The Impulsive Motion of Flat Plate in Generalized Second Grade Fluid with Anomalous Diffusion

Mohammad Tanzil Hasan,Md Shafiqul Islam,Mir Shariful Islam

doi:10.11648/j.ajam.20200806.15

Mohammad Tanzil Hasan, Md Shafiqul Islam + Show 1 more

Open Access

https://doi.org/10.11648/j.ajam.20200806.15

Copy DOI

Abstract

The flow adjacent to a wall rapidly set in motion for a generalized second-grade fluid with anomalous diffusion is examined. For the elucidation of such a fluid, the fractional-order derivative approach in the constitutive relationship model is presented because models based on ordinary differential equations have a relatively limited class of solutions, which does not provide compatible description of the complex systems in general. The current model of second-order fluid involving fractional calculus is based on the formal replacement of the first-order derivative in ordinary rheological constitutive equation by fractional derivative of a non-integer order. In addition, the time-fractional equation considered in this article describes the anomalous sub-diffusion. In this article, the velocity and stress field of generalized second-grade fluid with fractional anomalous diffusion are studied by fractional partial differential equations. Analytic solutions are given in closed form, from these differential equations in terms of the generalized G-functions or Fox's H-function with the discrete Laplace transform technique. Thus, many previous and classical results, namely, the solution of fractional diffusion equation obtained by Wyss, the classical Rayleigh’s time-space regularity solution, the relationship between velocity field and stress field obtained by Bagley and Torvik, are represented by particular cases of our proposed derivation.

Highlights

In the past few decades, non-Newtonian fluids have gained increasing popularity, mainly due to their concrete application in various fields such as material processing, chemical and nuclear industries, geophysics, oil reserving engineering and bioengineering [1]
The starting point of the fractional derivative model of viscoelastic fluid is generally a classical differential equation, which is adjusted by substituting the time derivative of an integer order by Riemann-Liouville fractional calculus operators
The classical problem of the plate moving impulsively in its own plane to the generalized second-order fluid is extended by replacing material time derivative of integer order with fractional-order and defining 0 < ≤ 1

Summary

Introduction

In the past few decades, non-Newtonian fluids have gained increasing popularity, mainly due to their concrete application in various fields such as material processing, chemical and nuclear industries, geophysics, oil reserving engineering and bioengineering [1]. The starting point of the fractional derivative model of viscoelastic fluid is generally a classical differential equation, which is adjusted by substituting the time derivative of an integer order by Riemann-Liouville fractional calculus operators. This generalization agrees one to define precisely non-integer order integrals or derivatives. Anomalous diffusion deviates from the standard Fichean description of Brownian motion Such anomalous behavior can be represented by Lévy processes whose main character is that the mean squared displacement is a nonlinear growth with respect to time, such as < x2 >∼ tγ. The velocity and stress field of the generalized second-grade fluid with fractional anomalous diffusion caused by a plate moving impulsively on its own plate is investigated. Exact analytic solutions of these differential equations are gained by applying the discrete Laplace transform technique of the sequential fractional derivatives method in terms of Fox’s Hfunction

Problem Formulation

Velocity Field

Stress Field

The Relationship Between Stress Field and Velocity Field

Conclusions