Analytical Solutions of Non-Linear Equations of Power-Law Fluids of Second Grade over an Infinite Porous Plate

Saeid Abbasbandy,Muhammet Yürüsoy,Hüseyin Güllüce

doi:10.3390/mca19020124

Abstract

The flow of an incompressible fluid of modified second grade past an infinite porous plate subject to either suction or blowing at the plate is studied. The model is a combination of power-law and second grade fluid in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. Equations of motion in dimensionless form are derived. Analytical solutions of the outcoming non-linear differential equations are found by using the homotopy analysis method (HAM), which is a powerful semi-analytical method. Effects of power-law index and second grade coefficient on the boundary layers are shown and solutions are contrasted with the usual second grade fluid solutions.

Highlights

The flow of an incompressible non-Newtonian fluid has important industrial applications, for example in the extrusion of a polymer sheet from a die or in the drawing of plastic films
Non-Newtonian fluid mechanics afford an excellent opportunity for studying many of the mathematical techniques which have been developed to solutions of non-linear equations
Normal stress effects can be expressed in second grade fluid model, a special type of Rivlin-Ericksen fluids, but this model is incapable in representing shear thinning/thickening behavior

Summary

1.INTRODUCTION

The flow of an incompressible non-Newtonian fluid has important industrial applications, for example in the extrusion of a polymer sheet from a die or in the drawing of plastic films. The below power-law fluid of second grade model is considered in this work. Where T* is the Cauchy stress tensor, p* is the pressure, I is the identity matrix, A1* and A2*are the first and second Rivlin-Ericksen tensors respectively, , m, 1 and 2 are material moduli that may be constants or depend on temperature For both models, when m=0, 1= 2=0, the fluid is Newtonian and represents the usual viscosity. Pakdemirli et al [16] solved the porous plate problem using perturbation method and presented analytical and numerical solutions of model (2). The flow of an incompressible fluid of modified second grade fluid past a porous plate is governed by a non-linear ordinary differential equation in a reasonably simple structure.

EQUATIONS OF MOTION

ANALYTIC SOLUTIONS BY THE HOMOTOPY ANALYSIS METHOD

CONCLUDING REMARKS