho }_f}\mathrm{,}$

$a_2=\frac{\left(1-\varphi \right)+\varphi {\left(\rho \beta \right)}_s/{\left(\rho \beta \right)}_f}{\left(1-\varphi \right)+\varphi {\rho }_s/{\rho }_f}\mathrm{,}$

$a_3=\frac{\left(ϵ+2\right)\left(1-\varphi \right)+3ϵ}{\left(\left(ϵ+2\right)-\left(ϵ-1\right)\varphi \right)\left(\left(1-\varphi \right)+\varphi {\rho }_s/{\rho }_f\right)}\mathrm{,}$

$a_4=\frac{\left({\kappa }_s+2{\kappa }_f\right)-2\varphi \left({\kappa }_f-{\kappa }_s\right)}{\left(\left({\kappa }_s+2{\kappa }_f\right)+2\varphi \left({\kappa }_f-{\kappa }_s\right)\right)\left(\left(1-\varphi \right)+\varphi {\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f\right)}\mathrm{,}$

$a_5=\frac{{\left(1-\varphi \right)}^{-2.5}}{\left(1-\varphi \right)+\varphi {\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f}\mathrm{,}$

$a_6=\frac{\left(ϵ+2\right)\left(1-\varphi \right)+3ϵ}{\left(\left(ϵ+2\right)-\left(ϵ-1\right)\varphi \right)\left(\left(1-\varphi \right)+\varphi {\left(\rho C_p\right)}_s/{\left(\rho C_p\right)}_f\right)}\mathrm{,}$

$a_7=\mathrm{1,}$

$a_8=\frac{\left(ϵ+2\right)-\left(ϵ-1\right)\varphi }{\left(ϵ+2\right)\left(1-\varphi \right)+3ϵ}\mathrm{.}$

The boundary conditions (8) in nondimensional form are

$\begin{array}{l}u=0,w=0,T=0,C=0,H_x=0,H_z=0\text{for}\text{all}1\ge y\ge 0\text{and}t\le 0,\\ u=1,w=0,H_x=1,H_z=1\text{at}y=0\text{for}t>0,\\ T=t\text{when}0<t\le 1\text{at}y=0\text{for}t>0,\\ T=1\text{when}t>1\text{at}y=0\text{for}t>0,\\ C=t\text{when}0<t\le 1\text{at}y=0\text{for}t>0,\\ C=1\text{when}t>1\text{at}y=0\text{for}t>0,\\ u\to 0,w\to 0,H_x\to 0,H_z\to 0,T\to 0,C\to 0\text{as}y\to \infty \text{for}t>0.\end{array}$(18)

Local skin friction coefficient:

The skin friction coefficient is a dimensionless quantity defined by the shear stress on the wall:

$C_f=\frac{{\tau }_w}{{\rho }_{nf}{u}_0^2}$(19)

where ${\rho }_{nf}$ and $u_0$ are the values of the nanofluid density and of the longitudinal velocity near the boundary layer’s edge. Accordingly, at the plate with $y=0$, the primary and secondary shear stresses ${\tau }_x$ and ${\tau }_z$ are given by:

${\tau }_x={\mu }_{nf}{\left(\frac{\partial u}{\partial t}\right)}_{y=0}\text{and}{\tau }_z={\mu }_{nf}{\left(\frac{\partial w}{\partial t}\right)}_{y=0}.$(20)

3. Numerical Method

This part is constructed to show the implementation of the Bivariate Spectral Relaxation Method (BI-SRM). The BI-SRM is a very effective tool that can be used in solving the highly nonlinear differential equations. The BI-SRM, a relatively new technique, improves upon the Spectral Relaxation Method (SRM)(see Motsa et al. [58] ), which is widely used for the numerical solution of the partial differential equations involved in the modeling of fluid flow. BI-SRM is introduced in this section in order to solve the system of the nonlinear partial differential Equations (12)-(17) along with the boundary conditions in (18). In order to make the spectral collocation method applicable, BI-SRM linearises the non-linear PDEs (12)-(17) to have

${\alpha }_1\frac{{\partial }^2{u}_{r+1}}{\partial y^2}+{\alpha }_2\frac{\partial u_{r+1}}{\partial y}+{\alpha }_3{u}_{r+1}-\frac{\partial u_{r+1}}{\partial t}=R_{u\mathrm{,}r}\mathrm{,}$(21)

${\beta }_1\frac{{\partial }^2{w}_{r+1}}{\partial y^2}+{\beta }_2\frac{\partial w_{r+1}}{\partial y}+{\beta }_3{w}_{r+1}-\frac{\partial w_{r+1}}{\partial t}=R_{w\mathrm{,}r}\mathrm{,}$(22)

${\gamma }_1\frac{{\partial }^2{T}_{r+1}}{\partial y^2}-\frac{\partial T_{r+1}}{\partial t}=R_{T\mathrm{,}r}\mathrm{,}$(23)

${\sigma }_1\frac{{\partial }^2{C}_{r+1}}{\partial y^2}+{\sigma }_2{C}_{r+1}-\frac{\partial C_{r+1}}{\partial t}=R_{C\mathrm{,}r}\mathrm{,}$(24)

${\delta }_1\frac{{\partial }^{2}H{x}_{r+1}}{\partial y^2}-\frac{\partial H{x}_{r+1}}{\partial t}=R_{Hx\mathrm{,}r}\mathrm{,}$(25)

