Consider the two-dimensional unsteady laminar MHD mixed convective flow of a nanofluid due to a stretching sheet situated at \(y = 0\) with stretching velocity \(u=ax\), where *a* is a constant. The temperature and nanoparticle volume fraction at the stretching surface are \(T_{w}\) and \(C_{w}\), respectively, and those of the ambient nanofluid are \(T_{\infty}\) and \(C_{\infty}\), respectively. The *x* and *y* directions are in the plane of and perpendicular to the sheet, respectively. The continuity, momentum, energy and concentration equations of unsteady, incompressible nanofluid boundary layer flow are as follows (see Yang [26]):

$$\begin{aligned}& \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} =0, \end{aligned}$$

(2.1)

$$\begin{aligned}& \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} =-\frac{1}{\rho_{nf}} \frac{\partial p}{\partial x}+\frac{\mu_{nf}}{ \rho_{nf}}\frac{\partial^{2} u}{\partial y^{2}} + g\beta_{T}(T-T_{\infty}) + g\beta_{C}(C-C_{\infty})-\frac{\sigma B_{0}^{2}}{\rho_{nf}}u, \end{aligned}$$

(2.2)

$$\begin{aligned}& \frac{\partial T}{\partial t} + u\frac{\partial T}{\partial x} + v\frac{\partial T}{\partial y} = \alpha_{nf}\frac{\partial^{2} T}{\partial y^{2}} +\frac{Q}{ (\rho c_{p})_{nf}}(T - T_{\infty})+ \frac{\rho_{f}D_{m}K_{T}}{ C_{s}(\rho c_{p})_{nf}}\frac{\partial^{2} C}{\partial^{2} y}, \end{aligned}$$

(2.3)

$$\begin{aligned}& \frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D_{m}\frac{\partial^{2} C}{\partial y^{2}} + \frac{D_{m}K_{m}}{T_{m}} \frac{\partial^{2} T}{\partial y^{2}} -R(C-C_{\infty}), \end{aligned}$$

(2.4)

where *t*, *u* and *v* are the time, the fluid velocity and the normal velocity components in the *x* and *y* orientations, respectively; \(\nu_{nf}\), *p*, \(\rho_{nf}\), *σ*, \(B_{0}\), \(\mu_{nf}\), *g* are the nanofluid kinematic viscosity, the pressure, nanofluid density, electrical conductivity, the uniform magnetic field in the *y* direction, the effective dynamic viscosity of the nanofluid and gravitational acceleration, respectively; \(\beta_{T}\), \(\beta_{C}\), *T*, *C*, \(\alpha _{nf}\), \((\rho cp)_{nf}\), *Q* are the volumetric thermal expansion coefficient, the solutal expansion coefficient, the temperature of the fluid in the boundary layer, fluid solutal concentration, the thermal diffusivity of the nanofluid, the nanofluid heat capacitance and the volumetric rate of heat generation, respectively; \(\rho_{f}\), \(D_{m}\), \(K_{T}\), \(C_{s}\), \((cp)_{nf}\), \(T_{m}\), *R* are the density of the base fluid, the mass diffusivity of concentration, thermal diffusion ratio, concentration susceptibility, specific heat of the fluid at constant pressure, mean fluid temperature and the chemical reaction parameter, respectively.

The boundary conditions are as follows:

$$ \begin{aligned} &t\geq0\mbox{:}\quad u=U_{w}(x)=ax, \qquad v=v_{w},\qquad T=T_{w}, \qquad C=C_{w} \quad \mbox{at }y=0, \\ &t\geq0 \mbox{:}\quad u=U_{\infty}(x)=a_{\infty}x,\qquad v=0,\qquad T = T_{\infty},\qquad C= C_{\infty}\quad \mbox{as }y\to \infty, \end{aligned} $$

(2.5)

and the initial conditions are

$$ \begin{aligned} &t< 0\mbox{:}\quad u(x,y,t)=0,\qquad v(x,y,t)=0, \\ &\hphantom{t<0\mbox{:}\quad}T(x,y,t)=T_{w}, \qquad C(x,y,t)=C_{w},\quad \forall{x,y}, \end{aligned} $$

(2.6)

where \(a_{\infty}\) (>0) is the stagnation flow rate parameter, \(a< 0\) for a shrinking surface and \(a > 0\) for a stretching surface. Here \(v_{w}\) is prescribed suction velocity (\(v_{w} < 0\)) or blowing velocity (\(v_{w} >0\)).

