Exact solutions to vorticity of the fractional nonuniform Poiseuille flows

Abstract

Closed-form expressions for the dimensionless velocity, shear stresses, and the flow vorticity fields corresponding to the isothermal unsteady Poiseuille flows of a fractional incompressible viscous fluid over an infinite flat plate are established. The fluid motion induced by a pressure gradient in the flow direction is also influenced by the flat plate that oscillates in its plane. The vorticity field is dependent on two spatial coordinate and time, and it is an arbitrary trigonometric polynomial in the horizontal coordinate. The exact solutions, obtained by generalized separation of variables and Laplace transform technique, are presented in terms of the Wright function and complementary error function of Gauss. Their advantage consists in the fact that the values of the fractional parameter can be chosen so that the predicted material properties by them to be in agreement with the corresponding experimental results. In addition, they describe motions for which the nontrivial shear stresses are influenced by history of the shear rates. It is found that the flow vorticity is stronger near the plate, but it could be attenuated in the case of fractional model.