Finite Element Study of MHD Impacts on the Rotating Flow of Casson Nanofluid with the Double Diffusion Cattaneo—Christov Heat Flux Model

Abstract

A study for MHD (magnetohydrodynamic) impacts on the rotating flow of Casson nanofluids is considered. The concentration and temperature distributions are related along with the double diffusion Cattaneo–Christov model, thermophoresis, and Brownian motion. The governing equations in the 3D form are changed into dimensionless two-dimensional form with the implementation of suitable scaling transformations. The variational finite element procedure is harnessed and coded in Matlab script to obtain the numerical solution of the coupled nonlinear partial differential problem. The variation patterns of Sherwood number, Nusselt number, skin friction coefficients, velocities, concentration, and temperature functions are computed to reveal the physical nature of this examination. It is seen that higher contributions of the magnetic force, Casson fluid, and rotational fluid parameters cause to raise the temperature like thermophoresis and Brownian motion does but causes slowing the primary as well as secondary velocities. The FEM solutions showing an excellent correlation with published results. The current study has significant applications in the biomedical, modern technologies of aerospace systems, and relevance to energy systems.

1. Introduction

Noteworthy endeavors have been made in recent years to explore nanofluids because of remarkable thermodynamic properties. Nanofluids can be utilized to cool the motors of vehicles, biomedical applications, high-transition gadgets, clothes washers machines, high-power microwaves, diode arrays of heavy-power laser, and various welding frameworks. In addition, significant advances in nano designing have opened up the chance of utilizing nanomaterials to treat various types of human body tumors, pharmacological medicines, artificial organs (lungs, heart surgery), and cancer therapy, etc. Nanofluids, presented by Choi and Eastman [1] in 1995, has gotten impressive consideration in present times. Makinde [2] examined numerically that the boundary layer flow induces in the nanofluid cause of a linear stretching surface with the influence of Brownian motion, and thermophoresis. Sarafraz et al. [3] explore the nanofluid flow as a potential coolant. Uddin et al. [4] deliberated about nanofluid through a permeable medium with thermal convective boundary conditions. Khan and Pop [5] provided nanoliquids flow through the Buongiorno method, which had passed over the stretching surface. Turkyilmazoglu [6] established a closed-form arrangement for the movement of different nanomaterials through a curved vertical surface.

The flow of boundary layer due to continuous stretching of sheet is an important type of flow happening in chemical industries and engineering [7]. These incorporate fluid metal, paper industry, lubricants, and fiber synthesis. The non-Newtonian liquids, including paints, colloidal and organic fluids, biopolymers, and food industry. Many researchers utilized different geometries to investigate several aspects of non-Newtonian fluid like Casson flow over a radially stretching sheet [8], Maxwell fluid [9], Micropolar Nanofluid [10], and Casson Nano-Fluid [11]. Nandy et al. [12] presented a study to investigate the impacts of velocity slip on nanofluid along with extending surface. Numerous current kinds of previous studies on non-Fourier energy equations are considered.

Magnetohydrodynamics (MHD) engagement has significant applications in the zone of drug, space science, various machines, and vitality generators. Tiwana et al. [13] investigate the MHD convective flow which is portrayed in the non-Newtonian liquid model. Naz et al. [14] delineated the impacts of MHD flow in a channel geometry. Hussain et al. [15] examined the heat transportation in MHD non-Newtonian boundary layer flow over a shrinking sheet. The heat transfer through a medium of porous consider flow over a exponentially shrinking surface studied by Ali et al. [16]. Khan et al. [17] investigated the magnetohydrodynamic axisymmetric buoyant nanofluid along with chemical reaction and radiation.

In recent years, the examination of liquid and problems of heat transport in the rotating frame is completely charming matter [18]. It is an aftereffect of their titanic applications in the assembling of crystal development, biomechanics, turbomechanics, food industry, gas turbine rotors, filtration process, and cosmic fluid dynamics [19]. The chief undertaking toward this way was made by Wang [20]. The effect of magnetohydrodynamics (MHD) in rotating fluid is concentrated by Takhar et al. [21]. Recently, published research articles on rotating flow are Powell–Eyring nanofluid [22], MHD ferrofluid [23], and viscous dissipation impacts on nanofluids [24]. The effects of Darcy Forchheimer in a porous medium are considered to study the velocity field within the examination of the rotating frame [25]. Hayat et al. [26] investigated the movement of nanoliquids distracted in a porous medium by applying the Darcy–Forchheimer model. The Hall effects in either the two-phase flow of dusty nanofluid were documented by Gireesha and co-workers [27]. Hayat et al. [28] examined the Darcy–Forchheimer 3D along with Arrhenius activation energy.

Christov [29] recommended effective speculation of Fourier law regarding thermal relaxation impact which is characterized as the time expected to develop steady-state heat transportation once the temperature gradient is introduced. Modeling of Cattaneo–Christov along with nanofluid was examined by [30]. Hayat et al. [31] examined the effects of heat flux Cattaneo–Christov model on variable thermal conductivity. Ibrahim et al. [32] represent the convective flow of Eyring–Powell nano-fluid by using Cattaneo–Christov model. Waqas [33] scrutinized the impact of thermal relaxation time on the boundary layer flow. Hayat et al. [34] describes the effect of Cattaneo–Christov heat flux in the stagnation point fluid flow over a nonlinear stretching sheet with fluctuating thickness.

The reason for this work is to generalize the work of Adnan et al. [35] through the incorporation of magnetohydrodynamics impacts with Cattaneo–Christov double diffusion for the time-dependent rotational flow of Casson nanofluids due to a horizontally stretching surface. The fundamental goal of this comprehensive examination is the upgrade of heat transportation with the double diffusion model of Cattaneo–Christov. As far as the authors know, these aspects of the problem have not been considered in previous examinations. The resulting nonlinear partial differential formulation is solved with utilization of widely validated finite element discretization because it solves boundary value problem adequately, rapidly, and precisely [36]. The validation of the numerical outcomes and Matlab code is confirmed in the face of previously available data for limiting cases. Furthermore, pictorial representations of some principal findings with a detailed discussion have also been presented.

2. Statement of the Problem

The time-dependent 3D magnetohydrodynamics of an incompressible Casson nanofluid flow over an extending sheet with a rotating frame of reference are considered as shown in Figure 1. Physically, we consider that the entire system is at rest in the time $t<0$; however, for $t=0$, the sheet is extended in the x-direction at $z=0$ with angular velocity $\Omega$. The mass and heat transfer component is examined through the heat flux model of Cattaneo–Christov double diffusion expressions. The framework is rotating with angular velocity ($\Omega$) along the z-direction. In the z-direction, $B_o$ (magnetic field) is applied, the instigated magnetic field is overlooked due to a small magnetic Reynolds number, and Ohmic dissipation and Hall’s current impacts are ignored since the field of magnetic is not too much strong [37]. Moreover, we assume that the concentration and temperature at the surface are ${\tilde{C}}_w,{\tilde{T}}_w$, respectively, and the ambient concentration and temperature are ${\tilde{C}}_{\infty },{\tilde{T}}_{\infty }$, respectively. The rheological model for the flow of a Casson fluid can be written as:

$${\tau }_{ij}=\left\{\begin{array}{c}\hfill 2\left({\mu }_B+\frac{p_y}{\sqrt{2\pi }}\right)e_{ij},\pi >{\pi }_c\\ \hfill 2\left({\mu }_B+\frac{p_y}{\sqrt{2{\pi }_c}}\right)e_{ij},\pi <{\pi }_c\end{array}\right.$$

(1)

In Equation (1), ${\pi }_{ij}$, $e_{ij}$, $p_y$, ${\mu }_B$, ${\pi }_c$, and $\pi =e_{ij}{e}_{ij}$ are Cauchy stress tensor, deformation rate components (i,j), yield stress of fluid, Casson fluid plastic dynamics viscosity, non-Newtonian based critical values of this product, and product of components of deformation rate with itself, respectively.

