Unsteady MHD Casson Dissipative Fluid Flow past a Semi-infinite Vertical Porous Plate with Radiation Absorption and Chemical Reaction in Presence of Heat Generation

In this problem, consider the effects of viscous dissipation and radiation absorption on unsteady two dimensional MHD free convective Casson fluid flow of a viscous, incompressible fluid over a vertical porous plate by means of heat and mass transfer in a uniform of a pressure grading. The following assumptions are listed below:

The rheological equation of state for the Cauchy stress tensor of Casson fluid can be written as

$\tau=\tau_{0}+\mu \alpha^{*}$             (1)

$\text { Equivalently } \tau_{i j}=\left\{\begin{array}{l}

2\left[\mu_{B}+\frac{p_{y}}{\sqrt{2 \pi}}\right] e_{i j} \pi>\pi_{c} \\

2\left[\mu_{B}+\frac{p_{y}}{\sqrt{2 \pi_{c}}}\right] e_{i j} \pi<\pi_{c}

\end{array}\right.$                     (2)

$\Rightarrow \pi=e_{i j} e_{i j}$           (3)

$e_{i j}=\text { the }(i, j)^{t h}$ component of deformation rate, ${{\tau }_{0}}$ = Casson yield stress, $\pi$ = the product based on the non-Newtonian fluid, $\pi_{c}$  =a critical value of this product, $\mu_{B}$ = plastic dynamic viscosity of the non-Newtonian fluid, $\mu$ = the dynamic velocity, $\alpha^{*}$ = shear rate.

The yield stress of fluid $=p_{y}=\frac{\mu_{B} \sqrt{2 \pi}}{\gamma}$   (4)

Some fluids require a gradually increasing shear stress to maintain a constant strain rate and are called Rheopectic, in the case of Casson fluid

$\text { Flow where } \tau>\tau_{c}, \mu=\mu_{B}+\frac{p_{y}}{\sqrt{2 \pi}}$             (5)

$\text { The Kinematic velocity } v=\frac{\mu}{\rho}=\frac{\mu_{B}}{\rho}\left[1+\frac{1}{\gamma}\right]$               (6)

Finally. the Casson fluid parameter is $\gamma=\frac{\mu_{B} \sqrt{2 \pi_{c}}}{p_{y}}$     (7)

Let the $\bar{x}$ -axis be taken along the porous plate in the vertical direction and $\bar{y}$ -axis in the direction of perpendicular to the flow. Let $\bar{x} \text { and } \bar{y}$ are the dimensional distance along the perpendicular to the plate and ${{\tau }_{1}}$ is the dimensional time. $\bar{u} \text { and } \bar{v}$ are the components of dimensional velocities along $\bar{x} \text { and } \bar{y}$ directions.  Suppose that transverse magnetic field of the uniform strength is to be applied normal to the plate. The radiate heat flux in the $\bar{x}$ - direction is considered insignificant in comparison with that in $\bar{y}$ - direction. Physical model of the problem has been shown in Figure 1. It is supposed that voltage is not applied which implies the nonappearance of an electric field. Induced magnetic field is supposed to be insignificant. C^* and g^* are the dimensional concentration and acceleration owing to gravity, $Q_{0}\left(T^{*}-T_{\infty}^{*}\right)$  is the amount of heat generated per unit volume. $Q_{0}$  is the constant where either $Q_{0}<0 \text { or } Q_{0}>0$ , g is acceleration owing to gravity. $g \beta\left(T^{*}-T_{\infty}^{*}\right)$  is the body force owing to non-uniform temperature, $g \beta^{*}\left(C^{*}-C_{\infty}^{*}\right)$ is the body force owing to non-uniform concentration. Viscous dissipation, Radiation absorption and permeability of porous medium are taken into account. The fluid contains constant thermal conductivity and kinematic viscosity, the Boussinesqu’s approximation has been considered for the flow. The homogeneous chemical reaction is of first order with rate constant between the diffusing species and the fluid is considered.

1.jpg

Figure 1. Physical model of the problem

From the above assumptions, the unsteady flow is governed by the following partial differential equations.

