On Fractional Viscoelastic Fluids Flowing over a Permeable Surface

Abstract

The viscoelastic fluid, which processes both the viscous and the elastics behaviours, is greatly encountered in engineering and industry fields. Many models have been proposed to describe the viscoelastic fluid. Among them, the fractional derivative typed model is an attractive option for it can best reveal the stress relaxation phenomenon with time and distance. In the present research, we introduce the spatial-fractional derivative to depict the momentum conservation equations for the viscoelastic fluid. Riemann-Liouville typed derivative is adopted. Assume the two-dimensional viscoelastic fluid is steady and incompressible. When it flows through a semi-infinite flat plate, the control governing equations can be deprived into a much simpler boundary layer equations. To make our problem much challenging, we use non-uniform boundary conditions subject to the boundary layer equations. This kind of boundary conditions are often describing the suction or blowing effects when fluids flow through a permeable flat surface. In industry, filtrating equipment are always equipped with the permeable channel walls. The fractional derivative model and the non-uniform boundary conditions have made the problem complex to solve. We notice the fact that it is hard to find out the accurate analytical solutions. Thus, the boundary layer equations are solved by a finite difference method after the coupled continuity equation and momentum equation are decoupled and linearized. The numerical technique used in this research can surely shed some lights on fractional calculus investigation in engineering.