Closed‐Form Solutions for a Nonlinear Partial Differential Equation Arising in the Study of a Fourth Grade Fluid Model

Taha Aziz,F M Mahomed

doi:10.1155/2012/931587

Abstract

The unsteady unidirectional flow of an incompressible fourth grade fluid bounded by a suddenly moved rigid plate is studied. The governing nonlinear higher order partial differential equation for this flow in a semiinfinite domain is modelled. Translational symmetries in variables t and y are employed to construct two different classes of closed‐form travelling wave solutions of the model equation. A conditional symmetry solution of the model equation is also obtained. The physical behavior and the properties of various interesting flow parameters on the structure of the velocity are presented and discussed. In particular, the significance of the rheological effects are mentioned.

Highlights

There has been substantial progress in the study of the behavior and properties of viscoelastic fluids over the past couple of years
This model is known to have interesting non-Newtonian flow properties such as shear thinning and shear thickening that many other non-Newtonian models do not exhibit. This model is capable of predicting the normal stress effects that lead to phenomena like “die-swell” and “rod-climbing” 14. With these facts in mind, we have considered a fourth grade fluid model in this study
The backward wave-front type travelling wave closed-form solution 4.12 and the conditional symmetry solution 5.15 best represent the physics of the problem considered in the sense that these solutions satisfy all the initial and the boundary conditions

Summary

Introduction

There has been substantial progress in the study of the behavior and properties of viscoelastic fluids over the past couple of years. The steady flow of a fourth grade fluid past a porous plate was treated by Marinca et al 21 with the help of the optimal homotopy asymptotic method OHAM Despite all of these works in recent years, the exact closed-form solutions for the problems dealing with the flow of fourth grade fluids are still rare in the literature. The Lie symmetry approach has become of great importance in the theory and applications of differential equations and widely applied by several authors to solve difficult nonlinear problems dealing with the flows of non-Newtonian fluids 24–28. Interest in the conditional symmetry approach has intensified This method has been used successfully to obtain new exact solutions for a number of interesting nonlinear PDEs 30–32. The influence of physically applicable parameters of the flow model are studied through several graphs and appropriate conclusions are drawn

Fundamental Equations

Flow Development

Travelling Wave Solutions

Backward Wave-Front Type Travelling Wave Solutions

Forward Wave-Front Type Travelling Wave Solutions

Conditional Symmetry Solution

Physical Interpretation of the Results

Concluding Remarks