Open Access

Effects of slip on nonlinear convection in nanofluid flow on stretching surfaces

Boundary Value Problems20162016:2

https://doi.org/10.1186/s13661-015-0506-2

Received: 20 October 2014

Accepted: 7 December 2015

Published: 4 January 2016

Abstract

We investigate the effects of momentum, thermal, and solute slip boundary conditions on nanofluid boundary layer flow along a permeable surface. The conventional no-slip boundary conditions at the surface are replaced by slip boundary conditions. At moderate to high temperatures, the temperature-concentration dependence relation is nonlinear and the Soret effect is significant. The governing partial differential equations are solved numerically. The influence of significant parameters on the fluid properties as well as on the skin friction, local Nusselt number, local Sherwood number, and the local nanoparticle Sherwood number are determined. We show, among other results, that the existence and uniqueness of the solutions depends on the slip parameters, and that the region of existence of the dual solution increases with the slip parameters.

Keywords

stagnation-point flownonlinear convectionpartial slipdual solutionsSoret effect

1 Introduction

Boundary layer flow over a stretching surface is important as it occurs in several engineering process, for example, materials manufactured by extrusion. During the manufacturing process, a stretching sheet interacts with the ambient fluid both thermally and mechanically. Several studies on the dynamics of boundary layer flow over a stretching surface have appeared in the literature (Crane [1]; Dutta et al. [2]). Recently an innovative technique used for improving heat transfer is to add ultra fine solid particles to a base fluid, Choi [3]. Recent literature shows a significant rise in applications of nanofluids such as in microchannels (Ebrahimi et al. [4], enzyme biosensors (Li et al. [5]), drug delivery (Shaw and Murthy [6]) biomimetic microsystems (Huh et al. [7]) etc. An impressive review of nanofluid research is given by Das et al. [8]. Kameswaran et al. [9] studied homogeneous-heterogeneous reactions in a nanofluid flow over a porous stretching sheet. Mabood et al. [10] studied the MHD boundary layer flow of nanofluids over a nonlinear stretching sheet.

A significant number of studies have applied the no-slip boundary conditions at the wall. However, the no-slip assumption is not applicable when fluid flows in micro and nano channels and must be replaced by slip boundary conditions (Aziz [11]). Nield and Kuznetsov [12] presented an analytic solution for convection flow in channel or circular ducts saturated with a rarefied gas in a slip-flow regime. The slip condition applies to corner flows and in the extrusion of polynomial melts from a capillary tube (Thompson and Troian [13]; Nguyen and Wereley [14]). Karniadakis et al. [15] showed that hydrodynamic and thermal slip occur simultaneously. The difference between the fluid velocity at the wall and the velocity of the wall itself is directly proportional to the shear stress. The proportionality constant is called the slip length (Maxwell [16]; Hak [17]).

Beavers and Joseph [18] investigated fluid flow over a permeable wall with a slip boundary condition. The effects of a second order velocity-slip and temperature-jump on basic gaseous fluctuating micro-flows were analyzed by Hamdan et al. [19]. The effects of partial slip on steady boundary layer stagnation-point flow of an incompressible fluid and heat transfer from a shrinking sheet was investigated by Bhattacharyya et al. [20]. This was extended in Bhattacharyya et al. [21] to unsteady stagnation-point flow of a Newtonian fluid and heat transfer from a stretching sheet with partial slip conditions. Niu et al. [22] investigated slip flow and heat transfer in a non-Newtonian nanofluid in a microtube. Khan et al. [23] analyzed the effects of hydrodynamic and thermal slip boundary conditions on double-diffusive free convective flow of a nanofluid along a semi-infinite flat solid vertical plate. Ibrahim and Shankar [24] studied the effects of velocity, thermal and solutal slip condition on the MHD boundary layer flow of a nanofluid past a permeable stretching sheet. Mabood et al. [25] studied the MHD slip flow over a radiation stretching sheet by using the optimal homotopy asymptotic method.

Some thermal systems such as those encountered in reactor safety, combustion and solar collectors operate at moderate to very high temperatures. In such cases, the temperature-concentration dependence relation is nonlinear and the Soret effect is of immense important. Partha [26] studied natural convection in a non-Darcy porous medium with a nonlinear temperature-concentration-dependent density relation.

