Consider steady, incompressible two dimensional boundary layer flow near the stagnation point in a permeable stretching or shrinking sheet. The x-axis is along the plate, the y-axis is measured normal to the plate. The temperature T, solute concentration S, and nanoparticle concentration C at the wall are denoted by \(T_{w}\), \(S_{w}\), and \(C_{w}\), respectively, and their ambient values are \(T_{\infty}\), \(S_{\infty}\), and \(C_{\infty}\), respectively, where \(T_{w}>T_{\infty}\), \(S_{w}>S_{\infty}\), and \(C_{w}>C_{\infty}\), and hence a momentum, thermal, solute, and nanoparticle concentration boundary layer form near the wall (Figure 1). We assume that hydrodynamic, thermal, and solute slip occur at the fluid-solid interface. Using the above assumption and the Oberbeck-Boussinesq approximation, the boundary layer equations are written as
$$\begin{aligned}& \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0, \end{aligned}$$
(1)
$$\begin{aligned}& \begin{aligned}[b] &u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y} \\ &\quad= u_{e} \frac{d u_{e}}{dx} + \nu\frac{\partial^{2} u}{\partial y^{2}}+ \frac{\nu}{K} (u_{e}-u) - \frac{(\rho_{p}-\rho_{f})Kg}{\mu}(C-C_{\infty}) \\ &\qquad{} + \frac{(1-S_{\infty})\rho_{f}Kg}{\nu} \bigl[\beta_{0}(T-T_{\infty})+ \beta _{1}(T-T_{\infty})^{2} +\beta_{2}(S-S_{\infty})+ \beta_{3}(S-S_{\infty})^{2} \bigr] , \end{aligned} \end{aligned}$$
(2)
$$\begin{aligned}& u\frac{\partial T}{\partial x}+ v\frac{\partial T}{\partial y} = \alpha_{m} \frac{\partial^{2} T}{\partial y^{2}}+ \tau D_{B} \frac{\partial T}{\partial y} \frac{\partial C}{\partial y} + \tau \frac{D_{T}}{T_{\infty}} \biggl(\frac{\partial T}{\partial y} \biggr)^{2}, \end{aligned}$$
(3)
$$\begin{aligned}& u\frac{\partial S}{\partial x}+ v\frac{\partial S}{\partial y} = D_{S} \frac{\partial^{2} S}{\partial y^{2}} + \biggl(\frac{D_{m} K_{T}}{T_{m}} \biggr) \frac{\partial^{2} T}{\partial y^{2}}, \end{aligned}$$
(4)
$$\begin{aligned}& u\frac{\partial C}{\partial x}+ v\frac{\partial C}{\partial y} = D_{B} \frac{\partial^{2} C}{\partial y^{2}} + \biggl(\frac{D_{T}}{T_{\infty}} \biggr) \frac{\partial^{2} T}{\partial y^{2}}, \end{aligned}$$
(5)
and the corresponding boundary conditions are
$$\begin{aligned} \begin{aligned} &u=cx+L(\partial u/ \partial y),\qquad v=0,\qquad T=T_{w}+k_{1}(\partial T/ \partial y),\qquad S=S_{w}+k_{2}(\partial S/ \partial y), \\ & C=C_{w}, \quad \mbox{at } y=0, \\ &u\rightarrow u_{e}(x)=ax,\qquad T\rightarrow T_{\infty},\qquad S\rightarrow S_{\infty},\qquad C\rightarrow C_{\infty}, \quad\mbox{as } y\rightarrow\infty, \end{aligned} \end{aligned}$$
(6)
u and v are the velocity components along the x- and y- directions, respectively, \(u_{e}(x)=ax\) is the ambient velocity of the fluid, a is a constant, ν, \(\rho_{f}\), and μ are the kinematic viscosity, density, and apparent viscosity of the base fluid, \(\rho_{P}\) is the density of the nanoparticle, K is the permeability of the porous medium, g is the acceleration due to gravity, \(\beta_{1}\) and \(\beta_{2}\) are the volumetric thermal expansion coefficient, respectively, \(\beta_{3}\) and \(\beta_{4}\) are the volumetric solute expansion coefficient, respectively, \(\alpha_{m}\) is the effective thermal diffusivity, \(\tau=\frac{(\rho c)_{p}}{(\rho_{c})_{f}}\), \((\rho c)_{p}\), and \((\rho c)_{f}\) are the volumetric heat capacity for nanoparticle and fluid, respectively, \(D_{B}\) and \(D_{T}\) are the Brownian and thermophoresis diffusion coefficients, respectively, \(D_{s}\) and \(D_{m}\) are the solute and mass diffusivities, respectively, \(K_{T}\) is the thermal diffusion ratio, c is a constant, L, \(k_{1}\), and \(k_{2}\) are the hydrodynamic, thermal, and solute slip factors, respectively.