In the free stream the momentum equation (2.2) becomes

$$ U_{\infty}\frac{dU_{\infty}}{dx} = - \frac{1}{\rho_{nf}}\frac {\partial p}{\partial x}- \frac{\sigma B_{0}^{2}}{\rho_{nf}}U_{\infty}. $$

(2.7)

Substituting (2.7) in (2.2) the momentum equation is written as

$$\begin{aligned} \begin{aligned}[b] \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} ={}&{ \nu_{nf}}\frac{\partial^{2} u}{\partial y^{2}}+U_{\infty}\frac{dU_{\infty}}{dx}+ (U_{\infty}- u) \frac{\sigma B_{0}^{2}}{\rho_{nf}} \\ &{}+g\beta_{T}(T-T{\infty})+g\beta _{C}(C-C{\infty}). \end{aligned} \end{aligned}$$

(2.8)

The effective dynamic viscosity of the nanofluid was given by Brinkman [27] as

$$ \mu_{nf}=\frac{\mu_{f}}{(1-\phi)^{2.5}}, $$

(2.9)

where *ϕ* is the solid volume fraction of nanoparticles, \(\mu_{f}\) is the dynamic viscosity of the base fluid. In equations (2.1)-(2.4),

$$\begin{aligned}& (\rho c_{p})_{nf}=(1-\phi) (\rho c_{p})_{f} + \phi(\rho c_{p})_{s}, \\& \rho_{nf}=(1-\phi)\rho_{f} + \phi\rho_{s},\qquad \nu_{nf}=\frac {\mu_{nf}}{\rho_{nf}}, \\& \alpha_{nf}=\frac{k_{nf}}{(\rho c_{p})_{nf}}, \qquad \frac{k_{nf}}{k_{f}}= \frac{(k_{s}+k_{f}) - 2\phi(k_{f} - k_{s})}{(k_{s}+k_{f}) + \phi(k_{f} - k_{s})}, \end{aligned}$$

(2.10)

where \(k_{nf}\) is the thermal conductivity of the nanofluid, \(k_{f}\) and \(k_{s}\) are the thermal conductivities of the fluid and of solid fractions, respectively, and \(\rho_{s}\) is the density of solid fractions, \((\rho c_{p})_{f}\) and \((\rho c_{p})_{s}\) are the heat capacity of the base fluid and the effective heat capacity of a nanoparticle, respectively, \(k_{nf}\) is the thermal conductivity of the nanofluid.

The continuity equation (2.1) is satisfied by introducing a stream function \(\psi(x,y)\) such that

$$ u=\frac{\partial\psi}{\partial y},\qquad v=-\frac{\partial \psi}{\partial x}. $$

(2.11)

We introduce the following non-dimensional variables (see Liao [28]):

$$ \begin{aligned} &\eta= \biggl[\frac{a_{\infty}}{\nu_{f}\xi} \biggr]^{\frac{1}{2}} y, \quad \xi =1-\exp(-\tau), \tau=a_{\infty}t,\qquad \psi= [{a_{\infty} \nu_{f}\xi} ]^{\frac{1}{2}} xf(\xi,\eta ),\\ &\theta(\xi,\eta)=\frac{T-T_{\infty}}{T_{w}-T_{\infty}},\qquad \varPhi (\xi,\eta)=\frac{C-C_{\infty}}{C_{w}-C_{\infty}}, \end{aligned} $$

(2.12)

where \(f(\xi,\eta)\) is a dimensionless stream function, \(\theta(\xi,\eta)\) is the dimensionless temperature and \(\phi(\xi,\eta)\) is the dimensionless solute concentration. By using (2.11) and (2.12), the governing equations (2.3), (2.4) and (2.8) along with the boundary conditions (2.5) are reduced to the following two-point boundary value problem:

$$\begin{aligned}& f'''+\phi_{1} \biggl[ \frac{\eta}{2}(1-\xi)f'' + \xi \bigl(ff''- {f}^{\prime 2}+1+\mathit{Ha}^{2}\bigl(1-f' \bigr)+\mathit{Gr}_{t} \theta+\mathit{Gr}_{c} \varPhi \bigr) \biggr] \\& \quad = \phi_{1} \xi(1-\xi)\frac{\partial f'}{\partial\xi}, \end{aligned}$$

(2.13)

$$\begin{aligned}& \theta'' + \frac{k_{f}}{k_{nf}}\mathit{Pr} \phi_{2} \biggl[\frac{\eta }{2}(1-\xi)\theta' +\xi \bigl(f\theta'+\delta\theta \bigr)+\frac{D_{f}}{\phi_{2}}\varPhi '' \biggr] =\frac{k_{f}}{k_{nf}}\mathit{Pr} \phi_{2}\xi(1-\xi)\frac{\partial \theta}{\partial\xi}, \end{aligned}$$

