A new finite element formulation for viscoelastic flows: Circumventing simultaneously the LBB condition and the high-Weissenberg number problem

Abstract

In this paper, we propose a new, fully consistent and highly stable finite element formulation for the simulation of viscoelastic flows. In our method, we have implemented equal order interpolants for all variables and a combination of classical finite element stabilization techniques (PSPG/DEVSS-TG/SUPG) with the log-conformation representation of the constitutive equation that has allowed us to obtain numerically stable solutions at high Weissenberg numbers. The validity of the presented FEM framework is testified by comparing the numerical results of our method to those of the literature in three benchmark tests: the 2-dimensional flows of a viscoelastic fluid in a square lid-driven cavity and past a cylinder in a channel, and the 3-dimensional flow in a cubic lid-driven cavity. We consider both direct steady-state and transient calculations using the Oldroyd-B and linear PTT models. In all cases, we can reach, and in some cases surpass, the maximum Weissenberg number values attainable by mixed finite element methods, but at a considerably lower computational cost and programming effort. In addition, we perform mesh-convergence tests illustrating that the proposed method is convergent and features almost 2nd order accuracy in space.