Numerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates

Abstract In recent years, non-Newtonian fluids have been widely applied in a number of engineering applications. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluids with fractional derivative constitutive equations, which can be used to approximate the response of many dilute polymeric liquids. Different to the general time fractional diffusion equation, the constitutive equation not only has a multi-term time derivative but also possesses a special time fractional operator on the spatial derivative, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will consider the finite difference method for its discretisation and propose a new scheme to approximate the time fractional derivative. Then we derive an implicit finite difference scheme and establish some new theoretical analysis of the stability and convergence. Furthermore, we present a numerical scheme to improve the time order. Finally, we present two numerical examples to show the effectiveness of our method and apply it to solve the generalized Oldroyd-B fluid model. Keywords Finite difference method Generalized Oldroyd-B fluid Fractional diffusion equation Multi-term time derivative Caputo fractional derivative