3. Governing Equations

Considering the above suppositions, the consistent mass, momentum, energy, and conservation of nanoparticles volume fraction equations in a Cartesian coordinate system (x, y, z) as follows [35,38,39,40]:

$$\begin{array}{c}{\tilde{u}}_x+{\tilde{v}}_y+{\tilde{w}}_z=0,\end{array}$$

(2)

$$\begin{array}{c}{\rho }_{nf}({\tilde{u}}_t+\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z-2\Omega \tilde{v})=-{\tilde{p}}_x+{\mu }_{nf}(1+\frac{1}{\beta }){\tilde{u}}_{zz}-{\sigma }_{nf}{B}_0^2\tilde{u},\end{array}$$

(3)

$$\begin{array}{c}{\rho }_{nf}({\tilde{v}}_t+\tilde{u}{\tilde{v}}_x+\tilde{v}{\tilde{v}}_y+\tilde{w}{\tilde{v}}_z+2\Omega \tilde{u})=-{\tilde{p}}_y+{\mu }_{nf}(1+\frac{1}{\beta }){\tilde{v}}_{zz}-{\sigma }_{nf}{B}_0^2\tilde{v},\end{array}$$

(4)

$$\begin{array}{c}{\rho }_{nf}({\tilde{w}}_t+\tilde{u}{\tilde{w}}_x+\tilde{v}{\tilde{w}}_y+\tilde{w}{\tilde{w}}_z)=-{\tilde{p}}_z+{\mu }_{nf}(1+\frac{1}{\beta }){\tilde{w}}_{zz},\end{array}$$

(5)

$$\begin{array}{c}{\tilde{T}}_t+\tilde{u}{\tilde{T}}_x+\tilde{v}{\tilde{T}}_y+\tilde{w}{\tilde{T}}_z+{\lambda }_1{\tilde{T}}_{\lambda }={\tilde{\alpha }}_{nf}{\tilde{T}}_{zz}+{\tilde{\tau }}^{*}\left\{{\tilde{D}}_B\left({\tilde{C}}_z{\tilde{T}}_z\right)+\frac{{\tilde{D}}_T}{{\tilde{T}}_{\infty }}\left({\tilde{T}}_z^2\right)\right\}+\frac{{\tilde{Q}}_o}{\rho C_p}(\tilde{T}-{\tilde{T}}_{\infty }),\end{array}$$

(6)

$$\begin{array}{c}{\tilde{C}}_t+\tilde{u}{\tilde{C}}_x+\tilde{v}{\tilde{C}}_y+\tilde{w}{\tilde{C}}_z+{\lambda }_2{\tilde{C}}_{\lambda }={\tilde{D}}_B{\tilde{C}}_{zz}+\frac{{\tilde{D}}_T}{{\tilde{T}}_{\infty }}{\tilde{T}}_{zz}.\end{array}$$

(7)

where

$$\begin{array}{c}{\tilde{T}}_{\lambda }={\tilde{u}}^2{\tilde{T}}_{xx}+{\tilde{v}}^2{\tilde{T}}_{yy}+{\tilde{w}}^2{\tilde{T}}_{zz}+2\tilde{u}\tilde{v}{\tilde{T}}_{xy}+2\tilde{v}\tilde{w}{\tilde{T}}_{yz}+2\tilde{u}\tilde{w}{\tilde{T}}_{xz}+(\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z){\tilde{T}}_x\\ +(\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z){\tilde{T}}_y+(\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z){\tilde{T}}_z,\\ {\tilde{C}}_{\lambda }={\tilde{u}}^2{\tilde{C}}_{xx}+{\tilde{v}}^2{\tilde{C}}_{yy}+{\tilde{w}}^2{\tilde{C}}_{zz}+2\tilde{u}\tilde{v}{\tilde{C}}_{xy}+2\tilde{v}\tilde{w}{\tilde{C}}_{yz}+2\tilde{u}\tilde{w}{\tilde{C}}_{xz}+(\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z){\tilde{C}}_x\\ +(\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z){\tilde{C}}_y+(\tilde{u}{\tilde{u}}_x+\tilde{v}{\tilde{u}}_y+\tilde{w}{\tilde{u}}_z){\tilde{C}}_z.\end{array}$$

Here, ($\tilde{u},\tilde{v}$, $\tilde{w}$) are components of velocity in directions ($x,y,z$), respectively, ${\rho }_{nf}$, ${\tilde{\alpha }}_{nf}$, ${\mu }_{nf}$, ${\sigma }_{nf}$, ${\lambda }_1$, and ${\lambda }_2$ are respectively the density, thermal diffusivity, dynamic viscosity, electrical conductivity, relaxation time of heat, and mass fluxes of the nanofluid. $\tilde{T}$ and $\tilde{C}$ are the fluid temperature and nanoparticle volume concentration, ${\tilde{D}}_B$ and ${\tilde{D}}_T$ are the Brownian and thermophoretic diffusion coefficient, respectively, Furthermore, t and C are respectively time and concentration of nanoparticles’ volume fraction. The current physical elaborated problem, characterized boundary conditions, are [38,41]:

$$\begin{array}{c}t<0:\tilde{u}=\tilde{v}=\tilde{w}=0,\tilde{T}={\tilde{T}}_{\infty },\tilde{C}={\tilde{C}}_{\infty },\end{array}$$

(8)

$$\begin{array}{c}t\ge 0:\tilde{u}={\tilde{u}}_w=\tilde{a}x,\tilde{v}=\tilde{w}=0,\tilde{T}={\tilde{T}}_w,\tilde{C}={\tilde{C}}_w,asz=0,\end{array}$$

(9)

$$\begin{array}{c}t\ge 0:\tilde{u}\to 0,\tilde{v}\to 0,\tilde{T}\to {\tilde{T}}_{\infty },\tilde{C}\to {\tilde{C}}_{\infty },asz\to \infty .\end{array}$$

(10)

