A Finite Difference Method for Solving Unsteady Fractional Oldroyd-B Viscoelastic Flow Based on Caputo Derivative

Abstract

In this paper, the effect of a fractional constitutive model on the rheological properties of fluids and its application in numerical simulation are investigated, which is important to characterize the rheological properties of fluids and physical characteristics of materials more accurately. Based on this consideration, numerical simulation and analytical study of unsteady fractional Oldroyd-B viscoelastic flow are carried out. In order to improve the degree of accuracy, the mixed partial derivative including the fractional derivative in the differential equation is converted effectively by integrating by parts instead of by direct discretization. Then, the stability, convergence, and unique solvability of the difference scheme are verified. The validity of the finite difference method is tested by making comparisons with analytical solutions for two kinds of fractional Oldroyd-B viscoelastic flow. Numerical results obtained using the finite difference method are in good agreement with analytical solutions obtained via the variable separation method. Viscoelastic characteristics of the unsteady Poiseuille flow are similar to the second-order fluid or integer-order Oldroyd-B fluid when the parameter is close to 0 or to 1. Oscillation characteristics of fractional viscoelastic oscillatory flow are similar to those of the classical viscoelastic fluid under the same condition. Compared with the previous research, the present work studies the rheological properties of fluids with the finite difference method, and the application of fractional constitutive models in describing the rheological properties of fluids is developed. Meanwhile, more cases reflecting the fractional-order characteristics are given. The present method can accurately capture the flow characteristics of unsteady fractional Oldroyd-B viscoelastic fluid and is applicable for the generalized fractional fluid.