Abstract
This study considered the effect of chemical reaction on transient magnetohydrodynamic flow of heat and mass transfer of a Casson nanofluid past a stretching sheet with nonuniform Heat Source/Sink. The highly nonlinear partial differential equations governing the fluid flow alongside its boundary conditions are formulated and suitable similarity variables are introduced to transform the nonlinear partial differential equations into a set of coupled ordinary differential equations. The results revealed that as the Casson fluid parameter increases, the yield stress reduces thereby reducing the velocity, temperature, and concentration profiles. Magnetic parameter, chemical reaction parameter, stretching ratio parameter and unsteadiness parameter can be used to adjust the fluid velocity, temperature and concentration distributions. The effects of local skin friction coefficients, local Nusselt and Sherwood numbers are also shown and considered using tables. The results revealed that the unsteadiness parameter, Casson fluid parameter and magnetic parameter reduces the momentum boundary layer thickness along $x$ and $y$ direction. The work provides physical insight into the thermofluidic flow phenomena of Casson nanofluid under the impacts of magnetic field, internal heat generation and chemical reaction.
Highlights
 Transient 3D MHD flow of heat and mass transfer of a Casson nanofluid is analyzed numerically.
 The flow process is numerically analyzed using Finite element analysis.
 Effects of nonuniform Heat Source/Sink and chemical reaction on the flow process is considered.
 Unsteadiness, Casson fluid and magnetic parameters reduce the momentum boundary layer thickness.
 The Nusselt number depreciated, and Sherwood number increases with chemical reaction parameter.
 The heat transfer rate is enhanced by varying the value of the magnetic field.
 Increase in unsteadiness parameter causes the skin friction, heat and mass transfer rate to increase.
1. Introduction
Chemical reactions with combined heat and mass transfer problems are important in many processes and have gotten a lot of attention in recent years. Heat and mass transfer occur simultaneously in processes such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower, the flow in a desert cooler, fruit tree groves, and electric power generation. A chemical reaction between a foreign mass and a fluid occurs in many chemical engineering processes. These processes are used in a variety of industrial applications, including polymer production, ceramics and glassware manufacturing, and food processing [1].
The flow application of non – Newtonian fluids are evident in polymer devolatization and processing, wire and fibre coating, heat exchangers, extrusion process, chemical processing equipment, etc. [2]. Combining heat transfer with the concept of stretching flow is vital in these areas of application. It is well known fact that the fluids appear in industrial and engineering processes are mostly non – Newtonian fluids [3]. The materials that falls into the category of non – Newtonian fluid include sugar solution, colloidal and suspension solution, honey lubricants, etc. the properties of such materials cannot be explored by simple Navier Stokes equations. In the category of non – Newtonian fluid, Casson has a distinctive features [4]. Casson fluids are Jelly, honey, protein, Human blood and fruit juices. Concentrated fluids like sauces, honey, juices, blood, and printing inks can be well described using this model [2]. Non – Newtonian fluid exhibits nonlinear relationship between the shear stress and rate of shear strain [5]. The study of heat transfer in the stretched flow is important because of its extensive application in chemical engineering. Several processes in chemical engineering including metallurgical process and polymer extrusion process involved cooling of molten liquid being stretched into a cooling system, glass fibre and paper production [6]. The quality and final product formation in such processes are dependent on the rate of cooling and stretching [7]. The combined heat and mass transfer problems with chemical reactions are of importance in many processes and therefore have received a considerable attention in recent years. The heat transportation analysis in view of the non – Newtonian fluids have achieved a great significance in technology and science applications like production of synthetic liquids, electro rheological liquid, biological liquids movement, plastic products manufacturing through extrusion, chemical in district (heating/cooling) etc. [8]. It has attracted the attention of new researchers due to their huge range of practical applications in engineering, science and industries [1 4]. Casson fluid is classified as the most popular non – Newtonian fluids which has several applications in food processing, metallurgy, drilling operations and bio – engineering operations [9]. Human blood, jelly, honey, concentrated fruit juices, tomato sauces are some examples of Casson fluid [1].