We offer a following set of transformation variables to proceed the analysis (see [38,42,43]):

$$\begin{array}{c}\left.\begin{array}{c}\hfill \eta =\sqrt{\frac{\tilde{a}x}{\xi \nu }}z,\widetilde{f^{\prime }}(\xi ,\eta )=\frac{\tilde{u}}{\tilde{a}x},\tilde{h}(\xi ,\eta )=\frac{\tilde{v}}{\tilde{a}x},\tilde{f}(\xi ,\eta )=\frac{\tilde{w}}{-\sqrt{\tilde{a}\nu \xi }},\xi =1-e^{-\tau }\\ \hfill \tau =\tilde{a}t,\tilde{\theta }(\xi ,\eta )={\left(\frac{({\tilde{T}}_w-{\tilde{T}}_{\infty })}{(\tilde{T}-{\tilde{T}}_{\infty })}\right)}^{-1},\tilde{\varphi }(\xi ,\eta )={\left(\frac{({\tilde{C}}_w-{\tilde{C}}_{\infty })}{(\tilde{C}-{\tilde{C}}_{\infty })}\right)}^{-1}\end{array}\right\}\end{array}$$

(11)

The continuity of Equation (2) is satisfied identically using the similarity transformations above. In light of Equation (11), Equations (3)–(10) reduce into the following nonlinear PDEs in the transformed coordinate system ($\xi ,\eta$):

$$\begin{array}{c}(1+{\beta }^{-1}){\tilde{f}}^{‴}+\frac{1}{2}\eta {\tilde{f}}^{″}-\frac{1}{2}\xi \eta {\tilde{f}}^{″}+\xi (2\lambda \tilde{h}+\tilde{f}{\tilde{f}}^{″}-{\tilde{f}}^{\prime 2}-M^2{\tilde{f}}^{\prime })=\xi \frac{\partial {\tilde{f}}^{\prime }}{\partial \xi }-{\xi }^2\frac{\partial {\tilde{f}}^{\prime }}{\partial \xi },\end{array}$$

(12)

$$(1+{\beta }^{-1}){\tilde{h}}^{″}+\frac{1}{2}\eta {\tilde{h}}^{\prime }-\frac{1}{2}\xi \eta {\tilde{h}}^{\prime }+\xi (\tilde{f}{\tilde{h}}^{\prime }-{\tilde{f}}^{\prime }\tilde{h}-2\lambda {\tilde{f}}^{\prime }-M^2\tilde{h})=\frac{\partial \tilde{h}}{\partial \xi }-{\xi }^2\frac{\partial \tilde{h}}{\partial \xi },$$

(13)

$$\begin{array}{c}{\tilde{\theta }}^{″}+\frac{1}{2}\eta Pr{\tilde{\theta }}^{\prime }-\frac{1}{2}Pr\xi \eta {\tilde{\theta }}^{\prime }+Pr\xi \tilde{f}{\tilde{\theta }}^{\prime }+PrNb{\tilde{\theta }}^{\prime }{\tilde{\varphi }}^{\prime }+NtPr{\left({\tilde{\theta }}^{\prime }\right)}^2-Pr\xi {\gamma }_T{\theta }_{rT}+Pr\xi Q_s\tilde{\theta }\\ =Pr\xi \frac{\partial \theta }{\partial \xi }-Pr{\xi }^2\frac{\partial \theta }{\partial \xi },\end{array}$$

(14)

$$\begin{array}{c}{\tilde{\varphi }}^{″}+\frac{1}{2}\eta Le{\tilde{\varphi }}^{\prime }-\frac{1}{2}Le\xi \eta {\tilde{\varphi }}^{\prime }+Le\xi \tilde{f}{\tilde{\varphi }}^{\prime }+\frac{Nt}{Nb}\left({\tilde{\theta }}^{″}\right)-Le\xi {\gamma }_C{\varphi }_{rc}=Le\xi \frac{\partial \varphi }{\partial \xi }-Le{\xi }^2\frac{\partial \varphi }{\partial \xi },\end{array}$$

(15)

$$\left.\begin{array}{c}\tilde{f}(\xi ,\eta )={\tilde{f}}^{\prime }(\xi ,\eta )=\tilde{h}(\xi ,\eta )=0,\tilde{\theta }(\xi ,\eta )=\tilde{\varphi }(\xi ,\eta )=1,at\eta =0,\hfill \\ {\tilde{f}}^{\prime }(\xi ,\eta )\to 0,\tilde{h}(\xi ,\eta )\to 0,\tilde{\theta }(\xi ,\eta )\to 0,\tilde{\varphi }(\xi ,\eta )\to 0,as\eta \to \infty .\hfill \end{array}\right\}$$

(16)

where ${\theta }_{rT}=\tilde{f}{\tilde{f}}^{\prime }{\tilde{\theta }}^{\prime }+{\tilde{f}}^2{\tilde{\theta }}^{″}$, and ${\varphi }_{rc}=\tilde{f}{\tilde{f}}^{\prime }{\tilde{\varphi }}^{\prime }+{\tilde{f}}^2{\tilde{\varphi }}^{″}$. The come into view parameters in Equations (12)–(15) are defined as:

$$\begin{array}{c}\beta ={\left(\frac{p_y}{{\mu }_B\sqrt{2{\pi }_c}}\right)}^{-1},\lambda ={\left(\frac{a}{\Omega }\right)}^{-1},M={\left(\frac{\rho a}{\sigma B_o^2}\right)}^{-1/2},Pr={\left(\frac{{\tilde{\alpha }}_{nf}}{\nu }\right)}^{-1},Le={\left(\frac{{\tilde{D}}_B}{\nu }\right)}^{-1}\\ Nb={\left(\frac{\nu }{\tau {\tilde{D}}_B({\tilde{C}}_w-{\tilde{C}}_{\infty })}\right)}^{-1},Nt={\left(\frac{\nu {\tilde{T}}_{\infty }}{\tau {\tilde{D}}_T({\tilde{T}}_w-{\tilde{T}}_{\infty })}\right)}^{-1},{\gamma }_T={\lambda }_1\tilde{a},{\gamma }_C={\lambda }_2\tilde{a},Q_s={\left(\frac{a\rho C_p}{{\tilde{Q}}_o}\right)}^{-1}.\end{array}$$

where $\beta$, $\lambda$, M, $Pr$, $Le$, $Nb$, $Nt$, ${\gamma }_T$, ${\gamma }_C$, and $Q_s$ are the Casson fluid parameter, rotating parameter, magnetic parameter, Prandtl number, Lewis number, Brownian motion, thermophoresis, thermal relaxation parameter, concentration relaxation parameter, and heat source, respectively. When $\tau \to \infty$, $\xi =1$, then the Equations (12)–(15) become:

$$\begin{array}{c}(1+{\beta }^{-1}){\tilde{f}}^{‴}+2\lambda \tilde{h}-{\tilde{f}}^{\prime 2}+\tilde{f}{\tilde{f}}^{″}-M^2{\tilde{f}}^{\prime }=0,\end{array}$$

(17)

$$\begin{array}{c}(1+{\beta }^{-1}){\tilde{h}}^{″}-{\tilde{f}}^{\prime }\tilde{h}+\tilde{f}{\tilde{h}}^{\prime }-2\lambda {\tilde{f}}^{\prime }-M^2\tilde{h}=0,\end{array}$$

(18)

$$\begin{array}{c}{\tilde{\theta }}^{″}+Pr\tilde{f}{\tilde{\theta }}^{\prime }+NbPr{\tilde{\theta }}^{\prime }{\tilde{\varphi }}^{\prime }+NtPr{\left({\tilde{\theta }}^{\prime }\right)}^2-Pr{\gamma }_T{\theta }_{rT}+Pr\xi Q_s\tilde{\theta }=0,\end{array}$$