In this paper, we analyze the effects of momentum, temperature and solute slip on stagnation-point flow over a permeable stretching or shrinking sheet. We transform the governing partial differential equations into similarity equations which are then solved numerically. The effects of physical parameters on the flow, heat, mass and nanoparticle concentration are determined and presented graphically. In the present paper, we mainly focus on the effect of slip parameters on the governing system along with the nonlinear thermal convection. To the best of authors knowledge such study has not been reported earlier in the literature.

2 Mathematical formulation

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.
Figure 1

Schematic diagram of the problem.

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.

3 Results and discussion

The system of ordinary differential equations (8)-(11) along with boundary conditions (12) are integrated numerically by first choosing suitable initial guess values for \(f(0)\), \(f''(0)\), \(\theta'(0)\), \(\chi'(0)\), and \(\phi'(0)\) to match the boundary conditions at ∞. Matlab bvp4c solver was used to integrate the system of equations. To verify the accuracy of the numerical results, we compared our results with those reported by Noghrehabadi et al. [27] as shown in Table 1. The results are in very good agreement, thus lending confidence to the accuracy of the present results.
Table 1

Comparison of the reduced Nusselt number \(\pmb{-\theta'(0)}\) and Sherwood number \(\pmb{-\phi'(0)}\) with Noghrehabadi et al. [ 27 ] for \(\pmb{Nc=1}\) , \(\pmb{Nr=3}\) , \(\pmb{\delta_{1}=1}\) , \(\pmb{Le=Pr=10}\) . (a)  \(\pmb{\delta_{3}=\delta_{4}=\lambda=\gamma= Ln=Sr = K_{p}=0}\) and (b) \(\pmb{\delta_{3}=\delta_{4}=\lambda=0}\) , \(\pmb{Ln=1}\) , \(\pmb{Sr=0.5}\) , \(\pmb{K_{p}=0.5}\)

\(\boldsymbol {\delta_{2}}\)

Nb

Nt

Noghrehabadi et al. [ 27 ]

Present (a)

Present (b)

