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.