(19)

$$\begin{array}{c}{\tilde{\varphi }}^{″}+\frac{Nt}{Nb}\left({\tilde{\theta }}^{″}\right)+Le\tilde{f}{\tilde{\varphi }}^{\prime }-Le{\gamma }_C{\varphi }_{rc}=0.\end{array}$$

(20)

subject to the boundary conditions (16)

Skin friction coefficient expressions, local Nusselt number, and Sherwood number are defined as:

$$\begin{array}{c}{Cf}_x={\left(\frac{\rho {\tilde{u}}^2}{{\tau }_w^x}\right)}^{-1},{Cf}_y={\left(\frac{\rho {\tilde{u}}^2}{{\tau }_w^y}\right)}^{-1},Nu={\left(\frac{\kappa ({\tilde{T}}_w-{\tilde{T}}_{\infty })}{x{q}_w}\right)}^{-1},Shr={\left(\frac{{\tilde{D}}_B({\tilde{C}}_w-{\tilde{C}}_{\infty })}{x{q}_m}\right)}^{-1}.\end{array}$$

(21)

where the skin friction tensor at wall are ${\tau }_w^x=\mu {\left[{\tilde{u}}_z\right]}_{z=0}$ (x-direction) and ${\tau }_w^y=\mu {\left[{\tilde{v}}_z\right]}_{z=0}$ (y-direction), the wall heat transfer is $q_w=-\kappa {\left[{\tilde{T}}_z\right]}_{z=0}$, and the mass flux from the sheet is $q_m=-{\tilde{D}}_B{\left[{\tilde{C}}_z\right]}_{z=0}$. By the aid of similarity transformation Equation (15), we get:

$$\left\{\begin{array}{c}\hfill C{f}_x{R{e}_x}^{1/2}=(1+{\beta }^{-1})f^{″}\left(0\right){\xi }^{-1/2},C{f}_y{R{e}_x}^{1/2}=(1+{\beta }^{-1})h^{\prime }\left(0\right){\xi }^{-1/2},\\ \hfill N{u}_x{R{e}_x}^{1/2}=-{\theta }^{\prime }\left(0\right){\xi }^{-1/2},Sh{r}_x{R{e}_x}^{1/2}=-{\varphi }^{\prime }\left(0\right){\xi }^{-1/2}\end{array}\right.$$

(22)

4. Finite Element Method Solutions

The transformed set of nonlinear partial differential Equations (12)–(15) is solved numerically utilizing the variational finite element method along with boundary conditions (Equation (16)) because Equations (12)–(15) cannot be solved analytically due to highly nonlinearity. This procedure is a great numerical computational methodology significant for solving the different types of real word problems [44] and problems of engineering [45]—for example, liquids with heat transportation [46], Bio-materials [47], rigid body dynamics [48], and various regions [49,50]. An astounding general description of variational finite elements method outlined by Reddy [51] and Jyothi et al. [52] summed up the basic steps involved in the FEM. Basically, the technique includes a continuous piecewise function for the solution and to get the functions parameters in an efficient way that minimizes the error [53]. The FEM solves boundary value problem adequately, rapidly, and precisely [54]. To reduce the order of nonlinear differential Equations (12)–(16), firstly we consider:

$$\begin{array}{c}{\tilde{f}}^{\prime }=\tilde{p},\end{array}$$

(23)

The set of Equations (12)–(16) thus reduces to

$$\begin{array}{c}(1+{\beta }^{-1}){\tilde{p}}^{″}+\frac{1}{2}\eta {\tilde{p}}^{\prime }-\frac{1}{2}\xi \eta {\tilde{p}}^{\prime }+\xi (2\lambda \tilde{h}+\tilde{f}{\tilde{p}}^{\prime }-{\tilde{p}}^2-M^2\tilde{p})=\xi \frac{\partial \tilde{p}}{\partial \xi }-{\xi }^2\frac{\partial \tilde{p}}{\partial \xi },\end{array}$$

(24)

$$\begin{array}{c}(1+{\beta }^{-1}){\tilde{h}}^{″}+\frac{1}{2}\eta {\tilde{h}}^{\prime }-\frac{1}{2}\xi \eta {\tilde{h}}^{\prime }+\xi (\tilde{f}{\tilde{h}}^{\prime }-\tilde{p}h-2\lambda \tilde{p}-M^2\tilde{h})=\xi \frac{\partial \tilde{h}}{\partial \xi }-{\xi }^2\frac{\partial \tilde{h}}{\partial \xi },\\ {\tilde{\theta }}^{″}+\frac{1}{2}Pr\eta {\tilde{\theta }}^{\prime }-\frac{1}{2}Pr\xi \eta {\tilde{\theta }}^{\prime }+Pr\xi \tilde{f}{\tilde{\theta }}^{\prime }+NbPr{\tilde{\theta }}^{\prime }{\tilde{\varphi }}^{\prime }+NtPr{\left({\tilde{\theta }}^{\prime }\right)}^2-Pr\xi {\gamma }_T({\tilde{f}}^2{\tilde{\theta }}^{″}+\tilde{f}\tilde{p}{\tilde{\theta }}^{\prime })+\end{array}$$

(25)

$$\begin{array}{c}Pr\xi Q_s\tilde{\theta }=Pr\xi \frac{\partial \tilde{\theta }}{\partial \xi }-Pr{\xi }^2\frac{\partial \tilde{\theta }}{\partial \xi },\end{array}$$

(26)

$$\begin{array}{c}{\tilde{\varphi }}^{″}+\frac{1}{2}Le\eta {\tilde{\varphi }}^{\prime }-\frac{1}{2}Le\xi \eta {\tilde{\varphi }}^{\prime }+Le\xi \tilde{f}{\tilde{\varphi }}^{\prime }+\frac{Nt}{Nb}{\left({\tilde{\theta }}^{\prime }\right)}^2-Le\xi {\gamma }_C({\tilde{f}}^2{\tilde{\varphi }}^{″}+\tilde{f}\tilde{p}{\tilde{\varphi }}^{\prime })=Le\xi \frac{\partial \tilde{\varphi }}{\partial \xi }-Le{\xi }^2\frac{\partial \tilde{\varphi }}{\partial \xi },\end{array}$$

(27)

$$\left.\begin{array}{c}\tilde{f}(\xi ,\eta )=\tilde{p}(\xi ,\eta )=\tilde{h}(\xi ,\eta )=0,\tilde{\theta }(\xi ,\eta )=\tilde{\varphi }(\xi ,\eta )=1,at\eta =0,\hfill \\ \tilde{p}(\xi ,\eta )\to 0,\tilde{h}(\xi ,\eta )\to 0,\tilde{\theta }(\xi ,\eta )\to 0,\tilde{\varphi }(\xi ,\eta )\to 0,as\eta \to \infty .\hfill \end{array}\right\}$$

(28)