${\omega }_1\frac{{\partial }^{2}H{z}_{r+1}}{\partial y^2}-\frac{\partial H{z}_{r+1}}{\partial t}=R_{Hz\mathrm{,}r}\mathrm{,}$(26)

subject to

$\begin{array}{l}{u}_{r+1}\left(0,y\right)=0,u_{r+1}\left(t,0\right)=1;u_{r+1}\left(t,\infty \right)=0,\\ w_{r+1}\left(0,y\right)=0,w_{r+1}\left(t,0\right)=0;w_{r+1}\left(t,\infty \right)=0,\\ T_{r+1}\left(0,y\right)=0,T_{r+1}\left(t,0\right)=1;T_{r+1}\left(t,\infty \right)=0,\\ C_{r+1}\left(0,y\right)=0,C_{r+1}\left(t,0\right)=1;C_{r+1}\left(t,\infty \right)=0,\\ H{x}_{r+1}\left(0,y\right)=0,H{x}_{r+1}\left(t,0\right)=1;H{x}_{r+1}\left(t,\infty \right)=0,\\ H{z}_{r+1}\left(0,y\right)=0,H{z}_{r+1}\left(t,0\right)=1;H{z}_{r+1}\left(t,\infty \right)=0.\end{array}$(27)

The variable coefficients in above equations are determined by

$\begin{array}{l}{\alpha }_1=\frac{a_1}{Re\left(1+\gamma \Delta T{T}_r\right)}\mathrm{,}{\alpha }_2=\frac{a_1\gamma \Delta T}{Re{\left(1+\gamma \Delta T{T}_r\right)}^2}\frac{\partial T_r}{\partial y}\mathrm{,}\\ {\alpha }_3=\frac{a_{1}K}{1+\gamma \Delta T{T}_r}-a_{3}M\left({\left(H{z}_r\right)}^2+1\right)\mathrm{,}{\beta }_1=\frac{a_1}{Re\left(1+\gamma \Delta T{T}_r\right)}\mathrm{,}\\ {\beta }_2=\frac{a_1\gamma \Delta T}{Re{\left(1+\gamma \Delta T{T}_r\right)}^2}\frac{\partial T_r}{\partial y}\mathrm{,}{\beta }_3=\frac{a_{1}K}{1+\gamma \Delta T{T}_r}-a_{3}M\left({\left(H{z}_r\right)}^2+1\right)\mathrm{,}\\ {\gamma }_1=\frac{a_4}{RePr}\mathrm{,}{\sigma }_1=\frac{a_7}{ScRe}\mathrm{,}{\sigma }_2=-a_{7}KcRe\mathrm{,}{\delta }_1=\frac{a_8}{R{e}_m}\mathrm{,}{\omega }_1=\frac{a_8}{R{e}_m}\mathrm{,}\\ R_{u\mathrm{,}r}=2R{w}_r-a_{2}ReGr{T}_r-ReGm{C}_{r+1}-a_{3}M\left(w_{r}H{x}_{r}H{z}_r\right)\mathrm{,}\\ R_{w\mathrm{,}r}=-2R{u}_{r+1}-a_{3}M{u}_{r+1}H{z}_{r}H{x}_r\mathrm{,}\end{array}$

$\begin{array}{l}{R}_{T\mathrm{,}r}=-a_5\frac{Ec}{Re\left(1+\gamma \Delta T{T}_r\right)}\left[{\left(\frac{\partial u_{r+1}}{\partial y}\right)}^2+{\left(\frac{\partial w_{r+1}}{\partial y}\right)}^2\right]\\ -a_{6}JeRe\left[\left(w_{r+1}^2+u_{r+1}^2\right)+{\left(w_{r+1}H{x}_r-u_{r+1}H{z}_r\right)}^2\right]\mathrm{,}\\ R_{C,r}=0,R_{Hx,r}=-\frac{\partial u_{r+1}}{\partial y},R_{Hz,r}=\frac{\partial w_{r+1}}{\partial y},\end{array}$

where $\left(r+1\right)$ and r denote the current and previous iterations, respectively. The Chebyshev spectral collocation is adopted to discretize the linear partial differential Equations (21)-(26) in both time (t) and space (y) directions. However, the spectral method is applicable only in the domain $\left[-\mathrm{1,1}\right]$, due to this, we have to convert the physical domain of the problem from $\left(t\mathrm{,}y\right)\in \left[\mathrm{0,1}\right]\times \left[\mathrm{0,}{L}_{\infty }\right]$ into $\left(\zeta \mathrm{,}\eta \right)\in \left[-\mathrm{1,1}\right]\times \left[-\mathrm{1,1}\right]$ by suitable linear transformations. The linear partial differential Equations (21)-(26)) are decoupled, so each of them can be solved separately and their solutions can be approximated by a bivariate Lagrange polynomial standard form as follows:

$U\left(\eta ,\zeta \right)={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}U}}\left({\eta }_p,{\zeta }_q\right)L_p\left(\eta \right)L_q\left(\zeta \right),$(28)

$W\left(\eta ,\zeta \right)={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}W}}\left({\eta }_p,{\zeta }_q\right)L_p\left(\eta \right)L_q\left(\zeta \right),$(29)

