Effects of MHD on the Unsteady Thin Flow of Non-Newtonian Oldroyd –B Fluid over an Oscillating Inclined Belt Through a Porous Medium

Abstract

The unsteady (MHD) thin film flow of an incompressible Oldroyd -B fluid over an oscillating inclined belt making a certain angle with the horizontal through a porous mediumis analyzed .The analytical solution of velocity field is obtained by a semi-numerical technique optimal homotopy asymptotic method (OHAM) .Finally the influence of various dimensionless parameters emerging in the model on velocity field is analyzed by graphical illustrations. I. Introduction In everyday life and in engineering flow of non-Newtonian fluid are frequently occurred .It is ubiquitous in nature and technologies .Therefore ,to understand there mechanics is essential in most application .Most of it problems appeared in several examples are polymer solution ,paints ,certain oils ,exocitic lubricants ,colloidal and suspension solutions , clay coatings and cosmetic products .In non-Newtonian fluids Thin film flows have a large varying of practical applications in nonlinear science and engineering industries . For modeling of non-Newtonian Fluid flow problem we used several model like (second grade, third grade , Maxwell fluid ,Oldroyd-B ) ..ect fluid model by which we got linear or nonlinear ordinary differential equation or partial differential equations which may be solved exactly, analytically or numerically. For solution of these problems different varieties of methods are used, in which (ADM,VAM,HAM,HPM,OHAM) are frequently used. In this work we have modeled a partial differential equations by using oldroyd-B fluid model. Solution of the problem is obtained by using OHAM. Fetecau et al. (3) studied the exact solutions of an Oldroyd-B fluid over a flat plate on constantly accelerating flow. In the following year, Fetecau et al. (4) obtained the exact solutions of the transient oscillating motion of an Oldroyd-B fluids in cylindrical domains. Haitao and Mingyu (5) investigated the series solution of an Oldroyd-B fluid by using the sine and Laplace transformations for the plane Poiseuille flowand plane couette flow. Lie L. (7) studied the effect of MHD flow for an Oldroyd-B fluid between two oscillating cylinders .Khan et al. Burdujan (2) discussed the solution for the flow of an incompressible Oldroyd-B fluid between two cylinders . Shahid et al. (12) studied the solution of the steady and unsteady flow of Oldroyd-B fluid by using Laplace and Fourier series. Shah et al. (13) studied the solution by using OHAM of thin film flow of third grade fluid on moving inclined plane. Marinca et al. (11) studied the approximate solution of non-linear steady flow of fourth grade fluid by using OHAM. Marinca et al (10) discussed the approximate solution of the unsteady viscous flow over a shrinking cylinder by using OHAM method . Kashkari (6) studied the OHAM solution of nonlinear Kawahara equation. For comparison HPM, VHPM and VIM method is used but OHAM ismore successful method. . Anakira et al. (1) the analytical solution of delay differential equation by using OHAM. Mabood et al. (8,9) investigated the approximate solution of non-linear Riccati differential equation by using OHAM . Taza Gul (14) studied the unsteady magnetohydrodynamics (MHD) thin film flow of an incompressible Oldroyd-B fluid over an oscillating inclined belt making a certain angle with the horizontal. In this paper , we study the unsteady (MHD) thin film flow of an incompressible Oldroyd -B fluid over an oscillating inclined belt making a certain angle with the horizontal in porous medium . the solution of velocity field is obtained by using a semi-numerical technique optimal homotopy asymptotic method (OHAM). Finally, we analyzed the influence of various dimensionless parameters emerging in the model on velocity field is analyzed by graphical illustrations.