Equation of continuity:

$\frac{\partial \bar{v}}{\partial \bar{y}}=0$            (8)

Equation of Momentum:

$\left. \begin{align}  & \frac{\partial \overline{u}}{\partial {{\tau }_{1}}}+\overline{v}\frac{\partial \overline{u}}{\partial \overrightarrow{y}}=\frac{-1}{\rho }\,\,\,\,\frac{\partial \overline{p}}{\partial \overline{x}}+\vartheta \,\,\,\,\left( 1+\frac{1}{\gamma } \right)\frac{{{\partial }^{2}}\overline{u}}{\partial {{\overline{y}}^{2}}}+g\,\,\,\,\beta (\overline{T}-{{\overline{T}}_{\infty }}) \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+g\overline{\beta }\left( \overline{C}-{{\overline{C}}_{\infty }} \right)-\frac{\vartheta }{\overline{K}}\,\,\,\overline{u}-\frac{\sigma }{\rho }\,\,\,\,B_{0}^{2}\overline{u} \\ \end{align} \right\}$           (9)

$\left.\begin{array}{rl}

\frac{\partial \bar{T}}{\partial \tau_{1}}+\bar{v} \frac{\partial \bar{T}}{\partial \bar{y}}=\frac{K}{\rho C_{p}} \frac{\partial^{2} \bar{T}}{\partial \bar{y}^{2}}+\frac{Q_{0}}{\rho C_{p}}(\bar{T}-\overline{T_{\infty}})-\frac{1}{K \rho C_{p}} \frac{\partial \bar{q}_{r}}{\partial \bar{y}} \\

+\frac{\vartheta}{C_{p}}\left(1+\frac{1}{\gamma}\right)\left(\frac{\partial \bar{u}}{\partial \bar{y}}\right)^{2}+\bar{R}(\bar{C}-\overline{C_{\infty}})

\end{array}\right\}$                     (10)

Equation of Concentration:

$\frac{\partial \bar{C}}{\partial \tau}+\bar{v} \frac{\partial \bar{C}}{\partial \bar{y}}=D \frac{\partial^{2} \bar{C}}{\partial \bar{y}^{2}}-K r(\bar{C}-\overline{C_{\infty}})$             (11)

The boundary conditions for velocity, temperature and concentration field are given by

$\left.\begin{array}{l}

\bar{u}=0, \bar{T}=\overline{T_{w}}+\varepsilon e^{\bar{n} \tau}(\overline{T_{w}}-\overline{T_{\infty}}), \bar{C}=\overline{C_{w}}+\varepsilon e^{\bar{m}}(\overline{C_{w}}-\overline{C_{\infty}}) \text { at } \bar{y}=0 \\

\bar{u}=U_{0}\left(1+\varepsilon e^{\bar{n} \tau}\right), \quad \bar{T} \rightarrow \bar{T}_{\infty}, \quad \bar{C} \rightarrow \overline{C_{\infty}} \quad A s \quad \bar{y} \rightarrow \infty

\end{array}\right\}$                   (12)

From the equation of the continuity it is obvious that the suction velocity at the plate surface is a function of time only and we shall take it in the form:

$\overline{v}=-{{v}_{0}}\left( 1+\varepsilon A{{e}^{\overline{n}\tau }} \right)$           (13)

From the equation of momentum equation gives

$-\frac{1}{\rho }\frac{\partial \overline{p}}{\partial \overline{x}}=\frac{d\overline{{{U}_{\infty }}}}{d\tau }+\frac{\vartheta }{\overline{K}}\overline{{{U}_{\infty }}}+\frac{\sigma }{\rho }B_{0}^{2}\overline{{{U}_{\infty }}}$           (14)

Eliminating $-\frac{1}{\rho }\frac{\partial \overline{p}}{\partial \overline{x}}$  in the Eq. (9) by using (14) then we get

$\left.\begin{array}{rl}

\frac{\partial \bar{u}}{\partial \tau}+\bar{v} \frac{\partial \bar{u}}{\partial y} & =\frac{d \bar{U}_{\infty}}{d t}+\vartheta\left(1+\frac{1}{\gamma}\right) \frac{\partial^{2} \bar{u}}{\partial \bar{y}^{2}}+g \beta\left(\bar{T}-\bar{T}_{\infty}\right) \\

& +g \bar{\beta}\left(\bar{C}-\bar{C}_{\infty}\right)+\frac{\vartheta}{K}(\overline{U_{\infty}}-\bar{u})+\frac{\sigma}{\rho} B_{0}^{2}(\overline{U_{\infty}}-\bar{u})

\end{array}\right\}$                    (15)

The radiative heat flux term by using the Rosseland approximation is given by

$\overline{{{q}_{r}}}=-\frac{4\overline{{{\sigma }_{S}}}}{3\overline{{{k}_{e}}}}\frac{\partial {{\overline{T}}^{4}}}{\partial \overline{y}}$          (16)