Chemical reactions, heat, and mass transfer effects on nonlinear magnetohydrodynamic boundary layer flow over a wedge with a porous medium were investigated by [10] Kandasamy and Palanimani (2007). The effect of magnetohydrodynamic three  dimensional Casson fluid flow past a porous linearly stretching sheet was presented by [4]. In [11], the effects of chemical reactions in a Casson nanofluid hydromagnetic free convection flow induced by a nonlinearly extending sheet immersed in a permeable medium under the influence of convective boundary conditions and thermal radiation using numerical simulations was carried out. The effect of chemical reaction on threedimensional MHD flow of couple stress Casson fluid past an unsteady stretching surface with convective boundary conditions was investigated by [12]. Investigation of the effects of an exothermic chemical reaction with Arrhenius activation energy, on MHD stagnation point flow of a Casson fluid over a nonlinearly stretching sheet was carried out by [13]. In [4], MHD three – dimensional Casson fluid flow past a porous linearly stretching sheet was carried out. It was obtained that the porosity parameter, Casson fluid parameter and magnetic parameter reduces the momentum boundary layer thickness in both directions. An unsteady MHD Casson fluid flow over a vertical cone and flat plate with non – uniform heat source/sink was studied by [14] and the results indicate that an increase in Casson fluid parameter is found to decelerate the fluid flow by increasing the plastic dynamic viscosity whereas it enhances the shear stress in flow regime. The unsteady magnetohydrodynamic free flow of a Casson fluid over an oscillating vertical plate with constant wall temperature was studied in [15]. The problem of a magnetohydrodynamic three dimensional Casson fluid flow past a porous linearly stretching sheet was investigated in [9]. Three dimensional MHD flow of couple stress Casson fluid past an unsteady stretching surface with the effect of chemical reaction and convective boundary conditions was considered by [12]. The heat and mass transfer of thermally radiating and chemically reacting MHD micropolar fluid flow past a permeable stretching sheet in a porous medium was studied by [16].] ethe Soret – Dufour characteristics in mixed convective radiated Casson fluid flow by exponentially heated surface was studied by [8], the result signifies the characteristics of temperature corresponding to convective and radiation values. This work therefore examines the numerical solution of transient MHD threedimensional heat and mass transfer of a Casson nanofluid over a stretching sheet through a porous medium using Finite Element Method (FEM).
2. Mathematical model of the problem
We consider a transient, threedimensional, heat and mass transfer, viscous incompressible, laminar, MHD boundary layer flow analysis of a Casson – nanofluid over an inclined stretching sheet in porous media was considered, as shown in Fig. 1. The fluid flow is caused by the multidirectional stretching of the sheet moving in its own plane with having surface velocities $u=\frac{ax}{1ct}+\gamma \frac{\partial u}{\partial z}$, $v=\frac{by}{1ct}+\gamma \frac{\partial v}{\partial z}$ and $w=0$, where $a$, $b$ (stretching rates) are positive constants. The fluid has no lateral motion as $z\to \infty $. Influences of Brownian and thermophoretic diffusion are considered in the transport equations. Physical properties of the fluid are assumed to be constant. The velocity components of $x$,$y$and $z$ are $u$, $v$and $w$ respectively. ${T}_{w}$ and ${T}_{\infty}$ are convective surface temperature and ambient temperature and ${C}_{w}$ and ${C}_{\infty}$ are the concentration of the fluid at the surface of the sheet and ambient concentration respectively.
Fig. 1Physical model and coordinate system
The constitutive equation of state for an isotropic and incompressible Casson fluid is as follows [1718]:
$\mathrm{w}$here ${\tau}_{ij}$ is the $(i,j)$th component of the stress tensor, ${\mu}_{B}$ is the plastic dynamic viscosity of the nonNewtonian fluid, ${P}_{z}$ is the yield stress of the fluid, $\pi $ is the product of the component of deformation rate with itself $\left(\pi ={e}_{ij},{e}_{ij}\right)$, ${e}_{ij}$ is the component of deformation rate, ${\pi}_{c}$ is the critical value of product based on nonNewtonian model, $\beta ={\mu}_{B}\frac{\sqrt{2{\pi}_{c}}}{{P}_{z}}$ is the Casson parameter. Under the above assumptions. Using the Oberbeck Boussinesq and boundary layer approximations, the governing equations describing the continuity, momentum, energy and concentration for the three – dimensional transient flow problem are can be written in Cartesian coordinate is given by [17]:
The associated boundary conditions are as follows:
where ${n}_{f}$ is the convective heat transfer coefficient, ${n}_{s}$ is the convective mass transfer coefficient ${T}_{f}$ and ${C}_{s}$ are the convective fluid temperature and concentration below the moving sheet:
Employing the Roseland diffusion approximation, the radiative heat flux ${q}_{r}$ is given by:
where ${\sigma}^{\mathrm{*}}$ represents the Stefan – Boltzmann constant, ${k}^{\mathrm{*}}$ denotes the Rosseland mean absorption coefficient. The temperature difference within the flow are assumed to be sufficiently small so that ${T}^{4}$, may be expressed as a linear function of $T$, and can be expanded by using Taylor’s series about ${T}_{\infty}$, as follows:
and neglecting the higher order terms beyond the first degree in $\left(T{T}_{\infty}\right)$, we get:
as used by other researchers such as [1922].