$T\left(\eta ,\zeta \right)={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}T}}\left({\eta }_p,{\zeta }_q\right)L_p\left(\eta \right)L_q\left(\zeta \right),$(30)

$C\left(\eta ,\zeta \right)={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}C}}\left({\eta }_p,{\zeta }_q\right)L_p\left(\eta \right)L_q\left(\zeta \right),$(31)

$H_x\left(\eta ,\zeta \right)={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}{H}_x}}\left({\eta }_p,{\zeta }_q\right)L_p\left(\eta \right)L_q\left(\zeta \right),$(32)

$H_z\left(\eta ,\zeta \right)={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}{H}_z}}\left({\eta }_p,{\zeta }_q\right)L_p\left(\eta \right)L_q\left(\zeta \right),$(33)

To estimate variables in Equations (28)-(33) at predetermined points in the indicated directions $\eta$ and $\zeta$ we introduce the Chebyshev-Gauss-Lobatto points

$\left\{{\eta }_p\right\}={\left\{\cos \left(\frac{\pi p}{N_{\eta }}\right)\right\}}_{p=0}^{N_{\eta }},\left\{{\zeta }_q\right\}={\left\{\cos \left(\frac{\pi q}{N_{\zeta }}\right)\right\}}_{q=0}^{N_{\zeta }}.$(34)

$N_{\eta }$ and $N_{\zeta }$, respectively, denote the number of grid points in the $\eta$ and $\zeta$ directions. Additionally, the discrete derivatives at the grid points are obtained by transforming the continuous time and space derivatives using Chebyshev-Gauss-Lobatto (CGL) (34) grid points. Moreover, the common Lagrange cardinal polynomials cardinal polynomials $L_q\left(\eta \right)$ and $L_p\left(\zeta \right)$ are defined in the form

$L_p\left(\eta \right)={\displaystyle \underset{\begin{array}{c}p=0\\ p\ne s\end{array}}{\overset{N_{\eta }}{\prod }}}\frac{\eta -{\eta }_s}{{\eta }_p-{\eta }_s},$(35)

where

$L_p\left({\eta }_s\right)={\delta }_{ps}=\{\begin{array}{cc}0& \text{if}p\ne s\mathrm{,}\\ 1& \text{if}p=s\mathrm{.}\end{array}$(36)

The definition of the function $L_q\left(\zeta \right)$ is identical to that of the function $L_p\left(\zeta \right)$. Solution procedure starts by the determination of the derivative of $L_p\left(\eta \right)$ and $L_q\left(\zeta \right)$ with respect to $\eta$ and $\zeta$, therefore, calculating the Chebyshev partial derivative of $U\left(\eta \mathrm{,}\zeta \right)$ with respect to $\zeta$ at any point $\left({\eta }_i\mathrm{,}{\zeta }_j\right)$ is given by

${\frac{\partial U}{\partial \zeta }|}_{\left({\eta }_i,{\zeta }_j\right)}={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}U}}\left({\eta }_q,{\zeta }_q\right)L_p\left(\eta \right){\frac{\text{d}{L}_q\left(\zeta \right)}{\text{d}\zeta }|}_{\left({\eta }_i,{\zeta }_j\right)}$(37)

$={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}U\left({\eta }_p,{\zeta }_q\right)L_p\left({\eta }_i\right)}}\frac{\text{d}{L}_q\left({\zeta }_j\right)}{\text{d}\zeta }$(38)

$={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\bar{D}}_{jq}U\left({\eta }_i,{\zeta }_q\right)}={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\bar{D}}_{jq}{U}_q}.$(39)

When $j,q=0,1,\cdots ,N_{\zeta }$, the “Chebyshev Differentiation Matrix” [59] is utilized to produce the matrix’s elements ${\bar{D}}_{jq}$. Partial derivatives of $U\left(\eta \mathrm{,}\zeta \right)$ with respect to $\eta$ are derived similarly. Similarly, at $\left({\eta }_i\mathrm{,}{\zeta }_i\right)$ CGL points, the partial derivative with respect to the space variable $\eta$ is app

${\frac{\partial U}{\partial \eta }|}_{\left({\eta }_i,{\zeta }_j\right)}={\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}{\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}U}}\left({\eta }_p,{\zeta }_q\right){\frac{\text{d}{L}_p\left(\eta \right)}{\text{d}\eta }{L}_q\left(\zeta \right)|}_{\left({\eta }_i,{\zeta }_j\right)}$(40)

$={\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}U}\left({\eta }_p,{\zeta }_j\right)\frac{\text{d}{L}_p\left({\eta }_i\right)}{\text{d}\eta }$(41)

$={\displaystyle \underset{p=0}{\overset{N_{\eta }}{\sum }}{D}_{ip}U\left({\eta }_p,{\zeta }_j\right)}=\left[D\right]U_j,$(42)

where i is the index nodes of $\eta$ and p is the index basis of $\eta$, such that

$U_j={\left[u\left({\eta }_0\mathrm{,}{\zeta }_j\right)\mathrm{,}u\left({\eta }_1\mathrm{,}{\zeta }_j\right)\mathrm{,}u\left({\eta }_2\mathrm{,}{\zeta }_j\right)\mathrm{,}\cdots \mathrm{,}u\left({\eta }_{N_{\eta }}\mathrm{,}{\zeta }_j\right)\right]}^{\top }\mathrm{.}$(43)

where $\top$ denotes the matrix transpose. Consequently, the derivative of second order with respect to $\eta$ is given by

