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# Dual solutions of stagnation-point flow of a nanofluid over a stretching surface

- Peri K Kameswaran
^{1}, - Precious Sibanda
^{1}Email author, - Chetteti RamReddy
^{2}and - Prabhala VSN Murthy
^{3}

**2013**:188

https://doi.org/10.1186/1687-2770-2013-188

© Kameswaran et al.; licensee Springer 2013

**Received:**10 May 2013**Accepted:**7 August 2013**Published:**22 August 2013

## Abstract

The paper discusses the effects of homogeneous-heterogeneous reactions on stagnation-point flow of a nanofluid over a stretching or shrinking sheet. The model presented describes mass transfer in copper-water and silver-water nanofluids. The governing system of equations is solved numerically, and the study shows that dual solutions exist for certain suction/injection, stretching/shrinking and magnetic parameter values. Comparison of the numerical results is made with previously published results for special cases.

## Keywords

- homogeneous-heterogeneous reactions
- nanofluids
- volume fraction model
- stretching or shrinking sheet
- magnetic field strength

## 1 Introduction

Problems involving fluid flow over stretching or shrinking surfaces can be found in many manufacturing processes such as in polymer extrusion, wire and fiber coating, foodstuff processing, *etc.* Crane [1] was the first to consider the steady two-dimensional flow of a Newtonian fluid driven by a stretching elastic flat sheet which moved in its own plane with velocity varying linearly with the distance from a fixed point. This study was subsequently extended by many authors to explore various aspects of heat transfer in a fluid surrounding a stretching sheet (Tsou *et al.* [2], Erickson *et al.* [3], Mucoglu and Chen [4], Grubka and Bobba [5], Karwe and Jaluria [6], Chen [7], Abo-Eldahab and El-Aziz [8], Salem and El-Aziz [9], Ali [10], Ishak *et al.* [11]).

The magnetohydrodynamic effect has important engineering applications in electrical motors. Heat transfer over a stretching or shrinking sheet subject to an external magnetic field, viscous dissipation and joule effects was studied by Jafar *et al.* [12]. They observed that the flow and heat transfer characteristics for a shrinking sheet were quite different from those of a stretching sheet. Lok *et al.* [13] analyzed MHD stagnation-point flow from a shrinking sheet. They found that dual solutions existed for small values of the magnetic parameter. The stagnation-point flow over a stretching or shrinking sheet in a nanofluid was investigated by Bachok *et al.* [14]. They showed that adding nanoparticles to a base fluid increased the skin friction and heat transfer coefficients. Recently, Narayana and Sibanda [15] investigated the laminar flow of a nanoliquid film over an unsteady stretching sheet. They noticed that the effect of an increase in the nanoparticle volume fraction was to reduce the axial velocity and free stream velocity in the case of a Cu-water nanoliquid. However, the opposite appeared to be true in the case of an ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$-water nanofluid. Kameswaran *et al.* [16] studied the effects of viscous dissipation and a chemical reaction in hydromagnetic nanofluid flow due to a stretching or shrinking sheet. They found that the velocity profiles decreased with an increase in the nanoparticle volume fraction, while the opposite was true in the case of temperature and concentration profiles. The study further showed that liquids with nanoparticle suspensions were better suited for effective cooling of the stretching sheet due to their enhanced conductivity and thermal properties.

Most chemically reacting systems involve both homogeneous and heterogeneous reactions (combustion, catalysis and biochemical systems). The simple combustion model helps us to understand the combustion phenomenon in many complex engineering applications such as in aircraft and rocket engines. A model for isothermal homogeneous-heterogeneous reactions in the boundary layer flow of a viscous fluid past a flat plate was presented by Merkin [17]. He modeled the homogeneous reaction by a cubic autocatalysis process and the heterogeneous reaction by a first-order process. Chaudhary and Merkin [18] analyzed homogeneous-heterogeneous reactions in boundary layer flow. They presented a numerical solution of the boundary layer equations near the leading edge of a flat plate. Ziabakhsh *et al.* [19] studied the diffusion of a chemically reactive species into a nonlinearly stretching sheet immersed in a porous medium. Chambre and Acrivos [20] studied isothermal chemical reactions on laminar boundary layer flow. The two-dimensional stagnation-point flow near an infinite permeable wall with a homogeneous-heterogeneous reaction was studied by Khan and Pop [21], while Khan and Pop [22] and Bachok *et al.* [23] studied the effects of homogeneous-heterogeneous reactions on fluid flow due to a stretching sheet. Recently, the effects of homogeneous-heterogeneous reactions in nanofluid flow due to a porous stretching sheet were studied by Kameswaran *et al.* [24]. They found that the concentration at the surface decreased with the strength of the heterogeneous reaction. In the case of a shrinking sheet, they showed that the velocity profiles decreased with increasing nanoparticle volume fraction in the case of a Cu-water nanofluid.