We assume that the temperature difference within the flow is sufficiently small such that ${{\overline{T}}^{4}}$  can be expressed as a linear function of the temperature; this is expanding in Taylor series about ${{\overline{T}}_{\infty }}$  and neglecting higher order terms to give:

$\overline{T}\cong 4\overline{T}\overline{T}_{\infty }^{3}-3\overline{{{T}_{\infty }}}$          (17)

By using Eqns. (16) and (17) into equation of energy (10) can be changed into following form:

$\left.\begin{array}{rl}

\frac{\partial \bar{T}}{\partial \tau}+v \frac{\partial \bar{T}}{\partial y}=\frac{K}{\rho C_{p}} \frac{\partial^{2} \bar{T}}{\partial \bar{y}^{2}}+ & \frac{Q_{0}}{\rho C_{p}}(\bar{T}-\overline{T_{\infty}})+\frac{16 \sigma_{S} T_{\infty}^{-3}}{3 \rho C_{p} k_{e}} \frac{\partial^{2} \bar{T}}{\partial \bar{y}^{-2}} \\

& +\frac{\vartheta}{C_{p}}\left(1+\frac{1}{\gamma}\right)\left(\frac{\partial \bar{u}}{\partial \bar{y}}\right)^{2}+\bar{R}(\bar{C}-\overline{C_{\infty}})

\end{array}\right\}$                          (18)

Now Non-dimensional quantities are defined as

$\left. \begin{align}  & f=\frac{\overline{u}}{{{U}_{0}}},\,v=\frac{\overline{v}}{{{V}_{0}}},\,\,\,\,\,\,{{U}_{\infty }}=\frac{\overline{{{U}_{\infty }}}}{{{U}_{0}}},y=\frac{{{V}_{0}}\overline{y}}{\vartheta },\,\,\,\,\,{{U}_{p}}=\frac{\overline{{{u}_{p}}}}{{{U}_{0}}} \\  & t=\frac{V_{0}^{2}{{\tau }_{1}}}{\vartheta },\overline{K}=\frac{K{{\vartheta }^{2}}}{V_{0}^{2}},\overline{T}={{\overline{T}}_{_{\infty }}}+\theta \left( {{\overline{T}}_{_{w}}}-{{\overline{T}}_{_{\infty }}} \right),\overline{n}=\frac{V_{0}^{2}n}{\vartheta } \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\overline{C}={{\overline{C}}_{_{\infty }}}+C\left( {{\overline{C}}_{w}}-{{\overline{C}}_{_{\infty }}} \right) \\ \end{align} \right\}$            (19)

After substituting the boundary conditions and non-dimensional variables in the governing Eqns. (11), (15) and (18) and taking into account Eq. (13) then we get

$\left.\begin{array}{r}

\frac{\partial f}{\partial t}-\left(1+\epsilon A e^{n t}\right) \frac{\partial f}{\partial y}=\frac{d U_{\infty}}{d t}+\left(1+\frac{1}{\gamma}\right) \frac{\partial^{2} f}{\partial y^{2}}+G r \theta+G m C \\

+\phi\left(U_{\infty}-U\right)

\end{array}\right\}$             (20)

$\left. \begin{align}  & \frac{\partial \theta }{\partial t}-\left( 1+\in A{{e}^{nt}} \right)\frac{\partial \theta }{\partial y}={{\left( \Pr  \right)}^{-1}}\left( 1+\frac{4R}{3} \right)\frac{{{\partial }^{2}}\theta }{\partial {{y}^{2}}}+\eta \theta +Ra\,\,\,C \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+Ec\left( 1+\frac{1}{\gamma } \right){{\left( \frac{\partial f}{\partial y} \right)}^{2}} \\ \end{align} \right\}$     (21)

$\frac{\partial C}{\partial t}-\left( 1+\in A{{e}^{nt}} \right)\frac{\partial C}{\partial y}={{\left( Sc \right)}^{-1}}\frac{{{\partial }^{2}}C}{\partial {{y}^{2}}}-Kr\,\,\,C$                (22)

Boundary conditions (12) are given by the following form

$\left.\begin{array}{l}

f=0, \theta=1+\varepsilon e^{n t}, C=1+\varepsilon e^{n t} \quad A t  y=0 \\

f=U_{\infty}=\left(1+\varepsilon e^{n t}\right), \theta \rightarrow 0, C \rightarrow 0 \quad A s  y \rightarrow \infty

\end{array}\right\}$                      (23)