4.1. Variational Formulations

Over a typical rectangular element ${\Omega }_e$, the associated variational form with Equations (23)–(27) is given by

$$\begin{array}{c}\underset{{\Omega }_e}{\int }{w}_{f1}\{f^{\prime }-p\}d{\Omega }_e=0,\end{array}$$

(29)

$$\begin{array}{c}\underset{{\Omega }_e}{\int }{w}_{f2}\left\{(1+{\beta }^{-1}){\tilde{p}}^{″}+\frac{1}{2}\eta {\tilde{p}}^{\prime }-\frac{1}{2}\xi \eta {\tilde{p}}^{\prime }+\xi (2\lambda \tilde{h}+\tilde{f}{\tilde{p}}^{\prime }-{\tilde{p}}^2-M^2\tilde{p})-\xi \frac{\partial \tilde{p}}{\partial \xi }+{\xi }^2\frac{\partial \tilde{p}}{\partial \xi }\right\}d{\Omega }_e=0,\end{array}$$

(30)

$$\begin{array}{c}\underset{{\Omega }_e}{\int }{w}_{f3}\left\{(1+{\beta }^{-1}){\tilde{h}}^{″}+\frac{1}{2}\eta {\tilde{h}}^{\prime }-\frac{1}{2}\xi \eta {\tilde{h}}^{\prime }+\xi (\tilde{f}{\tilde{h}}^{\prime }-\tilde{p}h-2\lambda \tilde{p}-M^2\tilde{h})-\xi \frac{\partial \tilde{h}}{\partial \xi }+{\xi }^2\frac{\partial \tilde{h}}{\partial \xi }\right\}d{\Omega }_e,\\ \underset{{\Omega }_e}{\int }{w}_{f4}\{{\tilde{\theta }}^{″}+\frac{1}{2}Pr\eta {\tilde{\theta }}^{\prime }-\frac{1}{2}Pr\xi \eta {\tilde{\theta }}^{\prime }+Pr\xi \tilde{f}{\tilde{\theta }}^{\prime }+NbPr{\tilde{\theta }}^{\prime }{\tilde{\varphi }}^{\prime }+NtPr{\left({\tilde{\theta }}^{\prime }\right)}^2-Pr\xi {\gamma }_T({\tilde{f}}^2{\tilde{\theta }}^{″}+\tilde{f}\tilde{p}{\tilde{\theta }}^{\prime })+\end{array}$$

(31)

$$\begin{array}{c}Pr\xi Q_s\tilde{\theta }-Pr\xi \frac{\partial \tilde{\theta }}{\partial \xi }+Pr{\xi }^2\frac{\partial \tilde{\theta }}{\partial \xi }\}d{\Omega }_e=0,\\ \underset{{\Omega }_e}{\int }{w}_{f5}\{{\tilde{\varphi }}^{″}+\frac{1}{2}Le\eta {\tilde{\varphi }}^{\prime }-\frac{1}{2}Le\xi \eta {\tilde{\varphi }}^{\prime }+Le\xi \tilde{f}{\tilde{\varphi }}^{\prime }+\frac{Nt}{Nb}{\left({\tilde{\theta }}^{\prime }\right)}^2-Le\xi {\gamma }_C({\tilde{f}}^2{\tilde{\varphi }}^{″}+\tilde{f}\tilde{p}{\tilde{\varphi }}^{\prime })\end{array}$$

(32)

$$\begin{array}{c}-Le\xi \frac{\partial \tilde{\varphi }}{\partial \xi }+Le{\xi }^2\frac{\partial \tilde{\varphi }}{\partial \xi }\}d{\Omega }_e=0.\end{array}$$

(33)

where $w_{fs}(s=1,2,3,4,5)$ are arbitrary weight functions or trial functions.

4.2. Finite Element Formulations

Let us divide the rectangular domain (${\Omega }_e$) into 4-noded (rectangular element) and (${\xi }_i,{\eta }_j$) be the domain grid points (see Figure 2). The length of plate and thickness of boundary layer is fixed at ${\xi }_{max}=2$, ${\eta }_{max}=5$, respectively. The finite model of the element can be obtained from Equations (29)–(33) by replacing the following form of finite element approximations:

$$\begin{array}{c}\hfill \tilde{f}=\sum _{j=1}^4{\tilde{f}}_j{\Psi }_j(\xi ,\eta ),\tilde{p}=\sum _{j=1}^4{\tilde{p}}_j{\Psi }_j(\xi ,\eta ),\tilde{h}=\sum _{j=1}^4{\tilde{h}}_j{\Psi }_j(\xi ,\eta ),{\tilde{\theta }}^{\prime }=\sum _{j=1}^4{\tilde{\theta }}_j^{\prime }{\Psi }_j(\xi ,\eta ),{\tilde{\varphi }}^{\prime }=\sum _{j=1}^4{\tilde{\varphi }}_j^{\prime }{\Psi }_j(\zeta ,\eta ),\end{array}$$

(34)

where ${\Psi }_j$ ($j=1,2,3,4$) are the linear interpolation functions for a rectangular element ${\Omega }_e$ (see Figure 2) and are given by:

$${\Psi }_1=\frac{({\xi }_{e+1}-\xi )({\eta }_{e+1}-\eta )}{({\xi }_{e+1}-{\xi }_e)({\eta }_{e+1}-{\eta }_e)},{\Psi }_2=\frac{(\xi -{\xi }_e)({\eta }_{e+1}-\eta )}{({\xi }_{e+1}-{\xi }_e)({\eta }_{e+1}-{\eta }_e)},$$

$${\Psi }_3=\frac{(\xi -{\xi }_e)(\eta -{\eta }_e)}{({\xi }_{e+1}-{\xi }_e)({\eta }_{e+1}-{\eta }_e)},{\Psi }_4=\frac{({\xi }_{e+1}-\xi )(\eta -{\eta }_e)}{({\xi }_{e+1}-{\xi }_e)({\eta }_{e+1}-{\eta }_e)}.$$

(35)

The model of finite elements of the equations thus developed is given by:

$$\left[\begin{array}{ccccc}\left[L^{11}\right]& \left[L^{12}\right]& \left[L^{13}\right]& \left[L^{14}\right]& \left[L^{15}\right]\\ [L^{21}]& \left[L^{22}\right]& \left[L^{23}\right]& \left[L^{24}\right]& \left[L^{25}\right]\\ [L^{31}]& \left[L^{32}\right]& \left[L^{33}\right]& \left[L^{34}\right]& \left[L^{35}\right]\\ [L^{41}]& \left[L^{42}\right]& \left[L^{43}\right]& \left[L^{44}\right]& \left[L^{45}\right]\\ [L^{51}]& \left[L^{52}\right]& \left[L^{53}\right]& \left[L^{54}\right]& \left[L^{55}\right]\end{array}\right]\left[\begin{array}{c}\left\{f\right\}\\ \left\{q\right\}\\ \left\{h\right\}\\ \left\{\theta \right\}\\ \left\{\varphi \right\}\end{array}\right]=\left[\begin{array}{c}\left\{r_1\right\}\\ \left\{r_2\right\}\\ \left\{r_3\right\}\\ \left\{r_4\right\}\\ \left\{r_5\right\}\end{array}\right]$$