This article presents a study of homogeneous-heterogeneous reactions on MHD nanofluid stagnation point flow due to a stretching or shrinking sheet. The transformed nonlinear conservation equations are solved numerically.

## 2 Mathematical formulation

*x*-axis along the sheet and the

*y*-axis normal to the sheet. The flow configuration and the coordinate system are shown in Figure 1. The velocity of the outer flow is of the form ${u}_{e}(x)={U}_{\mathrm{\infty}}x$ and the velocity of the stretching or shrinking sheet is ${u}_{w}(x)={U}_{w}x$, where ${U}_{w}$ and ${U}_{\mathrm{\infty}}$ are constants. The fluid is a water-based nanofluid containing copper (Cu) or silver (Ag) nanoparticles. The base fluid and the nanoparticles are in thermal equilibrium with no slip occurring between them. We assume the simple homogeneous-heterogeneous reaction model proposed by Chaudhary and Merkin [18] of the form

*a*and

*b*are the concentrations of the chemical species

*A*and

*B*, and ${k}_{c}$ and ${k}_{s}$ are the rate constants. We assume that both reaction processes are isothermal. Under these assumptions, the boundary layer equations governing the flow can be written in the dimensional form [18, 25]

*u*,

*v*are the velocity components in the

*x*and

*y*directions, respectively, ${D}_{A}$ and ${D}_{B}$ are the respective diffusion species coefficients of

*A*and

*B*, ${a}_{0}$ is a positive constant. The effective dynamic viscosity of the nanofluid was given by Brinkman [26] as

*ϕ*is the solid volume fraction of nanoparticles. The effective density of the nanofluids is given as

Here, the subscripts *nf*, *f* and *s* represent the thermophysical properties of the nanofluid, the base fluid and nanoparticles, respectively.

*ψ*such that

where $\psi ={({U}_{\mathrm{\infty}}{\nu}_{f})}^{\frac{1}{2}}xf(\eta )$, $f(\eta )$ is the dimensionless stream function and $\eta ={({U}_{\mathrm{\infty}}/{\nu}_{f})}^{\frac{1}{2}}y$.

*A*and

*B*are represented as

*M*, the Schmidt number

*Sc*, the measure of the strength of the homogeneous reaction

*K*, the ratio of diffusion coefficients

*δ*, the mass transfer parameter ${f}_{w}$, with ${f}_{w}>0$ for suction and ${f}_{w}<0$ for injection, the measure of the strength of the heterogeneous reaction ${K}_{s}$, the Reynolds number

*Re*and $\lambda ={U}_{w}/{U}_{\mathrm{\infty}}$ is the stretching parameter, with $\lambda >0$ for stretching and $\lambda <0$ for shrinking, respectively. These parameters are respectively defined as

*A*and

*B*to be of a comparable size. This leads us to making a further assumption that the diffusion coefficients ${D}_{A}$ and ${D}_{B}$ are equal,

*i.e.*, $\delta =1$ (Chaudhary and Merkin [18]). In this case we have, from equations (16), (17) and (18),

where ${\mathit{Re}}_{x}=x{u}_{e}(x)/{\nu}_{f}$ is the local Reynolds number.

## 3 Results and discussion

**Thermophysical properties of water,** **Cu-water and** **Ag-water nanofluids, Oztop and Abu-Nada** [27]

Properties → | ρ (kg/m | ${\mathit{C}}_{\mathit{p}}$ (J/kgK) | k (W/mK) |
---|---|---|---|

Pure water | 997.1 | 4,179 | 0.613 |

Cu | 8,933 | 385 | 401 |

Ag | 10,500 | 235 | 429 |

*et al.*[12], Bachok

*et al.*[23] and Wang [28] in Tables 2, 3 and 4.