$\left.\begin{array}{l}

G r=\frac{\vartheta \beta g\left(\bar{T}_{w}-\bar{T}_{\infty}\right)}{V_{0}^{2} U_{0}}, G m=\frac{\vartheta \beta^{*} g\left(\bar{C}_{w}-\bar{C}_{\infty}\right)}{V_{0}^{2} U_{0}}, M=\frac{\sigma B_{0}^{2}}{\rho V_{0}^{2}} \\

\eta=\frac{\vartheta Q_{0}}{\rho V_{0}^{2} C_{P}}, \phi=M+\frac{1}{K} \operatorname{Pr}=\frac{\rho \vartheta C_{p}}{k} E c=\frac{U_{0}^{2}}{C_{p}\left(\bar{T}_{w}-\bar{T}_{\infty}\right)} \\

R=\frac{4 \sigma \bar{T}_{\infty}^{-3}}{k_{e} k}, S c=\frac{\vartheta}{D}, R a=\frac{\bar{R} \vartheta\left(\bar{C}_{w}-\bar{C}_{\infty}\right)}{V_{0}^{2}\left(\bar{T}_{w}-\bar{T}_{\infty}\right)}, K r=\frac{k_{1} \vartheta}{V_{0}^{2}}

\end{array}\right\}$                           (24)

The problem posted in Eqns. (20), (21) and (22) subject to the boundary conditions presented in Eq. (23) it has been explored numerically through Multiple Regular Perturbation law. This is more economical and flexible from analytical point of view. The behaviors of velocity, temperature, concentration, Skin-friction, Nusselt number and Sherwood numbers has been discussed in detail for variation of thermo physical parameters. The solution is assumed to be as,

$\left.\begin{array}{l}

f=f_{0}(y)+\varepsilon e^{n t} f_{1}(y) \\

\theta=\theta_{0}(y)+\varepsilon e^{n t} \theta_{1}(y) \\

C=C_{0}(y)+\varepsilon e^{n t} C_{1}(y)

\end{array}\right\}$                 (25)

Substituting Eq. (25) in Eqns. (20), (21) and (22) then we get the subsequent pairs of equations for (f₀, θ₀, C₀) & (f₁, θ₁, C₁)

$\left( 1+\frac{1}{\gamma } \right)f_{0}^{''}+f_{0}^{'}-N{{f}_{0}}=-\phi -Gr\,\,{{\theta }_{0}}-Gm\,\,{{C}_{0}}$             (26)

$\left.\begin{array}{r}

\left(1+\frac{1}{\gamma}\right) f_{1}^{\prime \prime}+f_{1}^{\prime}-(N+n) f_{1}=-(\phi+n)-A f_{0}^{\prime}-G r \theta_{1} \\

-G m C_{1}

\end{array}\right\}$                        (27)

$\left.\begin{array}{rl}

(3+4 R) \theta_{1}^{\prime \prime}+3 \operatorname{Pr} \theta_{1}^{\prime}-3 \operatorname{Pr}(n-\eta) \theta_{1}=-3 \operatorname{Pr} A \theta_{0}^{\prime} \\

-6 E c\left(1+\frac{1}{\gamma}\right) P_{r} f_{0}^{\prime} f_{1}^{\prime}-3 \operatorname{Pr} R a C_{1}

\end{array}\right\}$                 (28)

$\left.\begin{array}{r}

(3+4 R) \theta_{0}^{\prime \prime}+3 \operatorname{Pr} \theta_{0}^{\prime}+3 \operatorname{Pr} \eta \theta_{0}=-3 \operatorname{Pr} E c\left(1+\frac{1}{\gamma}\right)\left(f_{0}^{\prime}\right)^{2} \\

-3 \operatorname{Pr} R a C_{0}

\end{array}\right\}$                         (29)

$C_{0}^{''}+Sc\,\,C_{0}^{'}-Sc\,\,Kr\,\,{{C}_{0}}=0$              (30)

$C_{1}^{''}+Sc\,C_{1}^{'}-Sc\,\,\,\left( Kr+n \right){{C}_{1}}=-Sc\,\,AC_{0}^{'}$            (31)

The corresponding boundary conditions can be written as

$\begin{aligned}

f_{0} &=0, f_{1}=0 \theta_{0}=1, \theta_{1}=1, C_{0}=1 C_{1}=1, \text { at } y=0 \\

f_{0}=1, f_{1} &=1, \theta_{0} \rightarrow 0, \theta_{1} \rightarrow 0, C_{0} \rightarrow 0, C_{1} \rightarrow 0 \text { as } y \rightarrow \infty

\end{aligned}$                             (32)

Here Eqns. (26) to (31) are still coupled and non-linear, whose exact solutions are not possible. So we expand ${{f}_{0}},\ {{f}_{1}}\ {{\theta }_{0}}\ {{\theta }_{1}}$ . Initialy we solve equation (30) and equation (31) by using Eq. (32) then we get

${{C}_{0}}={{e}^{-{{X}_{1}}y}}$              (33)

${{C}_{1}}={{e}^{-{{X}_{2}}y}}\left( 1-{{h}_{1}} \right)+{{e}^{-{{X}_{1}}y}}{{h}_{1}}$                 (34)

Now utilizing multi parameter perturbation law and presumptuous $E c<<1$.

