Steady Flow of a Power Law Fluids with Uniform Suction and Injection between Given Two Parallel Plates

1. INTRODUCTION

Non-Newtonian power law fluid refers to a specific category of fluid which exhibits variable viscosity under the action of applied force. It undoubtedly differs from the Newtonian fluid which follows the Newton‘s law of viscosity and bears constant viscosity under stress. The physical viscosity in non-Newtonian fluids could be depend on the magnitude of the shear stress. For example, shear-thickness (dilatants) fluids, Shear-thinning (Pseudoplastic) fluids, Bingham plastic etc. or even time-dependent (Thixotropic Liquid, rheopectic liquid) fluids. In reality and practically there are many fluids may present these non-Newtonian behaviors. e.g., Blood, Silicone oil, Printer ink, Polymers, Honey, Shampoo etc. Various types of study work are focused on the unsteady & steady investigation of the flow between parallel plates. However, these works are limited to the cogitation of the non-Newtonian fluid behavior [1, 2, 3]. The research analysis on the power law and viscoelastic fluid flow through allowance for the dissipation are presented [4, 5]. Analytical solutions of steady flow of non-Newtonian fluids are limited to very specific geometries and they offered long algebraic development. Thus, numerical approaches have been profusely used. The most used numerical investigation to solve the equation of fluid flow is finite and different method. The flow in duct of rectangular cross-section with uniform suction & injection has been examined by Rathy [6], Sai & Rao [7] and Erdogan & Imrak [8,9] Crane [10] & Terril and Thomas [11] started his work from the study stretching sheet by considering the non-Newtonian fluids under the assumption that velocity will vary in the direction of flow and that is must be linear to the distance from the specific point. Winter [12] investigated the flow in a pipe with uniform injection and suction shows a boundary layer character near the suction. Pinho [13] considering the asymptotic suction and Injection flows, has shown that this type of fluid expansion may lead to erroneous results.

We have focussed on the study of the steady flow of a power law fluid between two parallel plates given by the equations y = 0 and y = h. We assume that there is a constant pressure gradient in the x-direction (the direction of the main flow).We also assume that the velocity components u, v in the x and y directions are independent of the current length x measured parallel to the plates. Putting u0.x, we find from the equation of continuity

that v(x, y) is a constant v0 (say), so that were v0 is positive for injection at the plate y = 0. The only non-zero strain rate component has the value . Because of suction and injection the velocity profile would not be symmetrical about , but the maximum of velocity would shift. Let y = y0 be the point where u is maximum. Then u increases from when y increases from (y = 0 to y = y0) and then it decreases from as y increases from y = y0 so y = h. Hence as in 0 < y < yo and in All the stress components except sxy are zero. Then the equation of motion. gives and where, constant

Integrating equations (2.4) and (2.5) once and applying the condition when y = y0, we get and and obtain and Where the corresponding value of w and is a dimensionless number For solving equation (3.4) we substitute where being prime so each other, and get Putting z = 0, we obtain where Similarly substituting is equation (3.5) and integrating, we get Putting z = 1, we obtain Where Equation (3.8) and (3.12) may be regarded as two simultaneous equation for determining the values of z0 and . Then equations (3.4) and (3.11) will determine the velocity profile. All the integral entering into the problem can be analytically integrated for general values of p and q, but the simultaneous equations for z0 and would be too complicated to solve, hence in such situations numerical integration would be preferred.

We have found the velocity profiles for, 1 and 2 as particular examples to show the following work. (4.1) Case, (When) On integration of equations (3.7) and (3.11) & we get and The equations for determining z, and wmax are and Eliminating Wmax & we get A little consideration would show that in order to have only one value of z0 (which is physically feasible) between 0 and 1, C has to be between 0 and . If C is greater than , there would be more than one value of z0 and the flow of the type (laminar flow) is not possible. We have determined the value of z0, wmax for various values of C.

Equations (3.7) and (3. 11) give and The equations for deserting z0 and are and where Equation (4.2.3) and (4.2.4) can be solved for determining the values of s and z0, then wmax would be obtained from equation. Then from equation (4.2.1) & (4.2.2) we can determine the value of z and w for various values of W. When 00zz, the value of W for which z and w would be determined will lie between 0 and W0 determined by the equation. Similarly when 0zz1,Wlies between 0 and W1, where W1 is the root of the equation.

In this case w, z0 and wmax are given by the following equations. and

[a] The value of z0 and wmax for different value of n and c have been given in table 1.

[b] It is seen that for any value of c, z0 increases with n, that is the point of maximum velocity shifts more and more towards the wall z = 1 with the increases of n. We also see that the variation of z0 – [zo]c=0 with c, for any fixed n is almost linear. Hence for any fixed n, z0 increases with h and v0 decreases with the increase of . [c] It is found that for fixed value of decreases with the increases of C. Hence umax will decreases with the increase of . Therefore umax increases with the suction velocity v0 and h. Again it decreases with the increase of the pressure gradient for n < 2 and increases with the pressure gradient for n > 2. [d] It can be seen that for a fixed n, the velocity always decreases with the increases of C in the region 0 < z < ½ and in the neighbourhood of the wall z = 1, the velocity increases with the increases of C. Now C increases with v0 (the velocity of injection) and hence it is seen that the velocity in the neighbourhood of the wall z = 1, where the liquid is withdrawn, increases with the increase of suction velocity v0 where as it decreases in the region (0 < z < ½) with the increases of v0.

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Research Scholar, Department of Mathematics, Jai Prakash University Chapra (Saran) Bihar