We introduce the following similarity variables (see Ibrahim and Shankar [24]):
$$ \psi=\sqrt{a\nu}x f(\eta), \quad\eta=\sqrt{\frac{a}{\nu}} y, \quad \theta=\frac{T-T_{\infty}}{T_{w}-T_{\infty}},\quad \chi=\frac{S-S_{\infty}}{S_{w}-S_{\infty}}, \quad\phi= \frac{C-C_{\infty}}{C_{w}-C_{\infty}}, $$
(7)
where ψ is the stream function, which is defined in the usual way as \(u=\partial{\psi}/\partial{y}\) and \(v=-\partial{\psi}/\partial{x}\). Using the similarity variables, the governing equations are written as
$$\begin{aligned}& f'''+f f''+1-f^{\prime2}+K_{p} \bigl(1-f'\bigr) + \lambda \bigl[(1+\lambda_{1} \theta) \theta+Nc (1+\lambda_{2} \chi)\chi- Nr \phi \bigr]=0, \end{aligned}$$
(8)
$$\begin{aligned}& {\theta}''+Pr \bigl(f{\theta}'+ Nb { \theta}'{\phi}'+Nt {{\theta}'}^{2} \bigr)=0, \end{aligned}$$
(9)
$$\begin{aligned}& {\chi}''+Ln f{\chi}'+ Sr \theta''=0, \end{aligned}$$
(10)
$$\begin{aligned}& {\phi}''+Le f{\phi}'+ \biggl( \frac{Nt}{Nb} \biggr)\theta''=0, \end{aligned}$$
(11)
and the boundary conditions are written as
$$ \begin{aligned} &f'(0)=\alpha_{1}+ \alpha_{2}f''(0), \quad\theta (0)=1+ \alpha_{3}\theta'(0),\quad \chi(0)=1+\alpha_{4} \chi'(0),\quad \phi(0)=1, \\ &f'(\infty)=1, \quad\theta(\infty)=0,\quad \chi(\infty)=0, \quad \phi(\infty)=0. \end{aligned} $$
(12)
In the above equations, \(K_{p}={\nu}/{aK}\) is a parameter which is inversely proportional to the permeability K, \(\lambda=Gr_{x}/\hat{Re_{x}}^{2}\) is the mixed convection parameter, \(Gr_{x}\) is the Grashof number, \({Re_{x}}\) is the Reynolds number, Nc is the regular buoyancy parameter, \(\lambda_{1}\) and \(\lambda_{2}\) are the volumetric nonlinear thermal and solute constants, respectively, Nr is the buoyancy ratio parameter, Pr is the Prandtl number, Nb is the Brownian parameter, Nt is the thermophoresis parameter, Le is the Lewis number, \(\alpha_{1}\) is the velocity ratio or stretching ratio, \(\alpha_{2}\), \(\alpha_{3}\), and \(\alpha_{4}\) are momentum, thermal, and solute slip, respectively. These parameters are defined as
$$\begin{aligned}& Gr_{x}=\frac{(1-S_{\infty})\rho K g (T_{w}-T_{\infty})\beta_{0} x^{3}}{\nu^{3}},\qquad \hat{Re_{x}}= \frac{u_{e}(x)x}{\nu}, \qquad Nc=\frac{\beta_{2} (S_{w}-S_{\infty})}{\beta_{0} (T_{w}-T_{\infty})}, \\& \lambda_{1}=\frac{\beta_{1} (T_{w}-T_{\infty})}{\beta_{0}},\qquad \lambda_{2}= \frac{\beta_{3} (S_{w}-S_{\infty})}{\beta_{2}},\qquad Nr=\frac{(\rho_{p}-\rho_{f})(C_{w}-C_{\infty})}{\rho_{f}(1-S_{\infty})\beta_{0} (T_{w}-T_{\infty})}, \\& Pr=\frac{\nu}{\alpha_{m}},\qquad Nb=\frac{\tau D_{B}(C_{w}-C_{\infty})}{\nu},\qquad Nt=\frac{\tau D_{T} (T_{w}-T_{\infty})}{\nu T_{\infty}}, \\& Le=\frac{\alpha_{m}}{\varepsilon D_{B}},\qquad \alpha_{1}=c/a, \qquad \alpha_{2}=\sqrt{a/\nu}L,\qquad \alpha_{3}=k_{1} \sqrt{a/\nu}, \qquad \alpha_{4}=k_{2}\sqrt{a/\nu}. \end{aligned}$$
The parameters of practical interest are the skin friction, local Nusselt number \(Nu_{x}\), the local Sherwood number \(Sh_{x}\), and the local nanoparticle Sherwood number \(Nn_{x}\). These parameters are defined as
$$ C_{f}=\frac{x \tau_{w}}{\rho u_{e}^{2}}, \quad Nu_{x}= \frac{x q_{w}}{k(T-T_{\infty})}, \quad Sh_{x}=\frac{x q_{s}}{D_{S} (S-S_{\infty})},\quad Nn_{x}=\frac{x q_{n}}{D_{B} (C-C_{\infty})}, $$
(13)
where \(\tau_{w}=\mu (\frac{\partial u}{\partial y} )_{y=0}\), \(q_{w}=-k (\frac{\partial T}{\partial y} )_{y=0}\), \(q_{s}=-D_{S} (\frac{\partial S}{\partial y} )_{y=0}\), \(q_{n}=-D_{B} (\frac{\partial C}{\partial y} )_{y=0}\).
Using the above non-dimensionless and similarity transformation we get
$$ \begin{aligned} &Re_{x}^{1/2}C_{f}=f''(0), \qquad Re_{x}^{-1/2} Nu_{x}=-\theta'(0), \\ & Re_{x}^{-1/2}Sh_{x} = -\chi'(0), \qquad Re_{x}^{-1/2} Nn_{x}=-\phi'(0), \end{aligned} $$
(14)
where \(Re_{x}=u_{e}(x)x/\nu\) is the local Reynolds number.