$\left.\begin{array}{l}

f_{0}=f_{00}+E c f_{01}+0(\varepsilon)^{2} \ldots \ldots \ldots . \\

\theta_{0}=\theta_{00}+E c \theta_{01}+0(\varepsilon)^{2} \ldots \ldots \ldots . \\

f_{1}=f_{10}+E c f_{11}+0(\varepsilon)^{2} \ldots \ldots \ldots \ldots . \\

\theta_{1}=\theta_{10}+E c \theta_{11}+0(\varepsilon)^{2} \ldots \ldots \ldots \ldots .

\end{array}\right\}$                        (35)

By utilizing Eq. (35) in Eqns. (26) to (29) we get the subsequent set of differential equation

$\left( 1+\frac{1}{\gamma } \right)f_{00}^{''}+f_{00}^{'}-N{{f}_{00}}=-\phi -Gr{{\theta }_{00}}-Gm\,\,{{C}_{0}}$              (36)

$\left( 1+\frac{1}{\gamma } \right)f_{01}^{''}+f_{01}^{'}-\phi {{f}_{01}}=-{{G}_{r}}{{\theta }_{01}}$                        (37)

$\left.\begin{array}{r}

\left(1+\frac{1}{\gamma}\right) f_{10}^{\prime \prime}+f_{10}^{\prime}-(\phi+n) f_{10}=-(\phi+n)-A f_{00}^{\prime}-G r \theta_{10} \\

-G m C_{1}

\end{array}\right\}$                        (38)

$\left( 1+\frac{1}{\gamma } \right)f_{11}^{''}+f_{11}^{'}-\left( \phi +n \right){{u}_{11}}=-Af_{01}^{'}-{{G}_{r}}{{\theta }_{11}}$               (39)

$\left.\begin{array}{rl}

(3+4 R) \theta_{10}^{\prime \prime}+3 \operatorname{Pr} \theta_{10}-3 \operatorname{Pr}(n-\eta) \theta_{10} & =-3 A \operatorname{Pr} \theta_{00}^{\prime} \\

-3 \operatorname{Pr} R a C_{1}

\end{array}\right\}$                     (40)

$\left.\begin{array}{r}

(3+4 R) \theta_{11}^{\prime \prime}+3 \operatorname{Pr} \theta_{11}^{\prime}-3 \operatorname{Pr}(n-\eta) \theta_{11}=-3 A \operatorname{Pr} \theta_{01} \\

-6\left(1+\frac{1}{\gamma}\right) \operatorname{Pr} f_{00}^{\prime} f_{10}^{\prime}

\end{array}\right\}$                       (41)

$\left( 3+4R \right)\theta _{00}^{''}+3{{P}_{r}}\theta _{00}^{'}+3\eta {{P}_{r}}{{\theta }_{00}}=-3\Pr \,\,Ra\,\,{{C}_{0}}$                (42)

$\left( 3+4R \right)\theta _{00}^{''}+3\Pr \,\,\theta _{00}^{'}+3\eta \Pr \,\,\,{{\theta }_{00}}=-3\Pr \,\,Ra\,\,{{C}_{0}}$                 (43)

$(3+4 R) \theta_{01}+3 \operatorname{Pr} \theta_{01}^{\prime}+3 \eta \operatorname{Pr} \theta_{01}=-3 \operatorname{Pr}\left(1+\frac{1}{\gamma}\right)\left(f_{00}^{\prime}\right)^{2}$                     (44)

The related boundary conditions are given by

$\left.\begin{aligned}

&\left.\begin{array}{l}

f_{\infty}=0, f_{01}=0, f_{\infty}=0, f_{11}=0 \\

\theta_{00}=1, \theta_{01}=1, \theta_{10}=1, \theta_{11}=1

\end{array}\right\} \text { at } y=0\\

&\left.\begin{array}{l}

f_{\infty}=1, f_{01}=0, f_{10}=0, f_{11}=0 \\

\theta_{\infty}=0, \theta_{01}=0, \theta_{10}=0, \theta_{11}=0

\end{array}\right\} \text { As } y \rightarrow \infty

\end{aligned}\right\}$                      (45)