**Comparison of** ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ **from current results with Wang** [28] **and Jafar** et al. [12] **for various values of** λ **for the stretching sheet when** $\mathit{M}\mathbf{=}\mathit{K}\mathbf{=}\mathit{Sc}\mathbf{=}{\mathit{f}}_{\mathit{w}}\mathbf{=}{\mathit{K}}_{\mathit{s}}\mathbf{=}\mathbf{0}$ **and** $\mathit{\varphi}\mathbf{=}\mathbf{0}$

**Comparison of** ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ **from current results with Wang** [28] **and Jafar** et al. [12] **for various values of** λ **for the shrinking sheet when** $\mathit{M}\mathbf{=}\mathit{K}\mathbf{=}\mathit{Sc}\mathbf{=}{\mathit{f}}_{\mathit{w}}\mathbf{=}{\mathit{K}}_{\mathit{s}}\mathbf{=}\mathbf{0}$ **and** $\mathit{\varphi}\mathbf{=}\mathbf{0}$

**Comparison of** ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{(}\mathbf{0}\mathbf{)}$ **obtained by Bachok** et al. [23] **with the present results for particular values of** ${\mathit{K}}_{\mathit{s}}\mathbf{=}\mathit{K}\mathbf{=}\mathit{Sc}\mathbf{=}\mathbf{1}$**,** ${\mathit{f}}_{\mathit{w}}\mathbf{=}\mathit{M}\mathbf{=}\mathit{\varphi}\mathbf{=}\mathbf{0}$

λ | ${\mathit{f}}_{\mathbf{1}}^{\mathbf{\prime}\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ [23] | ${\mathit{f}}_{\mathbf{1}}^{\mathbf{\prime}\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ (Present) | ${\mathit{f}}_{\mathbf{2}}^{\mathbf{\prime}\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ [23] | ${\mathit{f}}_{\mathbf{2}}^{\mathbf{\prime}\mathbf{\prime}}\mathbf{(}\mathbf{0}\mathbf{)}$ (Present) |
---|---|---|---|---|

−1.15 | 1.0822 | 1.08223 | 0.1167 | 0.11670 |

−1.20 | 0.9325 | 0.93247 | 0.2336 | 0.23365 |

−1.21 | 0.8921 | 0.89209 | 0.2679 | 0.26789 |

−1.22 | 0.8451 | 0.84511 | 0.3088 | 0.30885 |

−1.23 | 0.7874 | 0.78744 | 0.3606 | 0.36062 |

−1.24 | 0.7066 | 0.70661 | 0.4357 | 0.43567 |

−1.242 | 0.6838 | 0.68376 | 0.4574 | 0.45738 |

−1.244 | 0.6551 | 0.65508 | 0.4849 | 0.48492 |

−1.246 | 0.6098 | 0.60983 | 0.5290 | 0.52903 |

Tables 2 and 3 give the coefficient ${f}^{\u2033}(0)$ for different parameter values. Table 2 gives a comparison of the present results with those obtained by Jafar *et al.* [12] and Wang [28] when $M=K={K}_{s}=\mathit{Sc}={f}_{w}=\varphi =0$, for different values of the stretching parameter. We observe that for increasing *λ*, the present results are in good agreement with results in the literature.

Table 4 gives the values of ${f}^{\u2033}(0)$ for selected *λ* when ${K}_{s}=K=\mathit{Sc}=1$, ${f}_{w}=M=\varphi =0$. We note here that with decreasing *λ* the first solution ${{f}_{1}}^{\u2033}(0)$ decreases while the second solution ${{f}_{2}}^{\u2033}(0)$ increases. These results are in good agreement with the results obtained by Bachok *et al.* [23] in the absence of the particular physical parameter.

Figure 2 shows the effects of both stretching and shrinking on the velocity profiles in the case of a Cu-water nanofluid. We observe that in both cases, the velocity profiles increase with the parameter *λ*. Further, we note that for a shrinking sheet, the velocity in the case of a Cu-water nanofluid is larger than that of a clear fluid. The opposite is, however, true for the case of a stretching sheet. The momentum boundary layer thickness decreases as *λ* increases and the flow has an inverted boundary layer structure when $\lambda <1$. The findings in the case of a clear fluid are similar to the results obtained by Jat and Chaudhary [29].