(36)

where $\left[W_{mn}\right]$ and $\left[b_m\right]$ ($m,n=1,2,3,4$) are defined as:

$$\begin{array}{ccc}\hfill L_{ij}^{11}& =& \underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e,L_{ij}^{12}=-\underset{{\Omega }_e}{\int }{\Psi }_i{\Psi }_{j}d{\Omega }_e,L_{ij}^{13}=L_{ij}^{14}=L_{ij}^{15}=L_{ij}^{21}==L_{ij}^{24}=L_{ij}^{25}=L_{ij}^{31}=0,\hfill \\ \hfill L_{ij}^{22}& =& -(1+{\beta }^{-1})\underset{{\Omega }_e}{\int }\frac{d{\Psi }_i}{d\eta }\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+\frac{\eta }{2}(1-\xi )\underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+\xi \underset{{\Omega }_e}{\int }\overline{\tilde{f}}{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e-\xi \underset{{\Omega }_e}{\int }\overline{\tilde{p}}{\Psi }_i{\Psi }_{j}d{\Omega }_e\hfill \\ & -& M^2\underset{{\Omega }_e}{\int }\xi {\Psi }_i{\Psi }_{j}d{\Omega }_e-\xi (1-\xi )\underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\xi }d{\Omega }_e,L_{ij}^{23}=-2\lambda \xi \underset{{\Omega }_e}{\int }{\Psi }_i{\Psi }_{j}d{\Omega }_e,L_{ij}^{32}=-2\lambda \xi \underset{{\Omega }_e}{\int }{\Psi }_i{\Psi }_{j}d{\Omega }_e,\hfill \\ \hfill L_{ij}^{33}& =& -(1+{\beta }^{-1})\underset{{\Omega }_e}{\int }\frac{d{\Psi }_i}{d\eta }\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+\frac{\eta }{2}(1-\xi )\underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+\xi \underset{{\Omega }_e}{\int }\overline{\tilde{f}}{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e-\xi \underset{{\Omega }_e}{\int }\overline{\tilde{p}}{\Psi }_i{\Psi }_{j}d{\Omega }_e\hfill \\ & -& M^2\underset{{\Omega }_e}{\int }\xi {\Psi }_i{\Psi }_{j}d{\Omega }_e-\xi (1-\xi )\underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\xi }d{\Omega }_e,L_{ij}^{34}=L_{ij}^{35}=L_{ij}^{41}=L_{ij}^{42}=L_{ij}^{43}=L_{ij}^{45}=L_{ij}^{51}=0,\hfill \end{array}$$

$$\begin{array}{ccc}\hfill L_{ij}^{44}& =& -\underset{{\Omega }_e}{\int }\frac{d{\Psi }_i}{d\eta }\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+\frac{1}{2}(1-\xi )Pr\eta \underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+Pr\xi \underset{{\Omega }_e}{\int }\overline{\tilde{f}}{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+PrNb\underset{{\Omega }_e}{\int }{\overline{\tilde{\varphi }}}^{\prime }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e\hfill \\ & +& PrNt\underset{{\Omega }_e}{\int }{\overline{\tilde{\theta }}}^{\prime }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+Pr{\gamma }_T\underset{{\Omega }_e}{\int }\xi {\overline{\tilde{f}}}^2\frac{d{\Psi }_i}{d\eta }\frac{d{\Psi }_j}{d\eta }d{\Omega }_e-Pr{\gamma }_T\xi \underset{{\Omega }_e}{\int }\overline{\tilde{f}}\overline{\tilde{p}}{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e-Pr\xi (1-\xi )\underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\xi }d{\Omega }_e,\hfill \\ \hfill L_{ij}^{52}& =& L_{ij}^{53}=0,L_{ij}^{54}=-\frac{Nt}{Nb}\underset{{\Omega }_e}{\int }\frac{d{\Psi }_i}{d\eta }\frac{d{\Psi }_j}{d\eta }d{\Omega }_e,L_{ij}^{55}=-\underset{{\Omega }_e}{\int }\frac{d{\Psi }_i}{d\eta }\frac{d{\Psi }_j}{d\eta }d{\Omega }_e+\frac{Le}{2}(1-\xi )\eta \underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e\hfill \\ & +& Le\xi \underset{{\Omega }_e}{\int }\overline{\tilde{f}}{\Psi }_i\frac{d{\Psi }_j}{d\eta }d{\Omega }_e-Le\xi (1-\xi )\underset{{\Omega }_e}{\int }{\Psi }_i\frac{d{\Psi }_j}{d\xi }d{\Omega }_e\hfill \end{array}$$

and

$$\begin{array}{c}{r}_i^1=0,r_i^2=-\underset{{\mathsf{\Gamma }}_e}{\oint }{\Psi }_i{n}_{\eta }\frac{\partial \tilde{p}}{\partial \eta }ds,r_i^3=-\underset{{\mathsf{\Gamma }}_e}{\oint }{\Psi }_i{n}_{\eta }\frac{\partial \tilde{h}}{\partial \eta }ds,r_i^4=-\underset{{\mathsf{\Gamma }}_e}{\oint }{\Psi }_i{n}_{\eta }\frac{\partial \tilde{\theta }}{\partial \eta }ds,\\ r_i^5=-\underset{{\mathsf{\Gamma }}_e}{\oint }{\Psi }_i{n}_{\eta }\frac{\partial \tilde{\varphi }}{\partial \eta }ds-\frac{Nt}{Nb}\underset{{\mathsf{\Gamma }}_e}{\oint }{\Psi }_i{n}_{\eta }\frac{\partial \tilde{\theta }}{\partial \eta }ds.\end{array}$$

(37)

where the known values to be considered are $\overline{\tilde{f}}={\sum }_{j=1}^4{\overline{\tilde{f}}}_j{\Psi }_j$, $\overline{\tilde{p}}={\sum }_{j=1}^4{\overline{\tilde{p}}}_j{\Psi }_j$, $\overline{\tilde{h}}={\sum }_{j=1}^4{\overline{\tilde{h}}}_j{\Psi }_j$, ${\overline{\tilde{\theta }}}^{\prime }={\sum }_{j=1}^4{\overline{\tilde{\theta }}}_j^{\prime }{\Psi }_j$, and ${\overline{\tilde{\varphi }}}^{\prime }={\sum }_{j=1}^4{\overline{\tilde{\varphi }}}_j^{\prime }{\Psi }_j$. The flow domain is divided into $101\times 101$ rectangular elements of similar size of grid points. Five functions can be assessed at each node, and 51,005 equations are obtained after assembly of all element equations. The obtained equations are nonlinear after applying boundary conditions which are solved by utilizing the Newton–Raphson method with the required precision of 0.000005.

5. Validation of Results

The skin friction coefficient $C{f}_{x}R{e}_x^{1/2}$ and $C{f}_{y}R{e}_y^{1/2}$ in x- and y-directions are compared with the already published article of Adnan et al. [35] for various values of $\lambda$ and $\beta$ when the magnetic field $M=0$ and $\xi =1$ (final steady state flow) that can be seen an excellent correlation in

Table 1. Comparison of skin friction coefficients for different values of $\lambda$ and $\beta$ when $M=0$ at $\xi =1$.