${\frac{{\partial }^{2}U}{\partial {\eta }^2}|}_{\left({\eta }_i,{\zeta }_j\right)}=D^2{U}_j.$(44)

Differentiation matrices of other dependent variables $W\mathrm{,}T\mathrm{,}C\mathrm{,}Hx\mathrm{,}Hz$ are approximated by collocation, similar to Equations (39) and (42). Using the equations (39), (42) and (44) in Equation (24) we arrive at

$\left[{\sigma }_1{D}^2+{\sigma }_{2}I\right]C_i-{\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}}{\bar{D}}_{ij}{U}_q=0,i=0,1,\cdots ,N_{\zeta }.$(45)

According to the definitions of the dependent variables $U\mathrm{,}W\mathrm{,}T\mathrm{,}C\mathrm{,}Hx\mathrm{,}Hz$ and their derivatives, the Equations (21)-(26) can be written in the following operator form

$\left[{\alpha }_1{D}^2+{\alpha }_{2}D+{\alpha }_{3}I\right]U_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}{U}_j=R_{u,i}$(46)

$\left[{\beta }_1{D}^2+{\beta }_{2}D+{\beta }_{3}I\right]W_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}{W}_j=R_{w,i},$(47)

${\gamma }_1{D}^2{T}_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}{T}_j=R_{T,i},$(48)

$\left[{\sigma }_1{D}^2+{\sigma }_{2}I\right]C_i-{\displaystyle \underset{q=0}{\overset{N_{\zeta }}{\sum }}}{\bar{D}}_{ij}{U}_q=0,$(49)

${\delta }_1{D}^{2}H{x}_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}H{x}_i=R_{Hx,i},$(50)

${\omega }_1{D}^{2}H{z}_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}H{z}_i=R_{Hz,i},$(51)

where

$R_{u,i}=R_{u,r}+D_{i{N}_{\zeta }}{U}_{N_{\zeta }},R_{w,i}=R_{w,r}+D_{i{N}_{\zeta }}{W}_{N_{\zeta }},$

$R_{T,i}=R_{T,r}+D_{i{N}_{\zeta }}{T}_{N_{\zeta }},R_{C,i}=R_{C,r}+D_{i{N}_{\zeta }}{C}_{N_{\zeta }},$

$R_{Hx,i}=R_{Hx,r}+D_{i{N}_{\zeta }}H{x}_{N_{\zeta }},R_{Hz,i}=R_{Hz,r}+D_{i{N}_{\zeta }}H{z}_{N_{\zeta }}.$

Therefore, for $i=\mathrm{0,1,}\cdots \mathrm{,}{N}_{\zeta -1}$, the boundary conditions in Equation (49) are established as in the following $N_{\zeta }\left(N_{\eta }+1\right)\times N_{\zeta }\left(N_{\eta }+1\right)$ matrix system:

$\left[\begin{array}{cccc}{Q}_{\mathrm{0,0}}& Q_{\mathrm{0,1}}& \cdots & Q_{\mathrm{0,}{N}_{\zeta }-1}\\ Q_{\mathrm{1,0}}& Q_{\mathrm{1,1}}& \cdots & Q_{\mathrm{1,}{N}_{\zeta }-1}\\ ⋮& ⋮& \ddots & ⋮\\ Q_{N_{\zeta }-\mathrm{1,0}}& Q_{N_{\zeta }-\mathrm{1,1}}& \cdots & Q_{N_{\zeta }-\mathrm{1,}{N}_{\zeta -1}}\end{array}\right]\left[\begin{array}{c}{C}_0\\ C_1\\ ⋮\\ C_{N_t-1}\end{array}\right]=\left[\begin{array}{c}{R}_{c\mathrm{,0}}\\ R_{c\mathrm{,1}}\\ ⋮\\ R_{c\mathrm{,}{N}_{\zeta -1}}\end{array}\right]$(52)

where

$\begin{array}{l}{Q}_{i\mathrm{,}i}={\sigma }_1{D}^2+{\sigma }_{2}I-{\bar{D}}_{ii}I\\ Q_{i\mathrm{,}j}=-{\bar{D}}_{ij}I\mathrm{,}\text{when}i\ne j\mathrm{.}\end{array}$(53)

The boundary condition (27) is imposed similarly for the remaining variables, and the bivariate Chebyshev collocation technique is performed similarly to produce

$\left[{\alpha }_1{D}^2+{\alpha }_{2}D+{\alpha }_{3}I\right]U_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}{U}_j=R_{u,i}$(54)

$\left[{\beta }_1{D}^2+{\beta }_{2}D+{\beta }_{3}I\right]W_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}{W}_j=R_{w,i}$(55)

${\gamma }_1{D}^2{T}_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}{T}_j=R_{T,i}$(56)

${\delta }_1{D}^{2}H{x}_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}H{x}_i=R_{Hx,i}$(57)

${\omega }_1{D}^{2}H{z}_i-{\displaystyle \underset{j=0}{\overset{N_{\zeta }-1}{\sum }}}{\bar{D}}_{ij}H{z}_i=R_{Hz,i}$(58)

