Finite Element Method for Two-Dimensional Unsteady MHD Couette Flow of a Generalized Oldroyd-B Fluid

Abstract

In recent years, non-Newtonian fluids have been widely applied in engineering and industry. Due to the complexity of these fluids, there are several constitutive equations of non- Newtonian fluids. One particular subclass of non-Newtonian fluids is the class of a generalized Oldroyd-B fluids with fractional derivative constitutive equations, which has been found to approximate the response of many dilute polymeric liquids. Although the analytic solution of these equations can be derived, it contains a multi-level sum of an infinite series and Fox H-functions, which is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. Different from the general time fractional diffusion equation, the constitutive equation is a novel multi-term time fractional diffusion equation and has a special time fractional operator on the spatial derivative, which is challenging to approximate. In this paper, we will consider the finite element method for the two-dimensional multi-term time fractional diffusion equation. Firstly, we utilise the L1 and L2 schemes to approximate the two different time fractional derivatives, respectively. Secondly, we apply the finite element method to the equation and establish the fully variational formulation. Then we adopt linear polynomials on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence of the scheme. Finally, two numerical examples are investigated to show the effectiveness of our method, in which a two-dimensional unsteady MHD Couette flow of a generalized Oldroyd-B fluid is considered.