$\mathit{\lambda }$	$\mathit{\beta }$	Adnan et  al. [35]		FEM (Our Results)
		$\mathit{C}{\mathit{f}}_{\mathit{x}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{\mathbf{1}/\mathbf{2}}$	$\mathit{C}{\mathit{f}}_{\mathit{y}}\mathit{R}{\mathit{e}}_{\mathit{y}}^{\mathbf{1}/\mathbf{2}}$	$\mathit{C}{\mathit{f}}_{\mathit{x}}\mathit{R}{\mathit{e}}_{\mathit{x}}^{\mathbf{1}/\mathbf{2}}$	$\mathit{C}{\mathit{f}}_{\mathit{y}}\mathit{R}{\mathit{e}}_{\mathit{y}}^{\mathbf{1}/\mathbf{2}}$
0.5	2.0	−1.394220	−0.628002	−1.394076	−0.630697
1.0	2.0	−1.622822	−1.025232	−1.624282	−1.025433
5.0	2.0	−2.927312	−2.633846	−2.925756	−2.636172
10.0	2.0	−4.007462	−3.797251	−4.003119	−3.803621
0.5	2.0	−1.394220	−0.628002	−1.394076	−0.630697
0.5	5.0	−1.247034	−0.561702	−1.248283	−0.562649
0.5	10.0	−1.193944	−0.537788	−1.195297	−0.538097
0.5	20.0	−1.166493	−0.525423	−1.167842	−0.525431

.

Table 2. Comparison of Nusselt number $-{\theta }^{\prime }\left(0\right)$ for $\lambda$ and Pr at $\xi =1$ when $M=Nt=Nb=0$, $\beta \to \infty$, ${\gamma }_c=0$, and ${\gamma }_T=Q_s=0$.

$\mathit{\lambda }$	Adnan et al. [35]			FEM (Our Results)
	Pr = 0.7	Pr = 2.0	Pr = 7.0	Pr = 0.7	Pr = 2.0	Pr = 7.0
0.0	0.455	0.911	1.894	0.4552	0.9108	1.8944
0.5	0.390	0.853	1.850	0.3901	0.8525	1.8500
1.0	0.321	0.770	1.788	0.3214	0.7703	1.7877
2.0	0.242	0.638	1.664	0.2420	0.6381	1.6642

shows the values of non-dimensional Nusselt number ($-{\theta }^{\prime }\left(0\right)$) is compared with different values of $\lambda$ and Pr under special cases like the absence of magnetic field $M=0$, Newtonian fluid ($\beta \to \infty$), no heat source ($Q_s=0$), pure fluid ($Nt=Nb=0$), $\xi =1$ (final steady state flow), and Fourier law (${\gamma }_T=0$). The results are found to be in excellent agreement with Adnan et al. [35].

Table 3. Comparison of Nusselt number $-{\theta }^{\prime }\left(0\right)$ for $\lambda$ and M at $\xi =1$ when ${\gamma }_c=0$, $Pr=2.0$, $Nt=Nb=0$, $\beta \to \infty$, and ${\gamma }_T=Q_s=0$.

$\mathit{\lambda }$	Abbas et al. [38]			FEM (Our Results)
	M = 0.5	M = 1.0	M = 2.0	M = 0.5	M = 1.0	M = 2.0
0.0	0.886	0.823	0.668	0.8862	0.8230	0.6682
0.5	0.841	0.800	0.663	0.8408	0.8003	0.6627
1.0	0.768	0.750	0.648	0.7684	0.7501	0.6483
2.0	0.641	0.643	0.603	0.6411	0.6429	0.6030
5.0	0.447	0.449	0.461	0.4467	0.4494	0.4612

demonstrates that the non-dimensional Nusselt number ($-{\theta }^{\prime }\left(0\right)$) is compared with different values of $\lambda$ and M when the fluid is pure and Newtonian with no heat source ($Q_s=0$), $\xi =1$ (steady flow), and absence of double diffusion heat flux model (${\gamma }_T={\gamma }_c=0$). It is noted in Table 3 that the comparison of the present results with the existing numerical results of Abbas et al. [38] is in good agreement. A brilliant relationship has been accomplished, which insists the validity of the FEM MATLAB code.

6. Results and Discussion

This portion gives some noteworthy results of the boundary value problem as finally comprised in Equations (12)–(16). A variational Galerkin method is utilized along with finite element discretization. A broad computational continuing is performed to see the reactions of velocities $(f^{\prime }(\xi ,\eta )$, $h(\xi ,\eta ))$, temperature $\theta (\xi ,\eta )$ and concentration $\varphi (\xi ,\eta )$ with the differing contributions of influential parameters. In addition, the outcomes for the Nusselt number as well as the coefficients of skin friction are additionally computed. For graphical results, one parameter varies while all the physical parameters are referred to constant values like $Nt=0.2$, $Nb=Q_s=0.2$, $\beta =\lambda =1.0$, $Pr=5$, $M=1.0$, $Le=9.0$, ${\gamma }_c=0.1$, and ${\gamma }_T=0.1$.

The effect of M (magnetic field) and $\beta$ (Casson fluid parameter) on $f^{\prime }(\xi ,\eta )$, $h(\xi ,\eta )$, $\theta (\xi ,\eta )$ and $\varphi (\xi ,\eta )$ is depicted in Figure 3a–d. As demonstrated in Figure 3a–d, the advancing contribution of M subsides the primary velocity $f^{\prime }(\xi ,\eta )$ and the magnitude of secondary velocity $h(\xi ,\eta )$. As a matter of fact, the interaction of M delivers an impeding force (Lorentz force) to halt the momentum of the flow in the $xy$-plane. The perception of Figure 3b shows that h experiences reverse flow because of Lorentz resistive force; therefore, a rise in the h profile is seen close to the sheet and afterward it gets zero. However, as opposed to the velocity, the temperature $\theta$ and concentration $\varphi$ display direct exceeding relation with parameter M (see Figure 3c,d). Similar dealings for progressing input of $\beta$ is observed (see Figure 3a–d). The velocities show a decline due to the resistance force created by tensile stress because of elasticity.

Figure 4a,b exhibit that both the components of velocity are diminished close to the sheet and afterward experience a variance when $\lambda$ is incremented. The primary velocity achieves its bigger value for $\lambda =0$. This is to make reference to the extending of the sheet along the x-axis being dependable to increase momentum toward this direction, but the y-direction momentum is denied any supporting element. In this way, Figure 4a,b individually shows that the $f^{\prime }$ is switched meagerly and the h is outstandingly switched. From Figure 4c,d, it is seen that both the $\theta$ and $\varphi$ are incremented with the increasing of $\lambda$. The rise in temperature is caused by dissipation produced by receded velocity near the sheet. In addition, the rotational factor directly strengthens the diffusion process, and it gives rise to the species concentration. Furthermore, Figure 4a–d are introduced to clarify the variation of $f^{\prime }$, h, $\theta$, and $\varphi$ as affected by $\tau$. It is to clarify that larger $\tau$ exemplifies the greater lapse of time after the jerk to the stretching sheet. Thus, Figure 4a reveals that $f^{\prime }(\xi ,\eta )$ is monotonically subsided against increasing $\tau$. Likewise, from Figure 4b, it is seen that the $h(\xi ,\eta )$ is diminished with reverse flow close to the sheet and afterward it changes at some distance away from the sheet. Both of the components of velocity become smoother for larger $\tau$. In opposition to velocity behavior, Figure 4c depicts that $\theta$ rises directly with $\tau$. This is because a longer lapse to the stretch for sheet makes the flow smoother and convection currents for heat transportation are established to raise the temperature in the boundary layer regime. The increased values of $\tau$ set the $\varphi (\xi ,\eta )$ into swaying pattern as delineated in Figure 4d. It is seen that the curve of $\varphi (\xi ,\eta )$ descends close to the sheet, and it ascends away from the sheet.