*λ*and

*M*. Furthermore, increasing the value of

*M*also causes thinning of the boundary layer. This implies an increase in the velocity gradient ${f}^{\u2033}(0)$. Thus the magnetic field enhances the fluid motion in the boundary layer in the case of a clear fluid. The same trend is observed in the case of a Cu-water nanofluid. We also observe in the case of injection that, with increasing magnetic parameter, the increment in the momentum boundary layer is more significant than in the case of suction.

*λ*varies from $\lambda =-1$ to $\lambda =1$ for both the clear fluid and the Cu-water nanofluid. However, beyond $\lambda =1$, the opposite is true for the clear fluid and the Cu-water nanofluid.

*K*and ${K}_{s}$ is shown in Figures 5 and 6, respectively. From Figure 5 we observe that the concentration at the surface decreases as the strength of the heterogeneous reaction increases. This is simply explained by the fact that the strength of the chemical reaction depends on the concentration. On the other hand, from Figure 6, we found that $g(0)$ decreases with increasing

*K*and ${K}_{s}$. These findings are similar to the results reported by Kameswaran

*et al.*[24].

*K*and

*λ*on the concentration when the other parameters are fixed. We note, as expected, that the wall concentration decreases as the strength of the homogeneous reaction increases. The level of decrease is, however, more significant in the case of Ag-water than for Cu-water. Figure 8 shows the influence of stretching on the concentration profiles. It can be seen that the concentration decreases when the sheet is shrunk and increases with stretching.

*i.e.*, the first and second solutions. The velocity and concentration profiles decrease with increasing ${f}_{w}>0$ in the case of the first solution. The opposite is, however, true in the case of the second solution. The far field boundary conditions are asymptotically satisfied, thus supporting the validity of the numerical solutions and the existence of dual solutions. In Figure 9(b), the concentration increases more rapidly with increasing suction/injection in the case of the second solution.

*M*causes thinning of the boundary layer. The magnetic field enhances the fluid motion at the boundary, and for clear fluids these results are similar to well-known results in the literature.

*λ*in the case of the first solution and increases in the case of the second solution. It is to be noted that momentum boundary layer thickness for the second solution is thicker than for the first solution. For the case of a clear fluid, the results are similar to those obtained by Bhattacharyya [30]. We also note that the velocity gradient at the surface increases with

*λ*, which is consistent with the results predicted from the computation of the skin friction coefficient.

*λ*is shown in Figure 12(a). The values of ${f}^{\u2033}(0)$ are positive when $\lambda <1$ and negative when $\lambda >1$. Physically, a positive ${f}^{\u2033}(0)$ implies that the fluid exerts a drag force on the plate and a negative sign implies the opposite. It is evident that dual solutions of equations (13) and (23) subject to the boundary conditions (16) and (24) exist when $\lambda <0$. There is a critical value of ${\lambda}_{c}<0$ for which the first and second solutions meet. This critical value depends on the values of the other embedded parameters, and we found, for instance, that ${\lambda}_{c}=-1.2465798095$ when $M=0$, ${\lambda}_{c}=-1.295662771$ when $M=0.1$ and ${\lambda}_{c}=-1.49444085032$ when $M=0.5$. These results show that $|{\lambda}_{c}|$ increases with

*M*. However, for ${\lambda}_{c}<\lambda \le -1$, the solution is not unique, there being two solutions for each

*λ*. Figure 12(b) shows that ${g}^{\prime}(0)$ increases with the magnetic parameter

*M*.

## 4 Conclusions

The effects of homogeneous-heterogeneous reactions in MHD nanofluid flow due to a stretching or shrinking sheet have been studied. The transformed governing nonlinear differential equations have been solved numerically. Dual solutions for the velocity and concentration distributions have been obtained for some values of the stretching/shrinking, suction/injection and magnetic parameters. The effects of physical and fluid parameters on the velocity, concentration and skin friction have been analyzed. It was observed that for both the cases of shrinking and stretching sheets, the fluid velocity increased with the magnetic parameter. The concentration at the surface decreased as the strength of heterogeneous reactions increased for both Cu-water and Ag-water nanofluids. The boundary layer thickness for the first solution is always thinner than that for the second solution.

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

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