Solve (36) – (44) subject to boundary Condition (45) we get;

$f(y, t)=\left\{\begin{array}{l}

A_{0} e^{-X_{4} y}+h_{3} \quad e^{-X_{3} y}+h_{4} e^{-X_{1} y}+E c A_{2} e^{-X_{4} y} \\

+E c A_{2} e^{-X_{4} y}+E c h_{11} e^{-X_{3} y}+E c h_{12} e^{-2 X_{4} y} \\

+E c h_{12} \quad e^{-2 X_{4} y}+E c h_{13} e^{-2 X_{3} y}+E c h_{14} e^{-2 X_{1} y} \\

+E c \quad h_{15} \quad e^{-\left(X_{3}+X_{4}\right) y}+E c \quad h_{16} \quad e^{-\left(X_{1}+X_{3}\right) y} \\

+\epsilon e^{m} A_{4} e^{-X_{6} y}+\epsilon e^{m} h_{21} e^{-X_{1} y}+\epsilon e^{m} h_{22} e^{-X_{2} y} \\

+\epsilon e^{m} h_{23} e^{-X_{3} y}+\epsilon e^{m} h_{24} e^{-X_{4} y}+\epsilon e^{m} h_{25} e^{-X_{5} y} \\

+\epsilon e^{n} E c \quad A_{6} e^{-X_{6} y}+\epsilon \quad e^{\pi} \quad E c \quad h_{42} e^{-X_{4} y} \\

+\epsilon \quad e^{m} E c \quad h_{44} e^{-2 X_{4} y}+\epsilon \quad e^{m} E c \quad h_{45} e^{-2 X_{3} y} \\

+\epsilon e^{m} E c \quad h_{46} e^{-2 X_{1} y}+\epsilon e^{m} E c h_{47} e^{-\left(X_{3}+X_{4}\right) y} \\

+\epsilon e^{m t} E c h_{48} e^{-\left(X_{1}+X_{3}\right) y}+\epsilon e^{\pi} E c h_{49} e^{-\left(X_{1}+X_{2}\right) y} \\

+\epsilon e^{\pi} E c h_{50} \quad e^{-X_{6} y}+\epsilon e^{\pi} E c \quad h_{51} e^{-\left(X_{2}+X_{4}\right) y} \\

+\epsilon e^{m} E c h_{52} e^{-\left(X_{2}+X_{3}\right) y}+\epsilon e^{m} E c h_{53} e^{-\left(X_{3}+X_{6}\right) y} \\

+\epsilon e^{m t} E c h_{55} e^{-\left(X_{1}+X_{5}\right) y}+\epsilon e^{m} E c h_{56} e^{-\left(X_{4}+X_{6}\right) y} \\

+\epsilon e^{m} E c h_{57} e^{-\left(X_{4}+X_{5}\right) y}+\epsilon e^{n t} E c h_{58} e^{-\left(X_{3}+X_{5}\right) y} \\

+\epsilon e^{m t} E c \quad h_{54} e^{-\left(X_{1}+X_{6}\right) y}+E c h_{17} e^{-\left(X_{1}+X_{4}\right) y} \\

+\epsilon e^{m t} E c h_{46} e^{-2 X_{1} y}+\epsilon e^{m} E c h_{43} e^{-X_{3} y}+\epsilon e^{\pi} \\

+\epsilon e^{m} E c h_{59} e^{-\left(X_{1}+X_{2}\right) y}

\end{array}\right]$                  (46)

$\theta(y, t)=\left\{\begin{array}{l}

\left(1-h_{2}\right) \quad e^{-X_{3} y}+h_{2} e^{-X_{1} y}+E c \quad A_{1} \quad e^{-X_{3} y} \\

+E c h_{5} e^{-2 X_{4} y}+E c h_{6} e^{-2 X_{3} y}+E c h_{7} \quad e^{-2 R X_{1} y} \\

+E c h_{8} e^{-\left(R X_{3}+X_{4}\right) y}+E c h_{9} e^{-\left(X_{1}+X_{3}\right) y}+E c h_{10} e^{-\left(X_{1}+X_{4}\right) y} \\

+E c h_{10} e^{-\left(X_{1}+X_{4}\right) y}+\epsilon e^{n t} \quad A_{3} e^{-X_{5} y}+\epsilon e^{n t} h_{18} e^{-X_{1} y} \\