Nur

Shr

Nur

Shr

Nur

Shr

0.0

0.1

0.1

0.952377

2.129394

0.95237683

2.12939377

1.00058819

2.66004285

0.3

0.520079

2.528638

0.52007905

2.52863816

0.54419785

3.46549450

0.5

0.321054

3.035142

0.32105433

3.03514247

0.31586598

4.38704894

0.2

0.1

0.505581

2.381871

0.50558141

2.38187064

0.54400672

2.85986019

0.3

0.273096

2.655459

0.27309580

2.65545946

0.28254157

3.33932740

0.5

0.168077

2.888339

0.16807658

2.88833918

0.16127263

3.75314398

0.3

0.1

0.252156

2.410019

0.25215609

2.41001880

0.27177120

2.86844013

0.3

0.135514

2.608819

0.13551419

2.60881871

0.13604735

3.19990440

0.5

0.083298

2.751875

0.08329860

2.75187540

0.07566950

3.45202615

1.0

0.1

0.1

2.751875

1.607430

0.71892800

1.60743180

1.14988755

3.00376688

0.3

0.392596

1.908809

0.39259606

1.90881247

0.63804840

3.78822390

0.5

0.242357

2.291156

0.24235674

2.29116127

0.37823785

4.77292460

0.2

0.1

0.381652

1.798019

0.38165211

1.79802095

0.63400147

3.28337188

0.3

0.206154

2.004545

0.20615392

2.00454704

0.33706394

3.78349666

0.5

0.126877

2.180339

0.12687726

2.18034274

0.19696830

4.23378793

10.0

0.1

0.1

0.412468

0.922099

0.41247939

0.92225151

1.20480363

3.13488626

0.3

0.225245

1.094883

0.22524897

1.09516618

0.67291837

3.91228460

0.5

0.139050

1.314098

0.13905033

1.31453566

0.40123365

4.91834227

0.2

0.1

0.218955

1.031454

0.21896995

1.03160059

0.66701497

3.44165247

0.3

0.118275

1.149881

0.11827923

1.15009326

0.35709376

3.94942212

0.5

1.204555

1.250672

0.07279486

1.25095500

0.21002703

4.41174227

0.3

0.1

0.109199

1.043650

0.10921014

1.04379192

0.33770638

3.47832715

0.3

0.058689

1.129708

0.05869191

1.12989321

0.17488396

3.84292518

0.5

0.036078

1.191622

0.03607706

1.19185182

0.10098316

4.12849783

A representative set of graphical results for the velocity, temperature, solute concentration and nanoparticle volume fraction as well as the skin friction, local Nusselt number, local Sherwood number and local nanoparticle Sherwood number is presented and discussed for different parametric values. We note that solutions of equations (8) and (11) exist for all values of \(\alpha_{1}>0\), while in the case of a shrinking surface (\(\alpha_{1}<0\)), the equations have a solution only in the range \(\alpha_{1}>{\alpha_{1}}_{\mathrm{crit}}\), where \({\alpha_{1}}_{\mathrm{crit}}\) is a critical value of \(\alpha_{1}\). This critical value depends on other parameter values. There are no solutions real when \(\alpha_{1}<{\alpha_{1}}_{\mathrm{crit}}\). Dual solutions of the boundary layer equations appear in the range \({\alpha_{1}}_{\mathrm{crit}}<\alpha_{1}\). As noted by Merkin [28], Postelnicu and Pop [29], the first solution is stable and physically realizable, while the second solution is unstable. In this study our primary focus is on the dual solutions and the effects of the slip coefficients and nonlinear volumetric thermal and solute constants.

The influence of the parameters Nr, Nc, λ, and \(K_{p}\) on the velocity profiles is shown in Figure 2. Dual solutions are observed in both instances. We note that the buoyancy ratio parameter has the effect of decelerating the fluid flow along the surface. This is reflected by the decrease in the fluid velocity in the vicinity of the surface. Hence the momentum boundary layer thickness decreases with an increase in Nr. The regular buoyancy ratio parameter Nc is similar to Nr and hence the momentum boundary layer thickness decreases with an increase in Nc. The parameter \(K_{p}\) is inversely proportional to the permeability of the medium K, hence the porous medium drag increases with K and so the velocity of the fluid increases with \(K_{p}\). The momentum boundary layer thickness increased with an increase in the mixed convection parameter. Dual solutions were obtained for the opposing flow when \(\lambda<1\).
Figure 2

Effect of the velocity as a function of (a) Nc , Nr ( \(\pmb{\lambda =1}\) , \(\pmb{K_{p}=0.1}\) ), (b) λ , \(\pmb{K_{p}}\) ( \(\pmb{Nc=Nr=0.1}\) ) with \(\pmb{\lambda_{1}=\lambda_{2}=Nt=Nb=0.1}\) , \(\pmb{Pr=0.5}\) , \(\pmb{Ln=Le=1}\) , \(\pmb{Sr=0.5}\) , \(\pmb{\alpha_{1}=-1.6}\) , \(\pmb{\alpha_{2}=0.5}\) , \(\pmb{\alpha_{3}=\alpha_{4}=0.1}\) .

Figure 3 shows the effect of the nanofluid Lewis number Ln, the Soret number Sr, the Lewis number Le and the Prandtl number Pr on the solute and the nanoparticle concentrations when the other parameters are fixed. The concentration boundary layer thickness decreases with an increase in the nanofluid Lewis number. This is because an increase in the nanofluid Lewis number causes a reduction in the mass diffusivity of the nanofluid which in turn reduces the velocity of the fluid as well as the solute concentration. The solute concentration boundary layer thickness decreases with an increase in Ln while the nanoparticle volume fraction decreases with an increase in the Lewis number. The Prandtl number enhances the viscous diffusion rate which in turn reduces the velocity of the fluid and increases the concentration of the nanoparticle.
Figure 3

Solute concentration as a function of Ln , Sr ( \(\pmb{Le=1}\) , \(\pmb{Pr=0.5}\) ) (a) and (b) nanoparticle concentration as a function of Le , Pr ( \(\pmb{Ln=1}\) , \(\pmb{Sr=0.5}\) ) with \(\pmb{K_{p}=\lambda=\lambda_{1}=\lambda_{2}=Nr=Nc=0.1}\) , \(\pmb{\alpha_{1}=-1.6}\) , \(\pmb{\alpha_{2}=0.5}\) , \(\pmb{\alpha_{3}=\alpha_{4}=0.1}\) .

