Behavior of micro-polar flow due to linear stretching of porous sheet with injection and suction

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Abstract

The boundary layer flow of a micro-polar fluid due to a linearly stretching sheet is investigated. The influence of various flow parameters like ‘suction and injection velocity through the porous surface’, ‘viscosity parameter causing the coupling of the micro-rotation field and the velocity field’ and ‘vortex viscosity parameter’ on ‘shear stress at the surface’, ‘fluid velocity’ and ‘micro-rotation’ are studied. The governing equations of the transformed boundary layer are solved analytically using homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly studied and a proper discussion is given for the obtained results. Comparison between the HAM and numerical solutions showed excellent agreement.

Introduction

Micro-polar fluids are fluids with internal structures in which coupling between the spin of each particle and the macroscopic velocity field is taken into account. Micro-polar fluids have been receiving great deal of researches focus due to their application in industrial processes. Such applications include the solidification of liquid crystals, cooling of a metallic plate in a bath, exotic lubricants, extrusion of metals and polymers drawing of plastic films, production of glass and paper sheets and colloidal suspension solutions. Micro-polar fluid theory was introduced by Eringen [1], [2] in order to describe some physical systems, which do not satisfy the Navier–Stokes equations. Many researchers have considered various problems in micro-polar fluids, Łukaszewicz [3] presented in his book mathematical aspects of the micro-polar fluid flow theory. The micro-polar flow on a moving flat plate was investigated by Ishak et al. [4], [5], [6]. Numerical solution was obtained by Hassanien and Gorla [7], who also included suction and injection. Hady [8] handled the same problem using a method of successive approximations. The Newtonian-fluid counterpart was studied by Gupta and Gupta [9] and recently Rahman et al. [10], [11], [12], [13], [14], [15] have investigated some new aspects of micro-polar fluids and their applications. In micro-polar fluids the two fields of micro-rotation and velocity are coupled through a viscosity parameter. When this parameter vanishes, the two fields are uncoupled. The velocity field is, then, governed by the Newtonian flow equations. Because of the nonlinear nature of micro-polar fluids, solving the related equations is generally more difficult to obtain. This is not only true for analytical solutions but also true for numerical solutions. Due to these facts, such flows have been a challenging research topic for mathematicians, physicists and engineers. In this study, the analytic results of variations of the shear stress at the surface, fluid velocity and the micro-rotation have been obtained by implementing HAM [16] and also compared with numerical results which are solved by Maple software using Fehlberg fourth–fifth order Runge–Kutta (RKF45) method with degree four interpolant [17], [18].

Section snippets

Problem formulation

An incompressible micro-polar fluid in steady two-dimensional motion, driven by a flat sheet that is linearly stretching away from a fixed point is assumed. Otherwise; the fluid would have been at rest. The governing boundary layer equations obtained by [19], as systems of nonlinear ordinary differential equation are given by:F‴+FF″-F′2=-εH′,H″-β2H=12β2F″,F(0)=Φ,F′(0)=1,F′(∞)=0,H(0)=0,H(∞)=0,where ‘prime’ denotes differentiation with respect to η; the similarity coordinate measuring distances

Homotopy analysis solution

Eqs. (1), (2) with their pertinent boundary conditions, Eqs. (3), (4), are solved by HAM [16]. We construct the zero-order deformation equations as(1-q)L1[F(η;q)-F0(η)]=qhH¯(η)I1[F(η;q),H(η;q)],(1-q)L2[H(η;q)-H0(η)]=qhH¯(η)I2[F(η;q),H(η;q)],where h is an auxiliary parameter, H¯(η) is an auxiliary function, F(η; q) and H(η; q) are mapping functions and q is an embedding parameter in the range [0, 1]. L1 and L1 are linear operators given byL1(F)=F‴+F″,L1(H)=H″-H,while I1 and I2 are nonlinear

Convergence of HAM solutions

As pointed by Liao, the convergence of series solution strongly depends upon the value of the auxiliary parameter h. It is worthwhile to be mentioned that for different values of flow parameters (Φ, β,  ε) a new h-curve should be plotted as using a unique h-curve for all cases may lead to a considerable error. Therefore, in this study, we have obtained admissible values of h for all cases; but only depicted the h-curve of F″(0) for Φ = 0, β = 0.6, ε = 0.1, for brevity. As shown in Fig. 1 the suitable

Results and discussion

Numerical and analytical results have been presented in Fig. 1, Fig. 2, Fig. 3, Fig. 4 and Table 1, Table 2, Table 3 to analyze the effects of flow parameters. Also Table 1, Table 2, Table 3 demonstrate the comparison between the analytical and numerical solutions along with relative errors. Fig. 2a and b illustrate the effect of different values of β on the micro-rotation. Because of the negligible effect of the parameter on the velocity, the related figure is not depicted. It is seen from the

Conclusions

This communication deals with the boundary layer flow of a micro-polar fluid due to a linearly stretching flat surface. Both numerical and HAM solutions have been obtained for the problem. The results are sketched and discussed for the fluid and flow parameters variations. The effects of different parameter on velocity and micro-rotation field are also discussed.

References (19)

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