Consider an unsteady two dimensional laminar flow of electrically conducting non-Newtonian Casson fluid over a stretching surface. It is assumed that the unsteady fluid flow, heat and mass transfer begins at *t* = 0. It is also assumed that the surface is being stretched with velocity *U*_{w}(*x*, *t*) = *ax*/1 - *ct* along the *x–*axis as shown Figure 1. Here *a* is the initial stretching rate and $B={B}_{0}/\sqrt{1-ct}$ is the magnetic field applied along the *y–* axis, which is perpendicular to the *x–* axis. Here *B*_{0} is the strength transverse magnetic field. The mass transfer *v*_{w}(*t*) assumed to be perpendicular to the stretching surface. When the sheet starts to move with velocity *U*_{w}(*x*, *t*) the surface temperature *T*_{w}(*x*, *t*) and concentration *C*_{w}(*x*, *t*) will suddenly raise to *T*_{∞}, *C*_{∞} respectively. Also, it is assumed that *T*_{w} > *T*_{∞}, which corresponds to an assisting flow.

Figure 1 Physical configuration and coordinate system

The rheological equation of state for the Cauchy stress tensor of Casson fluid can be written as [20, 21]
$${\tau}_{ij}=\left\{\begin{array}{ll}2({\mu}_{B}+{p}_{y}/\sqrt{2\pi}){e}_{ij},& \pi >{\pi}_{c}\\ 2({\mu}_{B}+{p}_{y}/\sqrt{2{\pi}_{c}}){e}_{ij},& \pi <{\pi}_{c}\end{array}\right.$$

Where *π = e*_{ij}e_{ij} and *e*_{ij} is the (*i,j*)^{th} component of the deformation rate with itself, *π*_{c} is the critical value of this product based on the non-Newtonian model, *µ*_{B} is the plastic dynamic viscosity of the non -Newtonian fluid and *p*_{y} is yield stress of the fluid.

Under above made assumptions the governing equations of the flow are given by,
$$\frac{{\mathrm{\partial}}_{ll}}{\mathrm{\partial}x}+\frac{\mathrm{\partial}v}{\mathrm{\partial}y}=0,$$(1)
$$\begin{array}{ll}\rho \left(\frac{{\mathrm{\partial}}_{u}}{\mathrm{\partial}t}+u\frac{{\mathrm{\partial}}_{ll}}{\mathrm{\partial}x}+v\frac{{\mathrm{\partial}}_{u}}{\mathrm{\partial}y}\right)& =\mu \left(1+\frac{1}{\beta}\right)\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{y}^{2}}\\ & +g(\rho {\beta}_{T})(T-{T}_{\mathrm{\infty}})+g(\rho {\beta}_{C})(C-{C}_{\mathrm{\infty}})\\ & -\sigma {B}^{2}u,\end{array}$$(2)
$$\begin{array}{ll}\left(\frac{\mathrm{\partial}T}{\mathrm{\partial}t}+u\frac{\mathrm{\partial}T}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}T}{\mathrm{\partial}y}\right)& =\alpha \frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}-\frac{1}{\rho {c}_{p}}\frac{\mathrm{\partial}{q}_{r}}{\mathrm{\partial}y}\\ & +\frac{{D}_{m}{K}_{T}}{{c}_{\mathrm{s}}{c}_{p}}\frac{{\mathrm{\partial}}^{2}C}{\mathrm{\partial}{y}^{2}},\end{array}$$(3)
$$\begin{array}{ll}\frac{\mathrm{\partial}C}{\mathrm{\partial}t}+u\frac{\mathrm{\partial}C}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}C}{\mathrm{\partial}y}& ={D}_{m}\frac{{\mathrm{\partial}}^{2}C}{\mathrm{\partial}{y}^{2}}-{k}_{l}(C-{C}_{\mathrm{\infty}})\\ & +\frac{{D}_{m}{K}_{T}}{{T}_{m}}\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}},\end{array}$$(4)

With the boundary conditions
$$u={U}_{w}(x,t),v={v}_{w}(t),T={T}_{w}(x,t),C={C}_{w}(x,t)$$(5) at *y* = 0, $$u\to 0,T\to {T}_{\mathrm{\infty}},C\to {C}_{\mathrm{\infty}},\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}y\to \mathrm{\infty}$$(6)

where *u* and *v* are the velocity components in the *x, y* directions, *t* refers to the time, $\beta ={\mu}_{B}\sqrt{2{\pi}_{c}}/{p}_{y}$ is the Casson parameter, *ρ* and *µ* are density and the dynamic viscosity of the Casson fluid respectively, *β*_{T} and *β*_{C} are the coefficients of volumetric expansion due to temperature and concentration differences respectively, *g* is the acceleration due to gravity, *T, C* are the fluid temperature and concentration, *σ* is the electrical conductivity, *α* is the thermal diffusivity, *D*_{m} is the species diffusivity, *K*_{T} is the thermal diffusion ratio, *c*_{s} is the concentration susceptibility, *c*_{p} is the specific heat at constant pressure, *k*_{l} is the chemical reaction parameter and *T*_{m} is the mean fluid temperature.

The quantity *q*_{r} on the right hand side of temperature equation (3) represents the radiative flux and is given by,
$${q}_{r}={\displaystyle \frac{-16{\sigma}^{\star}{T}_{\mathrm{\infty}}^{\star}}{3{k}^{\star}}\frac{\mathrm{\partial}T}{\mathrm{\partial}y},}$$(7)

In eqn. (7), *σ*^{*} is the Stephan-Boltzmann constant, *k*^{*} is the mass absorption coefficient, *T*_{∞} is the free stream temperature.