The effects of the nonlinear temperature and concentration coefficient on the velocity, temperature, solute concentration and nanoparticle concentration are shown in Figure 4. Dual solutions were obtained when \(\alpha_{1}=-1.6\) for all other parameter values. We observed that the effect of the convection nonlinearity (\(\lambda _{1}\) and \(\lambda_{2}\)) is to reduce the thermal and solute boundary layer thicknesses. A similar observation was made by Partha [26]. This may partly be due to the nonlinear term enhancing the solute and nanoparticle density gradients near the wall. A careful observation shows that the effect of \(\lambda_{1}\) is more significant compared with \(\lambda_{2}\), particularly with respect to the momentum boundary layer profiles.
Figure 4

Effect of \(\pmb{\lambda_{1}}\) and \(\pmb{\lambda_{2}}\) on (a) velocity, (b) temperature, (c) solute concentration, and (d) nanoparticle concentration with \(\pmb{K_{p}=\lambda=Nr=Nc=0.1}\) , \(\pmb{Le=Ln=1}\) , \(\pmb{Pr=Sr=0.5}\) , \(\pmb{\alpha_{1}=-1.6}\) , \(\pmb{\alpha_{2}=0.5}\) , \(\pmb{\alpha_{3}=\alpha_{4}=0.1}\) .

The slip coefficients have a significant influence on the velocity, temperature, solute, and nanoparticle concentration as clearly shown in Figure 5. The velocity of the nanofluid increases with momentum slip constant \(\alpha_{2}\). This is because momentum slip enhances the velocity at the fluid-solid interface. In the case of the no-slip condition, the fluid velocity adjacent to a solid surface is equal to the velocity of the stretching sheet i.e., \(f'(0)=\alpha_{1}=1\) while with an increase in other slip parameters, the momentum boundary layer thickness decreases. The thermal slip coefficient mainly affects the temperature profile and reduces the temperature at the surface. With thermal slip, more heat is transferred leading to a reduction in surface temperature and this reduces the velocity of the nanofluid. As a result both solute and nanoparticle concentration thickness decreases. The solute concentration is mainly affected by the solute slip coefficient and the thickness of the solute concentration decreases with increase of the solute slip coefficient.
Figure 5

Effect of \(\pmb{\alpha_{2}}\) , \(\pmb{\alpha_{3}}\) , and \(\pmb{\alpha_{4}}\) on (a) velocity, (b) temperature, (c) solute concentration, and (d) nanoparticle concentration with \(\pmb{K_{p}=\lambda=\lambda_{1}=\lambda_{2}=Nr=Nc=0.1}\) , \(\pmb{Le=Ln=1}\) , \(\pmb{Pr=Sr=0.5}\) , \(\pmb{\alpha_{1}=1}\) .

Figures 6 and 7 show the Nusselt number, Sherwood number, and density of the nanoparticle as a function of nonlinear thermal and solute constants for different values of \(K_{p}\). With an increase in the permeability parameter, the velocity of the fluid reduces, hence as \(K_{p}\) increases, the surface temperature gradient, solute, and the nanoparticle concentration at the surface decrease which helps to reduce the local Nusselt number, local Sherwood number, and local nanoparticle Sherwood number. It evident that nonlinearity of the temperature and solute concentration both enhance the local Nusselt number and the local Sherwood number. This may be due to the fact that the nonlinearity term increases the temperature and concentration of the solute at the surface. A similar behavior is observed in the case of the nanoparticle concentration. It is interesting to note that the nonlinear curves are highly influenced by the nonlinear convection parameter as compared to the nonlinear concentration parameter.
Figure 6

Local Nusselt number as a function Kp with \(\pmb{\lambda=1}\) , \(\pmb{Nt=Nb=Nr=Nc=0.1}\) , \(\pmb{Pr=0.5}\) , \(\pmb{Ln=Le=1}\) , \(\pmb{Sr=0.5}\) , \(\pmb{\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=0.1}\) .