(2.14)

$$\begin{aligned}& \varPhi '' + \mathit{Sc} \biggl[\frac{\eta}{2}(1-\xi) \varPhi ' + \xi \bigl(f\varPhi '-\gamma \varPhi \bigr)+\mathit{Sr} \theta'' \biggr] = \mathit{Sc} \xi(1-\xi) \frac{\partial \varPhi }{\partial\xi}. \end{aligned}$$

(2.15)

The boundary conditions are as follows:

$$ \begin{aligned} &f(\xi,0)=f_{w}, \qquad f'(\xi,0)= \lambda,\qquad \theta(\xi,0)=1,\qquad \varPhi (\xi,0)=1\quad \mbox{at } \eta=0, \xi \geq0, \\ &f'(\xi,\infty)=1,\qquad \theta(\xi,\infty)=0,\qquad \varPhi (\xi, \infty )=0\quad \mbox{as } \eta\rightarrow\infty, \xi\geq0, \end{aligned} $$

(2.16)

where primes denote differentiation with respect to *η*, \(\alpha_{f}=k_{f}/(\rho c_{p})_{f}\) and \(\nu_{f}=\mu_{f}/\rho_{f}\) are the thermal diffusivity and kinetic viscosity of the base fluid, respectively. Other non-dimensional parameters appearing in equations (2.13) to (2.15) are *Ha*, \(\mathit{Gr}_{t}\), \(\mathit{Gr}_{c}\), \(\mathit{Gr}_{c}\), *Pr*, *δ*, \(D_{f}\), *Sc*, *γ* and *Sr*, and they denote the Hartmann number, the local temperature Grashof number, the local concentration Grashof number, the Prandtl number, the dimensionless heat generation parameter, the Dufour number, the Schmidt number, the scaled chemical reaction parameter and the Soret number, respectively. These parameters are defined mathematically as

$$ \begin{aligned} &\mathit{Ha}^{2} = \frac{\sigma B_{0}^{2}}{a_{\infty}\rho_{nf}} ,\qquad \mathit{Gr}_{t}=\frac{g\beta_{T}(T_{w}-T_{\infty})}{a_{\infty}^{2}x}, \\ &\mathit{Gr}_{c}= \frac{g\beta_{C}(C_{w}-C_{\infty})}{a_{\infty}^{2}x} ,\qquad \mathit{Pr}=\frac{\nu_{f}}{\alpha_{f}},\qquad \delta= \frac{Q}{a_{\infty}(\rho c_{p})_{nf}}, \\ &D_{f} = \frac{D_{m}K_{T}(C_{w}-C_{\infty})}{C_{s}(C_{p})_{f}\nu_{f}(T_{w}-T_{\infty})}, \qquad \mathit{Sc}= \frac{\nu_{f}}{D_{m}}, \\ &\gamma=\frac{R}{a_{\infty}},\qquad \mathit{Sr}=\frac{D_{m}K_{T}}{T_{m}} \frac{(T_{w}-T_{\infty})}{\nu_{f} (C_{w} - C_{\infty})}. \end{aligned} $$

(2.17)

The boundary conditions are as follows:

$$ \begin{aligned} &f(\xi,0)=f_{w}, \qquad f'(\xi,0)= \lambda,\qquad \theta(\xi,0)=1, \qquad \varPhi (\xi,0)=1\quad \mbox{at } \eta=0, \xi \geq0, \\ &f'(\xi,\infty)=1,\qquad \theta(\xi,\infty)=0, \qquad \varPhi (\xi, \infty)=0\quad \mbox{as } \eta\rightarrow\infty, \xi\geq0. \end{aligned} $$

(2.18)

The nanoparticle volume fractions \(\phi_{1}\) and \(\phi_{2}\) are defined as

$$ \phi_{1}=(1-\phi)^{2.5} \biggl[1-\phi+ \phi \biggl( \frac{\rho_{s}}{\rho_{f}} \biggr) \biggr],\qquad \phi_{2}= \biggl[1-\phi+ \phi \frac{(\rho c)_{s}}{(\rho c)_{f}} \biggr]. $$

(2.19)

In equations (2.18), \(f_{w} =-v_{w}/\sqrt{a_{\infty}\nu_{f} \xi}\) represents suction (\(f_{w}>0\)) or injection (\(f_{w}<0\)) and *λ* (\(=a/a_{\infty}\)) is the stretching/shrinking parameter.