Also,
$$\begin{array}{ll}{U}_{w}(x,t)=\frac{ax}{1-ct},{v}_{w}(t)=\frac{{v}_{0}}{(1-ct{)}^{1/2}},& \\ {T}_{w}(x,t)={T}_{\mathrm{\infty}}+\frac{bx}{(1-ct{)}^{2}},{C}_{w}(x,t)={C}_{\mathrm{\infty}}+\frac{bx}{(1-ct{)}^{2}},\end{array}$$(8)

Here a, c are constants (*a* > 0 and *c* ≥ 0 with *ct* < 1). Also, *b* is a constant and has dimension temperature/concentration length (*b* = 0 refers to absence of buoyancy forces).

We now introduce the similarity transformation as,
$$\begin{array}{ll}\eta ={\left(\frac{a}{v(1-ct)}\right)}^{1/2}y,\psi (x,y)={\left(\frac{va}{1-ct}\right)}^{1/2}xf(\eta )& \\ T={T}_{\mathrm{\infty}}+\frac{bx}{(1-ct{)}^{2}}\theta (\eta ),C={C}_{\mathrm{\infty}}+\frac{bx}{(1-ct{)}^{2}}\varphi (\eta ),\end{array}$$(9)

Here *ψ*(*x*, *y*) is the stream function that satisfies the continuity equation (1) with
$$\begin{array}{l}u=\frac{\mathrm{\partial}\psi}{\mathrm{\partial}y}=\frac{ax}{1-ct}{f}^{\prime}(\eta ),\\ v=\frac{-\mathrm{\partial}\psi}{\mathrm{\partial}x}=-{\left(\frac{va}{1-ct}\right)}^{1/2}f(\eta ),\end{array}$$(10)

Using equations (7)–(10) the equations (2)–(5) can be transformed into
$$\begin{array}{l}\left(1+\frac{1}{\beta}\right)f-{f}^{\prime}({f}^{\prime}+A)-{f}^{\u2033}\left(\frac{1}{2}A\eta -f\right)\\ +\lambda \theta +{\lambda}^{\star}\varphi -M{f}^{\prime}=0\end{array}$$(11)
$$\begin{array}{l}\frac{1}{\mathrm{P}\mathrm{r}}\left(1+\frac{4R}{3}\right){\theta}^{\u2033}+Du{\varphi}^{\u2033}-\theta (2A+{f}^{\prime})\\ -{\theta}^{\prime}\left(\frac{1}{2}A\eta -f\right)=0,\end{array}$$(12)
$$\begin{array}{l}\frac{1}{Sc}{\varphi}^{\u2033}-Kr\varphi +Sr{\theta}^{\u2033}-\varphi (2A+{f}^{\prime})\\ -{\varphi}^{\prime}\left(\frac{1}{2}A\eta -f\right)=0,\end{array}$$(13)

The transformed boundary conditions are
$$\begin{array}{l}f(\eta )={f}_{w},{f}^{\prime}(\eta )=1,\theta (\eta )=1,\varphi (\eta )=1,\\ {f}^{\prime}(\eta )=1,\varphi (\eta )=1,\theta (\eta )=1\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{t}\phantom{\rule{thinmathspace}{0ex}}\eta =0\end{array}$$(14)
$${f}^{\prime}(\eta )\to 0,\theta (\eta )\to 0,\varphi (\eta )\to 0\phantom{\rule{thinmathspace}{0ex}}\mathrm{a}\mathrm{s}\phantom{\rule{thinmathspace}{0ex}}\eta \to \mathrm{\infty}$$(15)

Where primes denotes differentiation with respect to *η*, here *η* is the similarity variable, *A = c/a* is the unsteadiness parameter, *M = σB*_{0}^{2}/ *ρa* is the magnetic field parameter, *λ* = *gβ*_{T}b/a^{2} is the thermal buoyancy parameter, *λ** = *gβ*_{C}b/a^{2} is concentration buoyancy parameter, $R={\displaystyle \frac{4{\sigma}^{\star}{T}_{\mathrm{\infty}}^{3}}{k{k}^{\star}}}$ is the radiation parameter, $Dn={\displaystyle \frac{{D}_{m}{K}_{T}}{{c}_{s}{c}_{p}v}}$ is the Dufour number, $\mathrm{P}\mathrm{r}=\frac{\rho {c}_{p}v}{k}$ is the Prandtal number, *Sc = υ*_{f}/D_{m} is the Schmidt number, *Kr = k*_{l}(1 - *ct*)/*a* is the chemical reaction parameter and $Sr={\displaystyle \frac{{D}_{m}{K}_{T}}{{T}_{m}v}}$ is the Soret number.

The physical quantities of engineering interest are Skin friction coefficient (*Cf*_{x}), local Nusselt number (*Nu*_{x}) and Sherwood number (*Sh*_{x}).These are given by,
$$\begin{array}{ll}C{f}_{X}{\mathrm{R}\mathrm{e}}_{X}^{1/2}& =\left(1+\frac{1}{\beta}\right){f}^{\u2033}(0),N{u}_{X}{\mathrm{R}\mathrm{e}}_{X}^{-1/2}\\ & =-\left(1+\frac{4R}{3}\right){\theta}^{\prime}(0),S{h}_{X}{\mathrm{R}\mathrm{e}}_{X}^{-1/2}=-{\varphi}^{\prime}(0),\end{array}$$(16)

In the above equation Re_{x} = *xU*_{w}(*x*, *t*)/*υ* is the Reynolds number.

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