Figure 7

Effects of nonlinear thermal and solute constants on (a) local Sherwood number and (b) local density of nanoparticles as a function Kp with \(\pmb{\lambda=1}\) , \(\pmb{Nt=Nb=Nr=Nc=0.1}\) , \(\pmb{Pr=0.5}\) , \(\pmb{Ln=Le=1}\) , \(\pmb{Sr=0.5}\) , \(\pmb{\alpha_{1}=\alpha_{2}=\alpha_{3}=\alpha_{4}=0.1}\) .

Figure 8 shows the skin friction, local Nusselt number, local Sherwood number, and local nanoparticle Sherwood number for different values of the thermal and solute slip constants. Here \(\alpha_{2}=0.5\) is kept constant. As seen in Wang [30], in the case of the no-slip boundary conditions (i.e., \(\alpha_{2}=\alpha_{3}=\alpha_{4}=0\)), the solution is unique for \(\alpha_{1}>-1\), there exist dual solutions for \(-1.2465 \leq \alpha_{1} \leq-1\), and no solution is found when \(\alpha_{1}<-1.2465\). The first solution is shown by marker while the second solution by the dotted line. With slip conditions, the range of \(\alpha_{1}\) values for which a unique solution as well as dual solutions exists significantly increases. The increased range of values of \(\alpha_{1}\) depends principally on the momentum slip coefficient. The critical values of \(\alpha_{1}\) are shown in Table 2. The critical value of \(\alpha_{1}\) decreases from −1.902 to −1.899 when the thermal slip coefficient increases from 0.1 to 0.5. Similarly the critical \(\alpha_{1}\) decreases with an increase in the solute concentration. But the critical \(\alpha_{1}\) is enhanced by the momentum slip coefficient. It is observed that the lower limit of the critical \(\alpha_{1}\) is influenced more by the momentum slip parameter than by the thermal and concentration slip since it is part of the nanofluid velocity (see Figure 5(a)), which makes the system more unstable. A close observation shows that the dual solution region increases nonlinearly with increase of the momentum slip coefficient. The skin friction initially increases with momentum slip parameter but after a certain point (in this case \(\alpha_{1}=-1.56\)) it decreases nonlinearly.
Figure 8

Effect of the slip constants on (a) skin friction, (b) local Nusselt number, (c) local Sherwood number, and (d) local density of nanoparticle with \(\pmb{K_{p}=0.1}\) , \(\pmb{\lambda=\lambda_{1}=\lambda_{2}=1}\) , \(\pmb{Nt=Nb=Nr=Nc=0.1}\) , \(\pmb{Pr=0.5}\) , \(\pmb{Ln=Le=1}\) , \(\pmb{Sr=0.5}\) , \(\pmb{\alpha_{2}=0.5}\) .

Table 2

Value of critical velocity ratio \(\pmb{\alpha_{1}}\) for different momentum, thermal, and solute slip constants for \(\pmb{Kp=\lambda_{1}=\lambda_{2}=Nt=Nb=Nc=Nr=0.1}\) , \(\pmb{Pr=0.5}\) , \(\pmb{Le=Ln=1}\) , \(\pmb{Sr=0.5}\)

\(\boldsymbol {(\alpha_{1})}_{\mathbf{crit}}\)

\(\boldsymbol {\alpha_{2}}\)

\(\boldsymbol {\alpha_{3}}\)

