Study of the effects of rotation and magnetic field on thestagnation flow of a second grade fluid towards a

stretching/shrinking sheet using HPM and BPES protocols

Abstract

In the present investigation we have presented the effects of rotation and

magnetic field on the stagnation point flow of a second grade fluid towards a

porous stretching /shrinking sheet. The governing equations of second grade

fluid are simplified by using boundary layer approach and similarity

transformation. The reduced nonlinear coupled differential equations are

solved using the Boubaker Polynomials Expansion Scheme BPES and the

Homotopy Analysis Method HAM. The results of various physical

parameters are discussed through graphs. The convergence of the solution is

discussed by h-curves, homotopy pade approximation and residual error.

Keywords: Stagnation flow; stretching/shrinking sheet; rotating frame; HAM solutions; BPES solutions.

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1. Introduction

In a number of applications, rotation plays a significant role such as in cosmical fluid

dynamics. Similarly a great deal of metrology depends upon the dynamics of the

revolving fluid. The large scale and moderate motion of the atmosphere are greatly

affected by the vorticity of the earth’s rotation. In the case of infinite fluid rotating as a

rigid body about an axis, the amount of energy possessed by the fluid is infinite and it is

of great interest to know how small disturbances propagate in such a fluid. Greenspan

and Howard [1] have initiated the study of the dynamics of spin up of an incompressible,

homogenous Newtonian rotating fluid. They have presented a detailed mathematical and

physical analysis of the transient process by which the fluid adjusts to a small change in

the rotation rate of its boundary. It has been demonstrated that the Ekman boundary layer

is established on the horizontal boundary surfaces and it is primarily responsible for the

adjustment process. Furthermore, the Ekman layer produces a secondary interior

circulation throughout the fluid which transports angular momentum. After initiation of

Greenspan and Howard [1], a large number of studies have been presented by various

researchers keeping different fluids and its geometries for rotating flows [2-6].

The analysis of the effects of rotation and magnetic field in the fluid flows has been an

active area of research because of its geophysical and technological importance. It is well

known that a number of astronomical bodies (e.g. the Sun, Earth, Jupiter, magnetic stars,

Pulsars) possess fluid interiors and magnetic fields. Changes in rotation rate of such are

discussed elsewhere [7, 8].

Boundary layer behavior over a stretching surface is another area of research which has

attracted the attention of many researchers due to its important applications in

engineering. Such applications includes aerodynamics extrusion of plastic sheets, the

boundary layer along a liquid film condensation process, the cooling process of metallic

plate in a cooling bath and in the glass and polymer industries. After the initiation of

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Sakiadis [9] several authors consider the boundary layer over a stretching surface [10-

15].

In almost all the fields of engineering and scientific interest, stagnation point flow is an

important phenomenon. These flows may be characterized inviscid or viscous, steady or

unsteady, two dimensional or three dimensional, symmetric or asymmetric, normal or

oblique, homogenous or two fluids and forward or reverse. Some important studies to the

topic are described in refs. [16-22].

Motivated from the above analysis, the aim of this paper is to investigate the effects of

rotation and magnetic field on the stagnation point flow of a second grade fluid towards a

similarity transformation and then the reduced problem have been solved using the

Boubaker Polynomials Expansion Scheme BPES and the Homotopy Analysis Method

HAM. Homotopy analysis method is the successful tool for solving nonlinear ordinary

and partial differential equation and some recent related developments on the topic are

defined in [23-27]. The Boubaker Polynomials Expansion Scheme BPES is also aprotocol

which has been significantly tested in severel applied physics fields [28-38]. The solution

for both stretching and shrinking phenomena are presented. The results of the necessary

physical parameters are discussed through graphs. The convergence of the solution have

been discussed through h-curves, homotopy pade approximation and residual error.

