S. Nadeem*, M. A. Sadiq, Jung-il Choi and Changhoon Lee

Exponentially Stagnation Point Flow of

Non-Newtonian Nanofluid over an Exponentially

Stretching Surface

Abstract: The steady stagnation point flow of Jeffrey nanofluid over an exponential stretching surface under the boundary layer assumptions is discussed analytically.

The transport equations include the effects of Brownian motion and thermophoresis. The boundary layer coupled partial differential equations of Jeffrey nanofluid are simplified with the help of suitable semi-similar transformations. The reduced equations are then solved analytically with the help of homotopy analysis method (HAM). The convergence of HAM solutions have been discussed by plotting h-curve. The expressions for velocity, temperature and nano particle volume fraction are computed for some values of the parameters namely, Jeffrey relaxation and retardation parameters B and λ1, stretching/ shrinking parameter A, suction injection parameter vw,

Lewis number Le, the Brownian motion Nb, thermophoresis parameter Nt and Prandtl number Pr.

Keywords: stagnation point, Jeffrey nanofluid, porous stretching surface, boundary layer flow, series solutions, exponential stretching

MSC® (2010). 76A05, 76A10 *Corresponding author: S. Nadeem: Department of Mathematics,

Quaid-i-Azam University 45320, Islamabad 44000, Pakistan.

E-mail: snqau@hotmail.com

M. A. Sadiq: Department of Mathematics, Community College

Dammam, KFUPM. Saudi Arabia

Jung-il Choi, Changhoon Lee: Department of Computational Science and Engineering, Yonsei University, Seoul, Korea 1 Introduction

The dynamics of fluid flows due to stagnation point is important in many industrial and engineering applications The stagnation point flows can be viscous or inviscid, steady or unsteady, two dimensional or three dimensional, normal or oblique and forward or reversed etc. After the pioneering work done by Hiemenz [1] and

Homann [2], many researchers have highlighted various aspects of stagnation point flows. Mentions may be made to the interesting works [3–8]. The study of boundary layer flow due to stretching sheet is also important because of its applications, such as hot rolling wire drawn, extrusion of plastic sheets, paper production, glass blowing, metal spinning, drawing plastic films, the cooling of metallic plates in a cooling bath, polymer sheet extruded continuously from a dye and drawing of paper films etc.

The combined study of stagnation flows due to stretching sheet has been examined by various researchers. Some important studies on this topic are cited [9–14]. However, only a limited attention has been given to those studies in which exponentially stretching surface has been considered [15–20].

Recently, the study of nanofluids have achieved great importance because of its applications in industry and technology. The term nanofluid first introduced by Choi [21], means a fluid in which nano-sized particles are suspended in conventional heat transfer basic fluids.

The major advantage of nanofluid is to enhance the heat transfer as compared to conventional heat transfer liquids. Bachok et al. [22] have examined the boundary layer of nano fluids over a moving surface when the plate is assumed to move in the same or opposite directions to free stream. The boundary layer flow induced in a nano fluid due to a linearly stretching sheet is investigated by

Makinde and Aziz [23]. Kuznetsove and Nield [24] have examined the natural convection flow of nanofluid over a vertical plate. Some other important studies in this topic are cited [25–26].

In all the studies mentioned above, the exponentially stagnation point of a nano fluid towards an exponentially stretching surface is still unexplored. Therefore, the aim of the present analysis is to discuss the exponentially stagnation flow of a Jeffrey nano fluid over an exponentially stretching surface. The boundary layer equations of Jeffrey nano fluid are simplified by using semi-similar variables. The reduced equations are solved analytically with the help of homotopy analysis method [27–38]. The effects of embedded parameters on fluid velocity, temperature and nano particle concentration have been shown graphically.

DOI 10.1515/ijnsns-2011-0081 Int. J. Nonlinear Sci. Numer. Simul. 2014; 15(3–4): 171–180DE GRUYTER

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Download Date | 7/2/14 3:39 PM 2 Formulation of the problem

Consider the steady two dimensional flow of an incompressible stagnation point Jeffrey nanofluid over an exponentially stretching surface. We are considering Cartesian coordinate system in such a way that x-axis is taken along the stretching surface in the direction of the motion and y-axis is normal to it. The plate is stretched in the x-direction with a velocity Uw ¼ b expðx=lÞ defined at y ¼ 0 and due to stagnation point flow the velocity of the fluid is Uy ¼ a expðx=lÞ as y !y, where a and b are constants having the dimensions of velocity. The equations of Jeffrey nanofluid under the boundary layer approximations are given by the following equations [20]: ∂u ∂x þ ∂v ∂y ¼ 0; ð1Þ u ∂u ∂x þ v ∂u ∂y ¼ γ 1þ λ1 ∂2u ∂y2 þ γλ2 1þ λ1 u ∂3u ∂x∂y2 þ v ∂ 3u ∂y3 þ ∂u ∂y ∂2u ∂x∂y ∂u ∂x ∂2u ∂y2 ; ð2Þ u ∂T ∂x þ v ∂T ∂y ¼ α ∂ 2T ∂y2 þ γ cpð1þ λ1Þ ∂u ∂y 2 þ ρcp ρcf

DB ∂C ∂y ∂T ∂y þ DT

Ty ∂T ∂y 2" # þ γλ2 cpð1þ λ1Þ ∂u ∂y ∂ ∂y u ∂u ∂x þ v ∂u ∂y 1 ρcp ∂qr ∂y ; ð3Þ u ∂C ∂x þ v ∂C ∂y ¼ DB ∂ 2C ∂y2 þ DT

Ty ∂2T ∂y2 ; ð4Þ where ðu; vÞ are the velocity components in ðx; yÞ directions, ρf is the fluid density of base fluid, γ is the kinematic viscosity, λ1 is the ratio of relaxation to retardation time, λ2 is the retardation time, T is the temperature, C is the nano particle volume fraction, ðρcÞp is the effective heat capacity of nano particles, ðρcÞf is the heat capacity of the fluid, α ¼ k=ðρcÞf is the thermal diffusivity of the fluid, DB is the Brownian diffusion coefficient and DT the thermophoretic diffusion coefficient.