\(\boldsymbol {\alpha_{4}}\)

dual solution range

−1.902

0.5

0.1

0.1

−1.902, −1.18

−1.899

0.5

0.5

0.1

−1.899, −1.18

−1.898

0.5

0.5

0.5

−1.898, −1.18

−2.622

1

0.1

0.1

−2.622, −1.18

−3.407

1.5

0.1

0.1

−3.407, −1.18

The local Nusselt number, local Sherwood number, and the local nanoparticle density are nonlinear increasing functions of the momentum slip parameter for the first solution while they are a decreasing function for the case of the second solution. It is interesting to note that the generation of vorticity for the shrinking velocity is reduced by an increase in the momentum slip at the surface (when \(\alpha_{1}>1\)). The momentum slip parameter enhances the velocity at the surface which forces the solute and particle to move away from the surface. As a result, the local Nusselt number, local Sherwood number, and local nanoparticle Sherwood number increase nonlinearly with the momentum slip parameter but decrease with increases in the solute slip coefficient. It evident that the local Nusselt number decreases with increase of the partial thermal slip coefficient and this finding is similar to earlier results by Zheng et al. [31]. The nature of the second solution for the local Nusselt number, local Sherwood number, and local nanoparticle Sherwood number is quite similar and this mainly depends on the momentum slip coefficient rather than the other slip coefficient.

4 Conclusions

The effect of momentum, thermal, and solute slip on nonlinear convection boundary layer flow from a stretching and shrinking sheet has been investigated numerically. Analysis of stagnation-point slip flow from a shrinking sheet has shown that existence and uniqueness of the solution depends on the slip parameters, mainly the momentum slip and the velocity ratio parameter \(\alpha_{1}\). Dual solutions were obtained when the velocity ratio was less than a certain critical value. The region of existence of the dual solution increases with the slip parameters. The nonlinear temperature and concentration coefficients reduce the thermal and solute boundary layer thicknesses. The thermal slip coefficient reduces the momentum and thermal boundary layer thickness. The local Nusselt number, local Sherwood number, and local density of the nanoparticles increase nonlinearly with the convection coefficient.

Declarations

Acknowledgements

The authors are grateful to the University of KwaZulu-Natal for financial support.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal
(2)
Fluid Dynamics Division, School of Advanced Sciences, VIT University

