Flow between Parallel Plates for Non-Newtonian Fluid

1. INTRODUCTION

Various researchers have focused their mind towards non-Newtonian fluid with various perspectives. Ronald [1] explained about the field of rheology concerns the deformation and flow behaviour of fluids. The prefix ‗rheo‘ is for Greek word and refers to something that flows because of the particulate nature of blood. He expected the rheological behaviour of the blood to be somewhat more complex than a simple fluid such as water. He mentions that in order to understand the flow behaviour of blood, one must first define the relationship between shear stress and shear rate. Ishikawa et al. [2] found that the non-Newtonian pulsatile flow through a stenosed tube is different from Newtonian flow. The non-Newtonian properly strengthens the peaks of wall shear stress and wall pressure, weakens the strength of the vortex and reduces the vortex size and separated region. Therefore, they concluded that non-Newtonian flow is more stable than Newtonian flow. Hazem [3] studied the two-dimensional stagnation point flow of an incompressible non-Newtonian micropolar fluid with heat generation in the presence of uniform suction of blowing. He concluded that the effect of the suction velocity on the shear at wall depends on the value of the non-Newtonian parameter. Makinde [4] investigates the effects of hematocrit variation on the flow stability. He computes the hemodynamics analysis in large blood vessels and concluded that an increase in hematocrit towards the central core region of the artery has a stabilizing effect on the flow. In this pepper we investigate the velocity profile of non-Newtonian viscous flow between two parallel plates using shooting method technique.

We consider an incompressible viscous flow between two parallel plates.

y = +h y = 0 y = –hx y

Governing equation : With the boundary condition where  – density p – pressure

h – height of the channel t – time u – velocity components on x v0 – velocity components on y

Let 0

uy,yvh

Then, from equation (2.1), (2.2) & (3.1), we have Where

0020

ttpT,avxh



and taking 01

We assume that the fluid properties and the variables of this flow are independent of time i.e.

d0dt

Therefore,

Let the viscosity

21y0e



Then, equation (4.1) becomes and

Let the viscosity 0sechy,(1) then equations (4.1) becomes. and

Definition : Let f(x, y) be a function of two variables defined over a set . It is said that the function f(x, y) satisfies a lipschitz condition in variable y if the constant L > 0 exists such that the following property holds. 1212|f(x,y)f(x,y|L|yy| Whenever The constant L is called the Lipschitz constant for f. Theorem : [W.R.Derrik, (1976] The equations (4.2) and (4.4) above which satisfy the initial conditions (4.3) and (4.5) has a unique solution . Proof : The solution of the equation (4.2) and (4.4) are : and

21 21

22(1y) (1y)

T2y(1y)ea e1

 









T(sechytanhy)a 1sechy



 Let 1y

2 3

then the equation (4.2) becomes.

211

21

1 23

(1)223113(1)

1 2(1)eT/a e1

 

 







also, equation (4.4) becomes

1 23

3113

1

1 (sechtanh)T/a 1sech







subject to initial condition

1 2 3

(1)1 (1)0

k(1)

 



Where k are guessed such that 2(1)0 therefore,

11231 22123 33123

f(,,) f(,,) f(,,)

 



The

i j

fMaxL, For i = 1, 2, 3 & j = 1, 2, 3 are Lipschitz conditions and bounded. Hence by existence theorem the solution is unique. shooting method and the result is shown below graphically.

TTTI,0.5,1,2aaa

The one-dimensional flow of an incompressible non-Newtonian fluid between two parallel plates is presented. We proved the existence and the uniqueness of the steady state solution. The numerical results show that the velocity profile increases when either the viscosity or pressure component increases. Fig. 2 shows the viscosity behaviour, as the viscosity increases the velocity increases. At a constant viscosity the velocity profile also increases as any other fluid components (pressure and density) increases as shown in fig. 3. At low viscosity the fluid flow form a pulsatile, which is the nature of blood flow in an artery. As a result of that, this model can be use to describe the blood viscosity flow mechanics.

1. Renold, L.F. (1998). ―Basic Transport Phenomena in Biomedical Engineering‖, Philadelphia : Taylor and Francis. 2. Ishikawa, T, et. al. S. and R. (1998). ―Effect of Non-Newtonian property of Blood on flow through a Stenosed Tube‖, Fluid Dynamics Research. 22. Pp. 251-265. 3. Hazem, A.A. (2006). ―Investigation of non-Newtonian Micropolar Fluid Flow with Uniform Suction/Blowing and Heat Generation‖ Turkish J. End. Env. Sci. 30, pp. 359-365. 4. Makinde, O. D. (2008) ―Computational Hemodynamics Analysis in Large Blood Vessels : Effects of Hematocrit Variation on the flow stability‖. IMa Design in Biological System, University of Minnesota. 5. Olajuwon, B.I. (2007). ―On the flow of a Power Law fluid over a flat plate in the presence of a pressure gradient‖, Int. J. of Appl. Math and Mech. 3(1): pp. 1- 13. 6. William R.D., and I.G. Stanley (1976). ―Elementary Differential Equations with

Research Scholar, Department of Mathematics, J.P. University Chapra (Saran) Bihar