Effect of Couple Stresses on the MHD of a Non-Newtonian Unsteady Flow between Two Parallel Porous Plates

Abstract

In this paper the MHD of a Non-Newtonian unsteady flow of an incompressible fluid under the effect of couple stresses and a uniform external magnetic field is analysed by using the Eyring Powell model. In the first approximation the solution is obtained by using the Mathematica computational program with assuming a pulsatile pressure gradient in the direction of the motion. In the second order approximation a numerical solution of the non-linear partial differential equation is obtained by using a finite difference method. The effects of different parameters are discussed with the help of graphs in the two cases. In [4] an unsteady MHD non-Newtonian flow between two parallel fixed porous walls was studied using the Eyring Powell model [5], and in first approximation an exact solution of the velocity distribution was obtained if the pressure gradient in the direction of the motion is an arbitrary function of time. In second approximation a numerical solution was obtained when the pressure gradient is constant. A non-0932-0784 / 03 / 0400-0204 $ 06.00 c 2003 Verlag der Zeitschrift für Naturforschung, Tübingen · http://znaturforsch.com Newtonian fluid flow between two parallel walls, one of them moving with a uniform velocity under the action of a transverse magnetic field, was studied in [6].The present paper treats the flow of a pulsatile nonNewtonian incompressible and electrically conducting fluid in a magnetic field. Possible applications of these calculations are the flow of oil under ground, where there is a natural magnetic field and the earth is considered as a porous solid, and the motion of blood through arteries where the boundaries are porous.Couple stresses are the consequence of assuming that the mechanical action of one part of a body on another across a surface is equivalent to a force and moment distribution. In the classical nonpolar theory, moment distributions are not considered and the mechanical action is assumed to be equivalent to a force distribution only. The state of stress is measured by a stress tensor τ i j and a couple stress tensor M i j . The purpose of the present paper is to investigate the effect of couple stresses on the flow by obtaining the effect of the couple stress parameter besides other parameters entering the problem on the velocity distribution. The field equations are [7]:The continuity equationρ + ρv i,i = 0, Cauchy's first law of motion ρa i = T ji, j + ρ f i , and Cauchy's second law of motion M ji, j + ρ i + e i jk T jk = 0, where ρ is the density of the fluid, v i are the velocity components, a i the components of the acceleration, T i j is the second order stress tensor, M i j the second order couple stress tensor, f i the body force per unit volume, i the body Unauthenticated Download Date | 5/11/18 6:27 AM

Peristaltic Transport of a Third-Order Nano-Fluid in a Circular Cylindrical Tube with Radiation and Chemical Reaction

“…The basic equations governing the flow of an incompressible Nano-fluid are the following equations [25,26],…”

Unsteady magnetohydrodynamic flow of some non-Newtonian fluids with slip through porous channel

“…Eldabe et al [2] studied the one directional flow of non-Newtonian Powell-Eyring fluid with coupled stresses through parallel plates. Zueco and Beg [3] considered a pulsatile flow of Powell-Eyring fluid.…”

Slip Velocity Distribution on MHD Oscillatory Heat and Mass Transfer Flow of a Viscous Fluid in a Parallel Plate Channel

“…Mistra and Pal [18] investigated on the laminar pulsatile flow of blood under the influence of externally imposed body accelerations. Eldabe et al [7] studied the effect of couple stresses on the MHD of a nonNewtonian unsteady flow between two parallel porous plates. Zuecco and Beg [31] investigated on network numerical simulation applied to pulsatile non-Newtonian flow through a channel with couple stress and wall mass flux effects.…”