+\epsilon e^{n \pi} \quad h_{19} e^{-X_{2} y}+\epsilon e^{n t} h_{20} e^{-X_{3} y}+\epsilon \quad e^{\pi} E c A_{5} e^{-X_{6} y} \\

+\epsilon e^{n} E c \quad h_{26} \quad e^{-\left(X_{2}+X_{4}\right) y}+\epsilon e^{\pi} E c h_{27} e^{-\left(X_{3}+X_{4}\right) y} \\

+\epsilon \quad e^{n t} E c \quad h_{28} e^{-2 X_{4} y}+\epsilon e^{n t} E c \quad h_{29} \quad e^{-\left(X_{4}+X_{6}\right) y} \\

+\epsilon e^{\pi} \quad E c h_{30} \quad e^{-\left(X_{4}+X_{5}\right) y}+\epsilon e^{n t} \quad E c h_{31} e^{-\left(X_{1}+X_{3}\right) y} \\

+E c \quad h_{32} e^{-\left(X_{1}+X_{3}\right) y} \in e^{n t}+E c \quad h_{33} \quad e^{-2 X_{3} y} \in e^{n t} \\

+E c \quad h_{34} \quad e^{-\left(X_{3}+X_{6}\right) y} \in e^{n t}+E c h_{35} e^{-\left(X_{3}+X_{5}\right) y} \in e^{n t} \\

+E c \quad h_{36} \quad e^{-2 X_{1} y} \quad \in e^{n t}+E c h_{37} \quad e^{-\left(X_{1}+X_{2}\right) y} \in e^{n t} \\

+\epsilon e^{i t} \quad E c \quad h_{38} e^{-\left(X_{1}+X_{6}\right) y}+\epsilon \quad e^{n t} \quad E c h_{39} e^{-\left(X_{1}+X_{5}\right) y} \\

+\epsilon e^{\pi} E c h_{40} e^{-X_{3} y}+\epsilon e^{n t} E c h_{41} e^{-\left(X_{1}+X_{4}\right) y}

\end{array}\right\}$                  (47)

$C\left( y,t \right)={{e}^{-{{X}_{1}}y}}+\in {{e}^{nt}}\left( {{e}^{-{{X}_{2}}y}}\left( 1-{{h}_{1}} \right)+{{e}^{-{{X}_{1}}y}}{{h}_{1}} \right)$            (48)

​2.1 The skin-friction coefficient

${{\tau }_{w}}=\left[ 1+\frac{1}{\gamma } \right]{{\left[ \frac{\partial f}{\partial \overline{y}} \right]}_{y=0}}$                (49)

2.2 Nusselt number

$\left.\begin{array}{rl}

N_{u}= & -x\left[\frac{\partial \bar{T}}{\partial \bar{y}}\right]_{y=0}\left[\bar{T}_{w}-\bar{T}_{\infty}\right]^{-1} \\

& \Rightarrow N_{u} \operatorname{Re}_{x}^{-1}=-\left[\frac{\partial \theta}{\partial y}\right]_{y=0}

\end{array}\right\}$                  (50)

2.3 Sherwood number

$\left.\begin{array}{rl}

S_{h}=-x & {\left[\frac{\partial \bar{C}}{\partial \bar{y}}\right]_{y=0}\left[\bar{C}_{w}-\bar{C}_{\infty}\right]^{-1}} \\

& \Rightarrow S_{h} \operatorname{Re}_{x}^{-1}=-\left[\frac{\partial C}{\partial y}\right]_{y=0}

\end{array}\right\}$                (51)

$\tau_{w}=$$\begin{array}{l}

{\left[\begin{array}{c}

-X_{4} A_{0}-X_{3} h_{3}-X_{1} h_{4}-X_{4} E c A_{2}-X_{3} \operatorname{Ech}_{11} \\

-2 X_{4} \operatorname{Ec} h_{12}-2 X_{3} E c \quad h_{13}-2 X_{1} E c \\

-\left(X_{3}+X_{4}\right) \operatorname{Ec} h_{15}-\left(X_{1}+X_{3}\right) E c h_{16} \\

-\left(X_{1}+X_{4}\right) \operatorname{Ech}_{17}-X_{6} A_{4} \in e^{m t}-X_{1} h_{21} \in e^{m} \\

-X_{2} h_{22} \in e^{m t}-X_{3} \quad h_{23} \in e^{m}-X_{4} h_{24} \in e^{m} \\

-\epsilon e^{n t} X_{5} h_{25} \in e^{n t} X_{6} E c A_{6}-\epsilon e^{n t} X_{4} E c h_{12} \\