Substituting Eq. (7d) into Eq. (7c), we have:
The non – uniform heat generation/absorption $q\text{'}\text{'}\text{'}$ in Eq. (5), as used by [23] is expressed as:
Finding solution to the set of Eqs. (26), following similarity variables are introduced as given by [24]:
$\eta =z\sqrt{\frac{a}{\left(1ct\right){v}_{f}}},C={C}_{\infty}+\left({C}_{w}{C}_{\infty}\right)h\left(\eta \right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}T={T}_{\infty}+\left({T}_{w}{T}_{\infty}\right)\theta \left(\eta \right),$
where $\theta \left(\eta \right)$ is the dimensionless temperature and $h\left(\eta \right)$ is the dimensionless concentration:
The above Eq. (9) satisfies the continuity equation in Eq. (2).
The dimensional governing Eqs. (36) and the associated boundary conditions Eqs. (7) take the non – dimensional form:
And the boundary conditions are:
${g}^{\text{'}}=1+\gamma {g}^{\text{'}\text{'}},\text{at}\eta =0,$
where ${B}_{it}=\frac{{v}_{f}}{k}{\left(\frac{{v}_{f}}{a}\right)}^{1/2}$, ${B}_{ic}=\frac{{n}_{s}}{{D}_{B}}{\left(\frac{{v}_{f}}{a}\right)}^{1/2}$ are thermal and the solutal Biot Numbers respectively.
The non – dimensional variables are:
The local skin friction coefficients, the Nusselt number, which denotes the rate of heat transfer at the surface of the sheet, and the Sherwood number, which denotes the rate of mass transfer at the surface of the sheet, are the physical quantities of engineering interest considered in this problem. The following are the definitions for these quantities:
${\tau}_{wx}=\mu \left(1+\frac{1}{\beta}\right){\left(\frac{\partial u}{\partial z}\right)}_{z=0},{\tau}_{wy}=\mu \left(1+\frac{1}{\beta}\right)\left(\frac{\partial v}{\partial z}\right){}_{z=0}{}^{\mathrm{}}\mathrm{},$
where ${\tau}_{wx}$ and ${\tau}_{wy}$ are the wall shear stress along $x$ and $y$ – directions respectively ${q}_{w}$ and ${q}_{m}$ are the heat flux and mass flux at the surface respectively. Applying similarity variables into Eqs. (1618), the following are obtained:
where $\mathrm{R}\mathrm{e}=\frac{{u}_{w}x}{{v}_{f}}$ is the local Reynolds number based on the stretching velocity.
3. Method of solution
According to [22], the exact solution for the set of nonlinear differential equations alongside its boundary conditions are not possible and thus we employ a numerical method known as finite element method (FEM). This method has been applied by various researchers [2528].