The vectors $U_{N_{\zeta }}$, $W_{N_{\zeta }}$, $T_{N_{\zeta }}$, $C_{N_{\zeta }}$, $H{x}_{N_{\zeta }}$ and $H{z}_{N_{\zeta }}$ are the initial solutions derived from the supplied boundary conditions (27). After the boundary conditions are applied, Equations (54)-(58) are written as the following $N_{\zeta }\left(N_{\eta }+1\right)\times N_{\zeta }\left(N_{\eta }+1\right)$ matrices:

$\left[\begin{array}{cccc}{X}_{\mathrm{0,0}}& X_{\mathrm{0,1}}& \cdots & X_{\mathrm{0,}{N}_{\zeta }-1}\\ X_{\mathrm{1,0}}& X_{\mathrm{1,1}}& \cdots & X_{\mathrm{1,}{N}_{\zeta }-1}\\ ⋮& ⋮& \ddots & ⋮\\ X_{N_{\zeta }-\mathrm{1,0}}& X_{N_{\zeta }-\mathrm{1,1}}& \cdots & X_{N_{\zeta }-\mathrm{1,}{N}_{\zeta }-1}\end{array}\right]\left[\begin{array}{c}{U}_0\\ U_1\\ ⋮\\ U_{N_{\zeta }-1}\end{array}\right]=\left[\begin{array}{c}{R}_{u\mathrm{,0}}\\ R_{u\mathrm{,1}}\\ ⋮\\ R_{u\mathrm{,}{N}_{\zeta }-1}\end{array}\right]$(59)

$\left[\begin{array}{cccc}{Z}_{\mathrm{0,0}}& Z_{\mathrm{0,1}}& \cdots & Z_{\mathrm{0,}{N}_{\zeta }-1}\\ Z_{\mathrm{1,0}}& Z_{\mathrm{1,1}}& \cdots & Z_{\mathrm{1,}{N}_{\zeta }-1}\\ ⋮& ⋮& \ddots & ⋮\\ Z_{N_{\zeta }-\mathrm{1,0}}& Z_{N_{\zeta }-\mathrm{1,1}}& \cdots & Z_{N_{\zeta }-\mathrm{1,}{N}_{\zeta }-1}\end{array}\right]\left[\begin{array}{c}{W}_0\\ W_1\\ ⋮\\ W_{N_{\zeta }-1}\end{array}\right]=\left[\begin{array}{c}{R}_{w\mathrm{,0}}\\ R_{w\mathrm{,1}}\\ ⋮\\ R_{w\mathrm{,}{N}_{\zeta }-1}\end{array}\right]$(60)

$\left[\begin{array}{cccc}{F}_{\mathrm{0,0}}& F_{\mathrm{0,1}}& \cdots & F_{\mathrm{0,}{N}_{\zeta }-1}\\ F_{\mathrm{1,0}}& F_{\mathrm{1,1}}& \cdots & F_{\mathrm{1,}{N}_{\zeta }-1}\\ ⋮& ⋮& \ddots & ⋮\\ F_{N_{\zeta }-\mathrm{1,0}}& F_{N_{\zeta }-\mathrm{1,1}}& \cdots & F_{N_{\zeta }-\mathrm{1,}{N}_{\zeta }-1}\end{array}\right]\left[\begin{array}{c}{T}_0\\ T_1\\ ⋮\\ T_{N_{\zeta }-1}\end{array}\right]=\left[\begin{array}{c}{R}_{T\mathrm{,0}}\\ R_{T\mathrm{,1}}\\ ⋮\\ R_{T\mathrm{,}{N}_{\zeta }-1}\end{array}\right]$(61)

$\left[\begin{array}{cccc}{G}_{\mathrm{0,0}}& G_{\mathrm{0,1}}& \cdots & G_{\mathrm{0,}{N}_{\zeta }-1}\\ G_{\mathrm{1,0}}& G_{\mathrm{1,1}}& \cdots & G_{\mathrm{1,}{N}_{\zeta }-1}\\ ⋮& ⋮& \ddots & ⋮\\ G_{N_{\zeta }-\mathrm{1,0}}& G_{N_{\zeta }-\mathrm{1,1}}& \cdots & G_{N_{\zeta }-\mathrm{1,}{N}_{\zeta }-1}\end{array}\right]\left[\begin{array}{c}H{x}_0\\ H{x}_1\\ ⋮\\ H{x}_{N_{\zeta }-1}\end{array}\right]=\left[\begin{array}{c}{R}_{Hx\mathrm{,0}}\\ R_{Hx\mathrm{,1}}\\ ⋮\\ R_{Hx\mathrm{,}{N}_{\zeta }-1}\end{array}\right]$(62)

$\left[\begin{array}{cccc}{P}_{\mathrm{0,0}}& P_{\mathrm{0,1}}& \cdots & P_{\mathrm{0,}{N}_{\zeta }-1}\\ P_{\mathrm{1,0}}& P_{\mathrm{1,1}}& \cdots & P_{\mathrm{1,}{N}_{\zeta }-1}\\ ⋮& ⋮& \ddots & ⋮\\ P_{N_{\zeta }-\mathrm{1,0}}& P_{N_{\zeta }-\mathrm{1,1}}& \cdots & P_{N_{\zeta }-\mathrm{1,}{N}_{\zeta }-1}\end{array}\right]\left[\begin{array}{c}H{z}_0\\ H{z}_1\\ ⋮\\ H{z}_{N_{\zeta }-1}\end{array}\right]=\left[\begin{array}{c}{R}_{Hz\mathrm{,0}}\\ R_{Hz\mathrm{,1}}\\ ⋮\\ R_{Hz\mathrm{,}{N}_{\zeta }-1}\end{array}\right]$(63)

where

$X_{i,i}={\alpha }_{1,r}{D}^2+{\alpha }_{2,r}D+{\alpha }_{3,r}I-{\bar{D}}_{ii}I,X_{i,j}=-{\bar{D}}_{ij}I,\text{when}i\ne j,$

