Stagnation-point flow of a nanofluid towards a stretching sheet

doi:10.1016/j.ijheatmasstransfer.2011.07.021

International Journal of Heat and Mass Transfer

Volume 54, Issues 25–26, December 2011, Pages 5588-5594

https://doi.org/10.1016/j.ijheatmasstransfer.2011.07.021 Get rights and content

Abstract

This communication reports the flow of a nanofluid near a stagnation-point towards a stretching surface. The effects of Brownian motion and thermophoresis are further taken into account. The analytic solutions are developed by homotopy analysis method (HAM). Special emphasis has been given to the parameters of physical interest which include stretching ratio a/c, Prandtl number Pr, Lewis number Le, Brownian motion number Nb and thermophoresis number Nt. It is observed that reduced Nusselt number is an increasing function of ratio a/c. The comparison of the present results with the existing numerical solutions in a liming sense is also shown and this comparison is very good.

Introduction

The flow over a stretching surface has wide range of applications in engineering and several technological purposes. In particular in the extrusion of a polymer in a melt-spinning process, the extrudate from the die is generally drawn and simultaneously stretched into a sheet which is then solidified through quenching or gradual cooling by direct contact with water, cooling of a large metallic plate in a bath, which may be an electrolyte, etc. Crane [1] computed an exact similarity solution for the boundary layer flow of a Newtonian fluid towards an elastic sheet which is stretched with the velocity proportional to the distance from the origin. After this seminal work, the flows over the stretching surfaces have been examined by several researchers, incorporating various physical configurations [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].

The literature regarding the flows of nanofluids is in continuous growing. Regular fluids, such as oil, water, grease, etc. are poor heat conductors, since the thermal conductivity of such fluids has a significant impact on the heat transfer coefficient between the heat transfer medium and heat transfer surface. A reliable technique to improve the thermal conductivity is to incorporate nanoparticles in the base fluid (Choi [12]). Enhancement of heat transfer in electronic cooling, heat exchangers, double plane windows, etc. is an essential topic from an energy saving perspective. The low thermal conductivity of convectional heat transfer fluids such as water, oil and ethylene glycol mixture are poor heat transfer fluids, since the thermal conductivity of these fluids play important role on the heat transfer coefficient between the heat transfer medium and the heat transfer surface (Oztop and Abu-Nada [13]). Since the solid nanoparticles with typical length scales of 1–100 nm with high thermal conductivity are suspended in the base fluid (low thermal conductivity), have been shown to enhance effective thermal conductivity and the convective heat transfer coefficient of the base fluid. Choi [12] seems to be the first who used the term nanofluids to refer to the fluid with suspended nanoparticles. Choi et al. [14] showed that the addition of a small amount (less than 1% by volume) of nanoparticles to conventional heat transfer liquids increased the thermal conductivity of the fluid up to approximately two times. Several researchers Khanafer et al. [15], Maiga et al. [16], Tiwari and Das [17], Abu-Nada [18], Oztop and Abu-Nada [13], Muthtamilselvan et al. [19], Ghasemi and Aminossadati [20], etc. Recently, Nield and Kuznetsov [21] examined the natural convective heat transfer in the flow of a nanofluid past a vertical plate. They provided the numerical solution to the problem. In another paper Kuznetsov and Nield [22] analyzed the Cheng–Mincowcz problem for natural convective boundary layer flow in a porous medium filled with nanofluid taking into account the combined effects of heat and mass transfer in presence of Brownian motion and thermophoresis as proposed by Buongiorno [23]. Detailed review studies on nanofluids are published by Daungthongsuk and Wongwises [24], Wang and Mujumdar [25], [26], and Kakaç and Pramuanjaroenkij [27]. Very recently, boundary layer flow of a nanofluid past a stretching sheet is considered by Khan and Pop [28] using Buongiorno’s model. Up to date, no investigation is made which characterizes the stagnation-point flow of a nanofluid. Therefore, current investigation deals with the stagnation-point flow of a nanofluid towards a stretching surface. Combined effects of heat and mass transfer in presence of Brownian motion and thermophoresis are also taken into account. The solution of the resulting problem is derived by homotopy analysis method (HAM), which has been employed by several researchers [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. Graphical results are presented for the influence of emerging parameters.