The Garlekin global finite element model of the equations for the $i$th element thus formed is given in matrix form as follows:
where $\left[K{}_{\mathrm{}}{}^{mn}\mathrm{}\right]$, $\left\{{f}^{\left(e\right)}\right\}$,$\left\{{r}^{\left(e\right)}\right\}$, $\left\{{g}^{\left(e\right)}\right\}$, $\left\{{t}^{\left(e\right)}\right\}$, $\left\{{\theta}^{\left(e\right)}\right\}$, $\left\{{h}^{\left(e\right)}\right\}$ and $\left\{{S}^{m}\right\}$, {$m,n=$1,2,…,6} is the set of matrices defined as follows with ${\left[K{}_{\mathrm{}}{}^{mn}\mathrm{}\right]}_{3\times 3}$, ${\left[{S}^{m}\right]}_{2\times 1}$, {$m,n=$1,2,…, 6} defined as follows:
$+{A}^{\mathrm{*}}\underset{{\eta}_{e}}{\overset{{\eta}_{e+1}}{\int}}{\left[{N}^{\left(e\right)}\right]}^{T}\left[{N}^{\left(e\right)}\right]d\eta +\mathrm{P}\mathrm{r}E{c}_{x}\left(1+\frac{1}{\beta}\right)\underset{{\eta}_{e}}{\overset{{\eta}_{e+1}}{\int}}{\left[{N}^{\left(e\right)}\right]}^{T}\left[{N}^{\left(e\right)}\right]d\eta ,$
$+\mathrm{P}\mathrm{r}\underset{{\eta}_{e}}{\overset{{\eta}_{e+1}}{\int}}\left({\left[{N}^{\left(e\right)}\right]}^{T}\left[{N}^{\left(e\right)}\right]\left[{N}^{\left(e\right)}\right]\left({f}^{\left(e\right)}+s{g}^{\left(e\right)}\right)d\eta \right)$
$+Sc\underset{{\eta}_{e}}{\overset{{\eta}_{e+1}}{\int}}\left[{\left[{N}^{\left(e\right)}\right]}^{T}\left[{N}^{\left(e\right)}\right]\left[{N\text{'}}^{\left(e\right)}\right]\right]\left({f}^{\left(e\right)}+s{g}^{\left(e\right)}\right)d\eta +ScCr\underset{{\eta}_{e}}{\overset{{\eta}_{e+1}}{\int}}\left[{\left[{N}^{\left(e\right)}\right]}^{T}\left[{N}^{\left(e\right)}\right]\right]d\eta ,$
The element matrix given by Eq. (21) is of the order 18×18 and the whole domain is divided into a set of equations a matrix of order 9606×9606 is obtained. The system of equations obtained is nonlinear and therefore an iterative scheme (Gauss  Siedel method of successive relaxation) is used for solving it. After the imposition of the boundary conditions, the system of equations left is solved. Once the convergence criterion $\sum \left{\mathrm{\Theta}}_{i}^{t}{\mathrm{\Theta}}_{i}^{t1}\right\le 1{0}^{4}$ is satisfied, where $\mathrm{\Theta}$ stand for either $f$, $r$, $g$, $m$, $\theta $, $h$ and $t$ represents the iterative step, the iterative process is terminated.
4. Results and discussion
The simulations of the developed numerical solutions are presented in this section. Table 1 presents the values of the surface skin friction, Nusselt number and Sherwood number obtained for different variations of the governing parameters are tabulated. Casson fluid parameter is seen to increase in $x$ direction and decrease in $y$ direction. Magnetic parameter shows an opposite behaviour that is it decreases in both directions. The rate heat transfer and mass transfer are observed to increase for increasing values of Casson and Magnetic field parameters. Also, it is observed that as the chemical reaction parameter increases the surface skin fiction increases in $x$ and decreases in $y$ direction. The Nusselt number and Sherwood number are enhanced by various values of the observed parameters enhanced. The table shows that increasing Prandtl number increase the local Nusselt number and decreases the Sherwood number.
Fig. 2 show the effect of Casson parameter on the fluid velocities, $f\text{'}\left(\eta \right)$ and $g\text{'}\left(\eta \right)$, in $x$ and $y$ directions. It can be seen that as$$Casson parameter increases, the two velocities diminish due to the suppressing nature of the non – Newtonian nature of the fluid. This is also because as the Casson parameter is increased, the yield stress decreases, reducing the thickness of the momentum boundary layer. The effect of the unsteadiness parameter on the velocity profile is shown in Fig. 3. As the unsteadiness parameter increases, the fluid flow is met with more resistance, resulting in lower fluid velocities. The effect of the chemical reaction parameter on velocity profiles in both directions is depicted in Fig. 4. As the chemical reaction parameter is increased, the velocity profiles decrease. This is because the destructive chemical reaction reduces the thickness of the concentration boundary layer, which causes the concentration buoyancy effects to increase, causing the chemical reaction to increase and the fluid velocity to decrease. The effect of Casson parameter on the temperature profile is shown in Fig. 5. Increasing the value of Casson parameter causes the thermal boundary layer thickness to increase. The effect of a chemical reaction parameter on the temperature profile is shown in Fig. 6. As can be seen, as the chemical reaction parameter is increased, the temperature profile rises. This is because chemical energy is effectively converted to thermal energy, which warms the fluid and raises the temperature profile.
