Slip Effects on Fractional Viscoelastic Fluids

Muhammad Jamil,Najeeb Alam Khan

doi:10.1155/2011/193813

Abstract

Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no‐slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions pΨq, by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no‐slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is θ → 0. Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no‐slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.

Highlights

There are many fluids in industry and technology whose behavior cannot be explained by the classical linearly viscous Newtonian model
The departure from the Newtonian behavior manifests itself in a variety of ways: non-Newtonian viscosity shear thinning or shear thickening, stress relaxation, nonlinear creeping, development of normal stress differences, and yield stress 1
The Navier-Stokes equations are inadequate to predicted the behavior of such type of fluids; many constitutive relations of non-Newtonian fluids are proposed 2

Summary

Introduction

There are many fluids in industry and technology whose behavior cannot be explained by the classical linearly viscous Newtonian model. Fractional calculus is useful in the field of biorheology and bioengineering, in part, because many tissue-like materials polymers, gels, emulsions, composites, and suspensions exhibit power-law responses to an applied stress or strain 6, 7 An example of such power-law behavior in elastic tissue was observed recently for viscoelastic measurements of the aorta, both in vivo and in vitro 8, 9 , and the analysis of these data was most conveniently performed using fractional order viscoelastic models. The starting point of the fractional derivative model of non-Newtonian model is usually a classical differential equation which is modified by replacing the time derivative of an integer order by the so-called Riemann-Liouville/Caputo fractional calculus operators This generalization allows one to define precisely noninteger order integrals or derivatives. The difference among fractional Maxwell, ordinary Maxwell, and Newtonian fluid models is highlighted

The Differential Equations Governing the Flow

Statement of the Problem

Calculation of the Velocity Field

Newtonian Fluid with Slip Effects

Numerical Results and Conclusions