Figure 5a shows the implication of Buongiorno’s model parameters on the temperature field. The thermophoretic force causes the moving of the nanoparticles from the hotter area to the cooler area and, subsequently, a greater heat move happens in the boundary layer area. Similarly, the quicker random motion of species particles in nanofluids increased the Brownian forces to boost heat transportation. Thus, the rise of $\theta (\xi ,\eta )$ and improvement in the thermal boundary layer is reported in Figure 5a, respectively, when Nt and Nb are dynamically incremented. Figure 5b is presented to reveal the bringing down of temperature $\theta (\xi ,\eta )$ when ${\gamma }_T$ is apportioned higher input values. The greater values of ${\gamma }_T$ mean lesser thermal diffusion and hence the decline in temperature field occurs. In contradiction to the impact of ${\gamma }_T$, the incremented $Q_s$ boosts the temperature $\theta (\xi ,\eta )$ as displaced in Figure 5b.

As proved in Figure 6a, $\varphi (\xi ,\eta )$ is lessened when the Lewis number is augmented. Physically, the high Lewis number corresponds to bring down mass diffusivity, and, subsequently, the lesser species concentration in nanoliquid results. Figure 6a is also demonstrated to reveal the bringing down of $\varphi (\xi ,\eta )$ when ${\gamma }_c$ is designated higher input values. Figure 6b shows the ramifications of Buongiorno’s model parameters on the $\varphi (\xi ,\eta )$. The thermophoretic force causes the moving of the concentration layer from the lower area to the higher. Thus, the quicker random movement of species particles in nano liquids raised the Brownian forces to boost up the $\varphi (\xi ,\eta )$.

The computational outcomes for the $x,y$ directions coefficients of skin friction, Nusselt number, and Sherwood number are showed against the $\xi$ in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 to uncover the effects of different physical quantities. Figure 7a,b respectively show the enhancement in magnitudes of $C{f}_{x}R{e}_x^{1/2}$ and $C{f}_{y}R{e}_y^{1/2}$ when the M is progressed in value. The larger values of M indicate the increasing strength of the resistive force to provide resistance to the flow in both the primary and secondary directions. Figure 7c,d, in their respective order, demonstrate the meager reducing impacts of M on reduced Nusselt number and reduced Sherwood number. Both of these quantities become uniform against small $\xi$. Furthermore, Figure 7a,c,d respectively exhibit a similar trend for the Casson fluid parameter $\beta$, but opposing behavior is observed for $C{f}_{y}R{e}_y^{1/2}$ (see Figure 7b). The perceptions for Figure 8a portrays that increasing $\lambda$ created a significant increase in the magnitudes of primary skin friction $C{f}_{x}R{e}_x^{1/2}$. The diagrams for each estimation of $\lambda$ get uniform at small estimations of $\xi$$(\xi >0)$. Additionally, all the negative estimations of $C{f}_{x}R{e}_x^{1/2}$ show the reversal of primary flow at the surface. In Figure 8b, $C{f}_{y}R{e}_y^{1/2}$ shows negatively raised magnitude with higher $\lambda$.

Figure 9a,b and Figure 10a,b respectively draw sketches of reduced Nusselt number and reduced Sherwood number against magnetic field M, rotating parameter $\lambda$, and Casson fluid parameter $\beta$ for varying values of the combine parameters Nb, Nt, and heat source $Q_s$. It is revealed that increments in thermophoresis and Brownian motion parameters recede the wall heat transfer rate, but they boost the wall mass transfer rate. Similar results for wall heat transfer rate and wall mass transfer rate against the $Q_s$ are perceived (see Figure 9a,b and Figure 10a,b). Figure 9a,b and Figure 10a,b, in their respective order, demonstrate the meager reducing impacts of M and $\lambda$ on reduced Nusselt number, and reduced Sherwood number. Furthermore, it is seen that increments in $\beta$ recede the wall heat transfer rate and wall mass transfer rate.

Figure 11a shows the consolidated impact of the Nb (Brownian motion) and Nt (thermophoresis) on the reduced Nusselt number for two cases of Prandtl number, that is, $Pr=1.0$ and $Pr=5.0$, respectively. It is revealed that increments in thermophoresis and Brownian motion parameters recede the wall heat transfer rate but Pr = 5.0 boost the wall heat transfer rate. Similar results for wall mass transfer rate against the rotational parameter and Prandtl number are perceived (see Figure 11b). The reduced Sherwood number is delineated against different values of Lewis number Le for two cases of Lewis number, that is, $Le=10.0$ and $Le=15$, respectively. The Lewis number (Le) increases the wall mass transfer rate as disclosed in Figure 11b.

7. Conclusions

This computational and theoretical work addresses the 3D time-dependent magnetohydrodynamics rotational flow of Casson nanofluids across an extending sheet with double diffusion Cattaneo–Christove and heat source. The transformed 2D partial differential formulation is solved by variational Galerkin procedure. Numerical findings for velocity components, skin friction coefficients, temperature, Nusselt number, nano-particle volume fraction, and Sherwood number are computed for influential parameters. Some of the major outcomes are reported below:

• The progressing values of Casson fluid parameter $\beta$ and magnetic parameter M reduced the magnitude of secondary velocity $h(\xi ,\eta )$ and the primary velocity $f^{\prime }(\xi ,\eta )$ but concentration $\varphi$ and temperature $\theta$ are incremented.
• The concentration and temperature are incremented along with rising values of $\lambda$. Both components of velocity diminish near the surface when $\lambda$ is incremented.
• The decline of temperature is noted when thermal relaxation parameter ${\gamma }_T$ is progressively incremented, but improvement in temperature is reported when Nb, Nt, and $Q_s$ are increased.
• The higher values of Nb, concentration relaxation parameter ${\gamma }_C$, and Lewis number recede the nanoparticle concentration, but it increases against higher values of Nt.
• The Nusselt number attains lower values for higher values of Casson fluid parameter $\beta$, Brownian (Nb), thermophoresis (Nt), magnetic (M), heat source $Q_s$, and rotating ($\lambda$), but the reduced Nusselt number exhibits the opposite trend for the Prandtl number.
• The Sherwood number attains higher values for higher values of Lewis number (Le), Nb, and $Nt$, but the reduced Sherwood number exhibits opposite trend for magnetic, rotational, and Casson fluid parameter $\beta$ parameters.