References

  1. Crane, LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645-647 (1970) View ArticleGoogle Scholar
  2. Dutta, BK, Roy, P, Gupta, AS: Temperature field in the flow over a stretching sheet with uniform heat flux. Int. Commun. Heat Mass Transf. 12, 89-94 (1985) View ArticleGoogle Scholar
  3. Choi, SUS: Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer, DA, Wang, HP (eds.) Developments and Applications of Non-Newtonian Flows, vol. 99. ASME, New York (1995) Google Scholar
  4. Ebrahimi, S, Sabbaghzadeh, J, Lajevardi, M, Hadi, I: Cooling performance of a microchannel heat sink with nanofluids containing cylindrical nanoparticles (carbon nanotubes). Heat Mass Transf. 46, 549-553 (2010) View ArticleGoogle Scholar
  5. Li, H, Liu, S, Dai, Z, Bao, J, Yang, Z: Applications of nanomaterials in electrochemical enzyme biosensors. Sensors 9, 8547-8561 (2009) View ArticleGoogle Scholar
  6. Shaw, S, Murthy, PVSN: The effect of shape factor on the magnetic targeting in the permeable microvessel with two-phase Casson fluid model. J. Nanotechnol. Eng. Med. 2, 041003 (2011) View ArticleGoogle Scholar
  7. Huh, D, Matthews, BD, Mammoto, A, Montoya-Zaval, M, Hsin, NY, Ingber, DE: Reconstituting organ-level lung functions on a chip. Science 328, 1662-1668 (2010) View ArticleGoogle Scholar
  8. Das, SK, Choi, SUS, Yu, W, Pradeep, T: Nanofluids: Science and Technology. Wiley, Hoboken (2007) View ArticleGoogle Scholar
  9. Kameswaran, PK, Shaw, S, Sibanda, P, Murthy, PVSN: Homogeneous-heterogeneous reactions in a nanofluid flow due to a porous stretching sheet. Int. J. Heat Mass Transf. 57, 465-472 (2013) View ArticleGoogle Scholar
  10. Mabood, F, Khan, WA, Ismail, AIM: MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet: a numerical study. J. Magn. Magn. Mater. 374, 569-576 (2015) View ArticleGoogle Scholar
  11. Aziz, A: Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Commun. Nonlinear Sci. Numer. Simul. 15, 573-580 (2010) MathSciNetView ArticleGoogle Scholar
  12. Nield, DA, Kuznetsov, AV: Forced convection with slip-flow in a channel or duct occupied by a hyper-porous medium saturated by a rarefied gas. Transp. Porous Media 64, 161-170 (2006) MathSciNetView ArticleGoogle Scholar
  13. Thompson, PA, Troian, SM: A general boundary condition for liquid flow at solid surfaces. Nature 389, 360-362 (1997) View ArticleGoogle Scholar
  14. Nguyen, NT, Wereley, ST: Fundamentals and Applications of Microfluidics. Artech House, London (2009) Google Scholar
  15. Karniadakis, G, Beskok, A, Aluru, N: Microflows and Nanoflows. Fundamentals and Simulation. Springer, New York (2005) MATHGoogle Scholar
  16. Maxwell, JC: On stresses in rarefied gases arising from inequalities of temperature. Philos. Trans. R. Soc. Lond. 170, 231-256 (1879) MATHView ArticleGoogle Scholar
  17. Hak, GM: Flow physics. In: Gad-el-Hak, M (ed.) The MEMS Handbook. CRC Press, Boca Raton (2002) Google Scholar
  18. Beavers, GS, Joseph, DD: Boundary condition at a naturally permeable wall. J. Fluid Mech. 30, 197-207 (1967) View ArticleGoogle Scholar
  19. Hamdan, MA, Al-Nimr, MA, Hammoudeh, VA: Effect of second order velocity-slip/temperature-jump on basic gaseous fluctuating micro-flows. J. Fluids Eng. 132, 074503 (2010) View ArticleGoogle Scholar
  20. Bhattacharyya, K, Mukhopadhyay, S, Layek, GC: Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet. Int. J. Heat Mass Transf. 54, 308-313 (2011) MATHView ArticleGoogle Scholar
  21. Bhattacharyya, K, Mukhopadhyay, S, Layek, GC: Slip effects on an unsteady boundary layer stagnation-point flow and heat transfer towards a stretching sheet. Chin. Phys. Lett. 28, 094702 (2011) View ArticleGoogle Scholar
  22. Niu, J, Fu, C, Tan, W: Slip-flow and heat transfer of a non-Newtonian nanofluid in a microtube. PLoS ONE 7, e37274 (2012) View ArticleGoogle Scholar
  23. Khan, WA, Uddin, MJ, Md Ismail, AI: Hydrodynamic and thermal slip effect on double-diffusive free convective boundary layer flow of a nanofluid past a flat vertical plate in the moving free stream. PLoS ONE 8, e54024 (2013) View ArticleGoogle Scholar
  24. Ibrahim, W, Shankar, B: MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions. Comput. Fluids 75, 1-10 (2013) MATHMathSciNetView ArticleGoogle Scholar
  25. Mabood, F, Khan, WA, Uddin, MJ, Md Ismail, AI: Optimal homotopy asymptotic method for MHD slips flow over a radiating stretching sheet with heat transfer. Far East J. Appl. Math. 90, 21-40 (2015) View ArticleGoogle Scholar
  26. Partha, MK: Nonlinear convection in a non-Darcy porous medium. Appl. Math. Mech. 31, 565-574 (2010) MATHMathSciNetView ArticleGoogle Scholar
  27. Noghrehabadi, A, Pourrajab, R, Ghalambaz, M: Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature. Int. J. Therm. Sci. 54, 253-261 (2012) View ArticleGoogle Scholar
  28. Merkin, JH: On dual solutions occurring in mixed convection in a porous media. J. Eng. Math. 20, 171-179 (1985) MathSciNetView ArticleGoogle Scholar
  29. Postelnicu, A, Pop, I: Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge. Appl. Math. Comput. 217, 4359-4368 (2011) MATHMathSciNetView ArticleGoogle Scholar
  30. Wang, CY: Stagnation flow towards a shrinking sheet. Int. J. Non-Linear Mech. 43, 377-382 (2008) View ArticleGoogle Scholar
  31. Zheng, L, Zhang, C, Zhang, X, Zhang, J: Flow and radiation heat transfer of a nanofluid over a stretching sheet with velocity slip and temperature jump in porous medium. J. Franklin Inst. 350, 990-1007 (2013) MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Shaw et al. 2016