$Z_{i,i}={\beta }_{1,r}{D}^2+{\beta }_{2,r}D+{\beta }_{3,r}I-{\bar{D}}_{ii}I,Z_{i,j}=-{\bar{D}}_{ij}I,\text{when}i\ne j,$

$F_{i\mathrm{,}i}={\gamma }_{\mathrm{1,}r}{D}^2-{\bar{D}}_{ii}I\mathrm{,}{F}_{i\mathrm{,}j}=-{\bar{D}}_{ij}I\mathrm{,}\text{when}i\ne j\mathrm{,}$

$G_{i,i}={\delta }_{1,r}{D}^2-{\bar{D}}_{ii}I,G_{i,j}=-{\bar{D}}_{ij}I,\text{when}i\ne j,$

$P_{i,i}={\omega }_{1,r}{D}^2-{\bar{D}}_{ii}I,P_{i,j}=-{\bar{D}}_{ij}I,\text{when}i\ne j,$

$I$ is the $\left(N_{\eta }+1\right)\times \left(N_{\eta }+1\right)$ unit matrix. The system of above matrices (59)-(63) is iteratively solved to get the numerical solution for $u\left(\eta \mathrm{,}\zeta \right)\mathrm{,}w\left(\eta \mathrm{,}\zeta \right)\mathrm{,}T\left(\eta \mathrm{,}\zeta \right)\mathrm{,}Hx\left(\eta \mathrm{,}\zeta \right)$ and $Hz\left(\eta \mathrm{,}\zeta \right)$.

4. Results and Discussion

The results obtained by applying the Bivariate Spectral Relaxation Method (BI-SRM) to the problem of time-dependent MHD nanofluid flow driven by a semi-infinite vertical plate in a porous medium are carefully presented and thoroughly discussed in this section. As a check on the accuracy and convergent nature of the BI-SRM, we calculated the error of the method over orders see Figure 2 and Figure 3 . From these figures, it is observed that the BI-SRM achieves convergence in only 12 iterations with a maximum error of order 10⁻¹⁰ for the velocity and temperature profiles and 10⁻¹¹ for both $H_x\mathrm{,}{H}_z$ of the induction profiles. To understand the physics of the problem, we have shown in Figures 4 - 14 the effects of varying our governing parameters on velocity, temperature, concentration, induction fields as well as skin friction. Through all these figures, it is apparent that the mass volume fraction appearing leads to slowdown in the fluid motion. There are two potential explanations for this phenomenon. Firstly, it is worth

Figure 2

Figure 2. Residual error graphs.

Figure 3

Figure 3. Solution error graphs.

Figure 4

Figure 4. The velocity profiles for u and w against K and ϕ .

Figure 5

Figure 5. The velocity profiles for u and w against Re and ϕ .

Figure 6

Figure 6. The velocity profiles for u and w against R and ϕ .

noting that larger nanoparticles undergo less Brownian motion, which increases resistance in the current and effectively reduces velocity. Secondly, raising nanoparticle concentration may increase fluid viscosity, thus causing a thicker momentum boundary layer, which could ultimately decelerate fluid motion.

The influence of permeability K of the porous media on the velocity profiles is displayed in Figure 4 for different values of nanoparticle volume fractions. It is observed that raising the permeability coefficient of the porous medium reduces the resistance between the fluid and medium, leading to a thinner momentum boundary layer, hence increasing the primary and secondary flow. Figure 5 shows the influence of the Reynolds number Re on the velocity components u and w. The Reynolds number relates the fluid momentum to the viscous shear stress in the fluid system. Increasing the Reynolds number decreases the ratio 1/Re, causing the momentum force to become negligible compared to the viscous forces. This enhances the thickness of the momentum boundary layer so that the velocity components u and w increase as 1/Re decreases. See Fig. 6 for the effect of the rotation parameter R on the primary velocity u and the secondary velocity w. It is evident that as the rotation parameter R increases, the primary velocity u decreases. This phenomenon occurs due to the dominance of the Coriolis force exerted by the fluid medium in the direction of rotation. For the same reason, it is also worth noting that the secondary velocity increases with the rotation parameter. Figure 7 illustrates how the magnetic field M affects the primary and secondary velocity, respectively.

It can be seen that the primary flow velocity decreases significantly as the magnetic parameter M increases. In a particular case, secondary flow velocity increases near the plate as M increases and then decreases away from the plate as M increases. In practice, this occurs due to the Lorentz force created by the magnetic field which acts against the flow. Figure 8 and Figure 9 show the variations of the primary and secondary velocities with respect to the thermal Grashof number Gr and the concentration Grashof number Gm, which represent the ratio of the thermal and concentration buoyancy forces to the viscous force, respectively. As it is known that the thermal and concentration Grashof numbers increase, Gr and Gm increase the effectiveness of the thermal and concentration buoyancy forces, resulting in an increase in the thermal Grashof number, the mass Grashof number, or both, which increases the thickness of the momentum boundary layers, thus increasing the velocities.