Section snippets

Development of problems

We consider the two-dimensional stagnation-point flow of a nanofluid towards a stretching sheet kept at a constant temperature T_w and concentration C_w. The ambient temperature and concentration are respectively T_∞ and C_∞. The governing equations for conservation of mass, momentum, temperature and nanoparticles equations for nanofluids can be expressed as, see Khan and Pop [28] $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0,$ $u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = - \frac{1}{ρ_{f}} \frac{\partial p}{\partial x} + ν (\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}}),$ $u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = - \frac{1}{ρ_{f}} \frac{\partial p}{\partial y} + ν (\frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}}),$ $u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = α (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}}) + τ [D_{B} (\frac{\partial C}{\partial x} \frac{\partial T}{\partial})]$

Homotopy analysis solutions

The velocity f(η), the temperature θ(η) and the concentration field ϕ(η) in terms of the set of base functions ${η^{k} \exp (- n η) | k ⩾ 0, n ⩾ 0}$ can be expressed as $f (η) = a_{0, 0}^{0} + \sum_{n = 0}^{\infty} \sum_{k = 0}^{\infty} a_{m, n}^{k} η^{k} \exp (- n η),$ $θ (η) = \sum_{n = 0}^{\infty} \sum_{k = 0}^{\infty} b_{m, n}^{k} η^{k} \exp (- n η),$ $ϕ (η) = \sum_{n = 0}^{\infty} \sum_{k = 0}^{\infty} c_{m, n}^{k} η^{k} \exp (- n η),$ in which $a_{m, n}^{k}, b_{m, n}^{k}$ and $c_{m, n}^{k}$ are the coefficients. The initial guesses f₀, θ₀ and ϕ₀ of f(η), θ(η) and ϕ(η) are chosen as $f_{0} (η) = A η + (1 - A) (1 - \exp (- η)),$ $θ_{0} (η) = \exp (- η),$ $ϕ_{0} (η) = \exp (- η) .$ The auxiliary linear operators are selected as $L_{f} = \frac{d^{3} f}{d η^{3}} - \frac{df}{d η},$ $L_{θ} = \frac{d^{2} θ}{d η^{2}} - θ,$ $L_{ϕ}$

Convergence of the homotopy solutions

We note that the solutions (44), (45), (46) consist of $ℏ_{f}, ℏ_{θ}$ , and $ℏ_{ϕ}$ . As pointed out by Liao, these parameters play a vital role in controlling the convergence of the dericed series solution. In order to seek the permissible values of $ℏ_{f}, ℏ_{θ}$ , and $ℏ_{ϕ}$ of the functions $f^{″} (0), θ^{'} (0)$ , and $ϕ^{'} (0)$ the $ℏ_{f}, ℏ_{θ}$ , and $ℏ_{ϕ}$ curves are plotted at 15th-order of approximations. It is noticed from Fig. 1, Fig. 2 that the ranges for the admissible values of $ℏ_{f}, ℏ_{θ}$ , and $ℏ_{ϕ}$ are $- 0.9 ⩽ ℏ_{f} ⩽ - 0.4, - 1.0 ⩽ ℏ_{θ} ⩽ - 0.5, - 1.25 ⩽ ℏ_{ϕ} ⩽ - 0.2$

Results and discussion

The salient features of all the embedding physical parameters are highlighted in this section. Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 10 are plotted for temperature and concentration profiles for several values of parameters. In order to validate the present results, a comparison of present analytic results with the numerical solution is presented in Table 2, Table 3. The behavior of a/c (which denotes the ratio of free stream velocity to the velocity of the stretching

Conclusions

We have obtained an analytic solution of the steady boundary layer flow and heat transfer near the stagnation-point of a nanofluid towards a stretching sheet. The sheet is stretched in its own plane with a velocity u_w = cx and the velocity of the external flow is u_e = ax. The obtained similarity ordinary differential equations are solved using the well-known homotopy analysis method (HAM). It is found that with the increase of a/c both the local Nusselt and Sherwood numbers increase with the

Acknowledgements

We are grateful to the referee for his valuable comments and suggestions. Dr. T. Hayat as a visiting professor acknowledges the financial support of King Saud University via KSU-VPP-117.

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Stagnation-point flow of a nanofluid towards a stretching sheet

Abstract

Introduction

Section snippets

Development of problems

Homotopy analysis solutions

Convergence of the homotopy solutions

Results and discussion

Conclusions

Acknowledgements

Eur. J. Mech. B-Fluids

Phys. Lett. A

Int. J. Nonlinear Mech.

Commun. Nonlinear Sci. Numer. Simulat.

Nonlinear Anal. Real World Appl.

Int. J. Heat Mass Transfer

Int. J. Nonlinear Mech.

Int. Commun. Heat Mass Transfer

Int. J. Heat Fluid Flow

Int. J. Heat Mass Transfer

Int. J. Heat Fluid Flow

Int. J. Heat Mass Transfer

Int. J. Heat Fluid Flow

Commun. Nonlinear Sci. Numer. Simulat.

Int. Commun. Heat Mass Transfer

Int. J. Heat Mass Transfer

Renew. Sust. Eng. Rev.

Int. J. Heat Mass Transfer

Commun. Nonlinear Sci. Numer. Simulat.

Commun. Nonlinear Sci. Numer. Simulat.

Commun. Nonlinear Sci. Numer. Simulat.

J. Non-Newtonian Fluid Mech.