-X_{3} \in e^{m} E c h_{43}-2 \epsilon e^{m} X_{4} E c h_{44} \\

-2 \epsilon e^{m} X_{3} \quad E c h_{45}-2 \in e^{m} \quad X_{1} E c h_{46} \\

-\epsilon e^{m}\left(X_{3}+X_{4}\right) E c h_{47}-\epsilon e^{m}\left(X_{1}+X_{3}\right) E c h_{48} \\

-\epsilon \quad e^{n t}\left(X_{1}+X_{2}\right) E c h_{49}-\epsilon e^{m} X_{6} E c h_{50} \\

-\epsilon e^{m t}\left(X_{2}+X_{4}\right) E c h_{51}-\epsilon e^{m}\left(X_{2}+X_{3}\right) E c h_{52} \\

-\epsilon e^{m}\left(X_{3}+X_{6}\right) E c h_{53}-\epsilon e^{m}\left(X_{1}+X_{6}\right) E c h_{54} \\

-\epsilon e^{m}\left(X_{1}+X_{5}\right) E c h_{55}-\epsilon e^{m}\left(X_{4}+X_{6}\right) E c h_{56} \\

-\epsilon e^{m t}\left(X_{4}+X_{5}\right) E c h_{57}-\epsilon e^{m}\left(X_{3}+X_{5}\right) E c h_{58} \\

-\epsilon e^{m}\left(X_{1}+X_{2}\right) E c h_{59}

\end{array}\right]}

\end{array}$                (52)

$N_{u}=\left\{\begin{array}{l}

-X_{3}\left(1-h_{2}\right)-X_{1} \quad h_{2}-X_{3} \quad E c A_{1}-2 X_{4} E c h_{5} \\

-2 \quad X_{3} E c h_{6}-2 X_{1} \quad E c h_{7}-\left(X_{3}+X_{4}\right) E c h_{8} \\

-\left(X_{1}+X_{3}\right) \quad E c \quad h_{9}-\left(X_{1}+X_{4}\right) \quad E c h_{10} \\

-\varepsilon e^{m} X_{5} A_{3}-\varepsilon e^{m} \quad X_{1} h_{18}-\varepsilon e^{m} \quad X_{2} h_{19} \\

-\varepsilon e^{\pi t} X_{3} h_{20}-\varepsilon e^{i t} X_{6} E c A_{5}-\varepsilon e^{i t}\left(X_{2}+X_{4}\right) E c h_{26} \\

-\varepsilon e^{i t}\left(X_{3}+X_{4}\right) E c \quad h_{27}-2 \varepsilon e^{m} X_{4} \quad E c \quad h_{28} \\

-\varepsilon e^{i t}\left(X_{4}+X_{6}\right) E c h_{29}-\varepsilon e^{i t}\left(X_{4}+X_{5}\right) E c h_{30} \\

-\varepsilon e^{i t}\left(X_{1}+X_{3}\right) E c h_{31}-\varepsilon e^{i t}\left(X_{1}+X_{3}\right) E c h_{32} \\

-2 \varepsilon e^{\lambda t} X_{3} E c h_{33}-\varepsilon e^{i t}\left(X_{3}+X_{6}\right) \quad E c \quad h_{34} \\

-\varepsilon e^{i t}\left(X_{3}+X_{5}\right) E c h_{35}-2 \varepsilon e^{m} X_{1} \quad E c h_{36} \\

-\varepsilon e^{i t}\left(X_{1}+X_{2}\right) E c h_{37}-\varepsilon e^{m}\left(X_{1}+X_{6}\right) E c h_{38} \\

-\varepsilon e^{i t}\left(X_{1}+X_{5}\right) E c h_{39}-\varepsilon e^{i t} X_{3} \operatorname{Ec} h_{40} \\

-\varepsilon e^{i t}\left(X_{1}+X_{4}\right) E c h_{41}

\end{array}\right\}$               (53)

$S_{h}=-X_{1}-\varepsilon e^{n t} X_{2}\left(1-h_{1}\right)-\varepsilon e^{n t} X_{1} h_{1}$         (54)

This paper is an extended of the work of Rajakumar et al. [28] with Casson fluid parameter γ in the obscene of Casson fluid parameter γ i.e., $(\gamma \rightarrow \infty)$  our results are in good agreement with the results of Rajakumar et al. [28] it was shown from the Table 2.

Table 2. Effect of various physical parameters on Skin friction and Nusselt number

The authors are cordially thankful to the honorable reviewers for their constructive comments to improve the quality of the paper. There is no conflict of interest in between the co-authors and there is no funding from any funding agencies.