Figure 10 shows the velocity profiles of the primary and secondary components, u and w, respectively, for different values of the Prandtl number Pr. In fact, the ratio 1/Pr is a decreasing function of Pr, so in our fluid case, increasing Pr causes highly viscous flow. Therefore, both velocity components increase with increasing Pr. Figure 11 illustrates the effect of the chemical reaction parameter on the velocity components u and w. The results indicate that increasing the chemical reaction parameter leads to a reduction in both primary and secondary velocities, this is due to a decrease in fluid concentration.

Figure 7

Figure 7. The velocity profiles for u and w against M and ϕ .

Figure 8

Figure 8. The velocity profiles for u and w against Gr and ϕ .

Figure 9

Figure 9. The velocity profiles for u and w against Gm and ϕ .

Figure 10

Figure 10. The velocity profiles for u and w against Pr and ϕ .

Figure 11

Figure 11. The velocity profiles for u and w against Kc and ϕ .

Figure 12

Figure 12. The velocity profiles for u and w against Sc and ϕ .

Figure 12 shows the effect of Schmidt number Sc on the primary and secondary velocity profiles. As Sc increases, both u and w tend to decrease. This is because an increase in Sc leads to a decrease in molecular diffusivity, resulting in a thicker momentum boundary layer. The fluid velocity then decreases in both directions.

The primary and secondary velocity patterns for different values of the Reynolds magnetic number (Re_m) which estimates the induction of magnetic diffusion by a magnetic field are plotted in Figure 13 . Here is roughly estimated by the magnetic Reynolds number. As Re_m increases, the primary velocity rises, while the secondary speed decreases. Physically, an increase in Re_m leads to enhanced magnetic diffusion generates Lorentz drag, which interacts with fluid motion, resulting in a decrease in the primary velocity profile. Conversely, the secondary velocity exhibits the opposite trend for a while and then decreases due to the same indicated reasons. On the other hand, the side effects of the magnetic Reynolds number Re_m on both axially induced magnetic fields, Hx and Hz, are shown in Figure 14 . It can be seen that both Hx

Figure 13

Figure 13. The velocity profiles for u and w against Re m and ϕ .

Figure 14

Figure 14. The magnetic profiles for Hx and Hz against Re m and ϕ .

and Hz are decreasing functions in Re_m. This is because as Re_m decreases, the mass diffusivity increases and this accelerates the flow regime.

Figure 15 illustrates the temperature distribution for different values of Reynolds number Re and mass volume fraction $\varphi$. The figure shows that as the Reynolds number increases, the heat flux cells become smaller and smaller in size. An increasing Reynolds number indicates that the inertial forces become dominant compared to the viscous force. This causes the fluid to accelerate, as a result, more heat is transported by convection. It also can be seen that increasing the amount of mass volume fraction has a positive effect on the thermal conductivity of the system. Figure 16 and Figure 17 are plotted to illustrate the impacts of Schmidt number Sc and Prandtl number Pr on temperature distribution, respectively. From Figure 16 , it is very noticeable that iso-thermal cells become an increasing function of Sc. This means that by increasing Sc, the temperature is well distributed over the flow area. Conversely, it is clear that from Figure 17 , the effects of the Prandtl number on the temperature distribution is to produce a completely opposite behavior. For different values of the Schemdit number Sc at Re = 2 and Re = 5, the distribution of the concentration was plotted in Figure 18 . We found that the concentration distribution is strongly time dependent near the edge of the plate. When both Sc and Re are increasing, the concentration occupies a narrow region of the solution domain. Figure 19 illustrates the effect of the Eckert number Ec in the temperature distribution field. It is seen that increasing of Ec enhances the temperature distribution over the region domain. This demonstrates how the Ec number plays a significant role in the temperature distribution.

Figure 15

Figure 15. Temperature distribution for different values of Re and ϕ .

Figure 16

Figure 16. Temperature distribution for different values of Re and Sc.

Figure 17

Figure 17. Temperature distribution for different values of Re and Pr.

Figure 20 and Figure 21 are plotted to illustrate the impact of the Reynolds magnetic number on both x-axis magnetic field H_x and y-axis magnetic field H_z. From these figures we can see that as the Reynolds magnetic number, the orientation and direction of the magnetic lines change from positive to negative or vice versa, see the first surface plot. The two separated groups of cells in the second contour plot are evidence of this. When the magnetic field is oriented in a

Figure 18

Figure 18. Concentration distribution for different values of Re and Sc.

Figure 19

Figure 19. Temperature distribution for different values of Re and Ec.

Figure 20

Figure 20. X-axis magnetic filed induction distribution for different values of Re and Re m

Figure 21

Figure 21. Z-axis magnetic filed induction distribution for different values of Re and Re m

direction opposite to the normal vector of the domain by changing the magnetic polarity, the flux is negative, or when an arrangement of magnets takes place.

Table 2 and Table 3 display the primary and secondary skin friction under the influence of M, K, Gr, Gm, R, Sc, Ec, Pr, Kc, Re, and Re_m for various values of the

Table 2

Table 2. Skin friction coefficient for primary velocity u.