Fig. 2Effect of Casson parameter on velocity profile
a)$x$ – direction
b)$y$ – direction
Fig. 3Effect of unsteadiness parameter on velocity profile
a)$x$ – direction
b)$y$ – direction
Fig. 4Effect of chemical reaction parameter on velocity profile
a)$x$ – direction
b)$y$ – direction
The temperature profile decreases as the unsteadiness parameter increases, as shown in Fig. 7. The reason for this is that as the rate of heat loss by the fluid increases, so does the unsteadiness parameter. In practice, as the unsteadiness parameter rises, the thermal boundary layer shrinks, causing the fluid temperature to plummet. The effect of Casson parameter on the concentration profile is shown in Fig. 8. The concentration boundary layer thickness is reduced as the Casson parameter is increased. Physically, increasing the Casson parameter causes a rise in the viscosity of the fluid, which causes the fluid to slow down and saturate the nanoparticles near the stretching sheet. Fig. 9 depicts the effect of the chemical reaction parameter on the concentration profile. With a higher value of the chemical reaction parameter, the concentration profile is heavily influenced and decelerated. This is because destructive chemical reactions prevent diffusion, lowering the concentration profile.
Fig. 5Effect of Casson parameter on the temperature profile
Fig. 6Effect of chemical reaction parameter on the temperature profile
Fig. 7Effect of unsteadiness parameter on temperature profile
Fig. 8Effect of Casson parameter on the concentration profile
Fig. 9Effect of Chemical reaction parameter on the concentration profile
Fig. 10Effect of unsteadiness parameter on temperature profile
Table 1Skin friction coefficients, Nusselt number and Sherwood number of various values of important parameters where, A*=B*=Ec=γ=0.1, K1=R=0.5, Bic=Bit=1.0
$\beta $  M  Cr  $\lambda $  Cfx  Cfy  Nu  Sh 
0.1  0.5  0.1  0.5  –5.106011  31.858097  0.519260  –1.661414 
0.2  –3.831593  13.220039  0.685547  –0.8490386  
0.3  –3.292098  7.590677  0.794603  –0.383889  
0.4  –2.982865  4.964475  0.874366  –0.067096  
0.5  –2.780840  3.436612  0.936103  0.181934  
0.5  0.0  0.1  0.5  –2.557637  4.182315  0.815603  –0.022393 
0.5  –2.780840  3.436612  0.936104  0.181934  
1.0  –2.997403  2.850816  1.040385  0.335933  
1.5  –3.155939  2.367540  1.132998  0.459937  
2.0  –3.320810  1.955465  1.216798  0.564124  
0.5  0.5  0.0  0.5  –2.766751  3.265498  0.936166  0.023647 
0.3  –2.816733  3.781496  0.935751  0.525625  
0.5  –2.865144  4.176897  0.935104  0.930996  
1.0  –3.076173  5.835048  0.931645  2.405443  
1.5  –1.084563  0.460314  0.962170  –16.102385  
0.5  0.5  0.1  1.0  –2.929721  3.654077  0.191477  –0.342295 
1.5  –3.113362  3.046629  1.023925  –0.172345  
2.0  –3.282556  2.539055  1.119785  –0.031366  
2.5  –3.445025  2.099725  1.205966  0.089965  
3.0  –3.587719  1.371037  1.285665  0.196654 
5. Conclusions
Transient analysis of a threedimensional, heat and mass transfer, viscous incompressible, laminar, MHD boundary layer flow analysis of a Casson – nanofluid over a stretching sheet in porous media was considered. The system of highly nonlinear partial differential equations is reduced to a system of ordinary differential equations through appropriate existing similarity transformation. The resulting equations are solved through Garlekin finite element method. The results obtained through FEM are verified by other numerical solutions via Runge– Kutta fourth  order integration scheme.
The following are the findings from the study:
1) The momentum boundary layer reduces as the Casson parameter increases and the velocity profile is enhanced.
2) The unsteadiness parameter can be used to adjust the thickness of the momentum boundary layer and as well as the thermal boundary layer.
3) The concentration profiles are highly influenced and are decelerated with higher value of chemical reaction parameter.
4) The Nusselt number depreciated, and Sherwood number increases with the higher values of chemical reaction parameter.
4) The heat transfer rate can be enhanced by varying the value of the magnetic field.
5) Increase in the unsteadiness parameter also increases the skin friction coefficients, heat transfer rate, and as well as the mass transfer rate.
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About this article
The authors have not disclosed any funding.
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
The authors declare that they have no conflict of interest.