2. Mathematical formulation

We consider an incompressible, stagnation point flow of a second grade fluid

towards a porous stretching/shrinking sheet in a rotating frame in the presence of the

magnetic field. We are choosing the cartesian coordinate system in which (u; v;w) are the

velocity components along x; y and z-axis with Ω being the angular velocity of the

rotating fluid in the z-direction. The potential stagnation flow at infinity is taken by u =

ax, w = -az, v=0 (in which a is the strength of the stagnation flow) where as the velocities

on the shrinking(stretching) rate are considered u = bx, w =-W, v=0, here b > 0 is the

stretching rate (shrinking if b < 0). A constant magnetic field B0 is imposed along z-axis

vertical to the sheet. The governing boundary layer equations for the flow problem are:

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We introduce the similarity variables and non-dimensional variables as follows:

Making use of Eq. (4); the incompressibility condition is automatically satisfied and the momentum equations take the form:

where , , and prime denotes the derivative with respect to .

The boundary conditions for the problem under consideration are:

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where

3. HAM Solution

To seek HAM solution of Eqs. (5) and (6) we select:

as the initial approximations of f, h and:

as auxiliary linear operators which satisfy:

where Ci (i = 1..5) are arbitrary constants. If p [0, 1] is an embedding parameter and

(i = 1.. 2) are non-zero auxiliary parameters then the zeroth order and mth order

deformation problems are:

Zeroth order deformation problems:

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where:

mth order deformation problems:

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where :

The symbolic software MATHEMATICA isused to get the solutions of Eqs. (21) and

(22) up to first few order of approximations. It is found that the solution for f, h are given

by:

in which the coefficients and of and can be found by using given

boundary conditions and by the initial guess approximations in Eqs. (8) and (9) and

numerical data is presented through graphs.

4. BPES Solution

To seek Boubaker polynomials expansion scheme BPES [28-38] solution of Eqs. (5) and

(6) we set:

(29)

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and:

(30)

where are the 4k-order Boubaker polynomials, are minimal positive roots,

is a prefixed integer, and are unknown pondering real coefficients.

The main advantage of these formulations is the fact of verifying the four boundary

conditions in Eq. 1, in advance to problem resolution.

Due to the properties of the Boubaker polynomials [29-32] , and since are roots,

the following conditions stand, for i.e. :

(31)

By introducing expressions (29) and (30) in Eq. (5-8), and by majoring and integrating

along the interval , and are confined, through the coefficients and

, to be weak solutions of the system (32):

(32)

which gives:

(33)

with

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The set of solutions and is the one which minimizes the global

Minimum Square function :

(34)

The correspondent solutions are represented along with the HPM solution (27-28).

5. Convergence of the analytic solution

The analytic series solutions given in Eqs. (27) and (28) of the considered problems are

found by homotopy analysis method. These expressions contain non-zero auxiliary

parameters (i = 1.. 2) which can adjust and control the convergence of the solutions.

To ensure the convergence of the solutions in the admissible range of the values of the

auxiliary parameters one can draw h curves. For the present cases, 20th order -curves

are plotted in Figs. (1) and (2). It is seen from these figures that the ranges for the values

of are -0.7 ≤ ≤ -0.2 and -0.95 ≤ ≤ -0.25. The residual error of the problem is

calculated and shown in Fig (3).

6. Results and discussion

This section deals with the influence of stretching/shrinking parameter

α, magnetic field parameter M, rotation parameter and suction parameter s on and h.

The velocities and h against for various values of α are presented in Figs (4) and (5).

It is observed that magnitude of and h increases with the increase in α. However, the

layer thickness decreases. The variation of M on and h are shown in Figs (6) and (7). It

is observed that the magnitude of increases with the increase in M whereas h decreases

with the increase in M however boundary layer thickness decreases for both the velocity

cases. The magnitude of increases while the magnitude of h decreases with the

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increases in (see Figs (8) and (9)). Figures (10) and (11) show that the suction causes

boundary layer reduction.

It is observed from Figs (12) and (13) that increases with the increase in β whereas h

increases with increase in β up to β = 0.09 then it gives the reverse behavior.

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