M	K	Gr	Gm	R	Ec	Pr	Sc	Kc	Re	Rem	$-C_{f\mathrm{,}x}\left(\varphi =0.35\right)$	$-C_{f\mathrm{,}x}\left(\varphi =0.4\right)$	$-C_{f\mathrm{,}x}\left(\varphi =0.45\right)$
10	0.5	10	10	1	0.5	1.7	0.5	1	0.5	0.5	4.3251	4.4641	4.7435
											4.8302	4.9784	5.2754
											5.3038	5.4617	5.7763
	1										4.2670	4.3968	4.6637
	1.5										4.2082	4.3287	4.5829
		15									4.0050	4.1769	4.4831
		20									3.6748	3.8826	4.2177
			15								2.9961	3.0720	3.2836
			20								1.6214	1.6329	1.7755
				2							4.3351	4.4749	4.7541
				3							4.3701	4.5122	4.7922
					1						4.0937	4.2121	4.4572
					1.5						3.8952	3.9971	4.2153
						3					4.0543	4.1754	4.4241
						4					3.8767	3.9834	4.2104
							1				4.3400	4.4801	4.7607
							1.5				4.3545	4.4958	4.7776
								2			4.3397	4.4799	4.7604
								3			4.3540	4.4952	4.7770
									1		2.8157	2.9008	3.0787
									1.5		2.1457	2.2164	2.3607
										1	4.1546	4.2765	4.5344
										1.5	4.0186	4.1321	4.3792

Table 3

Table 3. Skin friction coefficient for secondary velocity w.

M	K	Gr	Gm	R	Ec	Pr	Sc	Kc	Re	Rem	$C_{f\mathrm{,}z}\left(\varphi =0.35\right)$	$C_{f\mathrm{,}z}\left(\varphi =0.4\right)$	$C_{f\mathrm{,}z}\left(\varphi =0.45\right)$
10	0.5	10	10	1	0.5	1.7	0.5	1	0.5	0.5	2.7835	2.8359	2.9276
											2.8552	2.9123	3.0114
											2.9255	2.9869	3.0928
	1										2.8016	2.8563	2.9507
	1.5										2.8201	2.8770	2.9743
		15									2.8559	2.8993	2.9833
		20									2.9314	2.9649	3.0406
			15								3.0405	3.0965	3.1901
			20								3.3038	3.3632	3.4589
				2							3.1750	3.2497	3.3655
				3							3.5540	3.6504	3.7901
					1						2.8029	2.8539	2.9444
					1.5						2.8231	2.8730	2.9624
						3					2.8022	2.8520	2.9409
						4					2.8138	2.8624	2.9497
							1				2.7788	2.8310	2.9225
							1.5				2.7742	2.8263	2.9175
								2			2.7789	2.8312	2.9227
								3			2.7745	2.8266	2.918
									1		2.1384	2.1875	2.2688
									1.5		1.8064	1.8479	1.9173
										1	2.7337	2.7747	2.8518
										1.5	2.6694	2.7028	2.7702

nanoparticle mass fraction. From Table 2 , it is clear that increases in M, R, Sc, and Kc have been found to enhance the primary skin friction. On the other hand, the Primary skin friction reduces with increases in K, Gr, Gm, Ec, Pr, Re, and Re_m. Secondary skin friction is enhanced by increases in M, K, Gr, Gm, R, Ec and Pr while it is diminished by increases in Sc, Kc, Re, and Re_m. It is also observed that both the primary and secondary skin friction coefficient increases with an increase in the nanoparticle mass fraction $\varphi$. Table 2 and Table 3 represent the skin friction for various leading parameter variations. This involved setting up the values R = 1, K = 0.5, Gr = 10, Gm = 10, M = 10, Pr = 0.71, Sc = 0.5, Je = 2, Kc = 1, Re_m = 0.5, and Re = 0.5 which were then scattered around the range for computing reasons.

5. Conclusion

In this investigation, we carried out attributes of the convective flow of nanofluid flow with a vertical rotating plate imbedded in a porous medium by incorporating-induced magnetic field influence. The current model of the flow is utilized to stimulate the physical phenomena of the flow that occur inside the washing machine of the new generation, which is one of the frequent engineering utilizations in designing engineering products. The findings acquired are crucial in improving the performance of the new generation washing machine by adjusting the physical parameter values to get the best performance out of it. This paper uses the BI-SRM technique to numerically examine the effects of temperature-dependent viscosity, an induced magnetic field, nanofluid natural convection flow with heat and mass transfer, and a first-order chemical reaction through a semi-infinite, vertical rotating plate embedded in a porous media. Analysis of the momentum equations and induction equations, which incorporate variable viscosity and variable magnetic field, is the major objective. The problem investigates the primary physical parameters, which are then utilized in the governing equations. The most significant engineering outcomes are highlighted and well explained. The key findings of the current research lead to the following significant conclusions:

In terms of future study, the authors recommend incorporating cross-diffusion effects, which are common in many real applications. Entropy generation analysis may also be used to improve the thermodynamic optimization of the model’s system.

Data Availability

All required data is available in the text of the paper.

Acknowledgments

The African Union and the Pan African University’s Institute for Basic Sciences, Technology, and Innovation (PAUSTI) were responsible for supporting this work, which the first author would like to acknowledge.

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