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

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.