K-Version of finite element method in 2D-polymer flows: Upper convected Maxwell model

Abstract

In this paper, a new mathematical framework based on h, p, k and variational consistency (VC) of the integral forms is utilized to develop a finite element computational process for steady two dimensional polymer flows with upper convected Maxwell constitutive model (UCMM). Alternate forms of choices of dependent variables in the governing differential equations (GDEs) are considered and it is concluded that u, v, p, τ choice yielding strong form of the GDEs is meritorious. Since, the differential operators for all possible choices of dependent variables are non-linear, Galerkin method and Galerkin method with weak form are variationally inconsistent (VIC). The coefficient matrices in these processes are non-symmetric and hence may have partial or completely complex basis and the resulting computational processes may yield spurious solutions. Furthermore, since the VC of VIC integral forms cannot be restored through any mathematically justifiable means, the computational processes in these approaches may remain spurious. Least squares processes utilizing GDEs in u, v, p, τ or any other variables are always variationally consistent. The coefficient matrices are symmetric, positive definite and hence always have real basis and thus naturally yield computational processes that are free of spurious solutions. The theoretical solution of GDEs are generally of higher order global differentiability. Numerical simulations of such solutions in which higher order global differentiability characteristics of the theoretical solutions are preserved, undoubtedly require local approximations in scalar product spaces H k , p ( Ω ¯ xy e ) containing higher order global differentiability local approximations. LSP with local approximation in H k , p ( Ω ¯ xy e ) spaces provides an excellent mathematical and computational framework in which it is possible to incorporate desired characteristics of the theoretical solution in the computational process. Numerical studies are presented for fully developed flow between parallel plates and a lid driven square cavity. M1 fluid is used in all numerical studies. The range of applicability of UCMM or lack of it is examined for both model problems for increasing De. A regularization of the velocity of the lid at the corners where stationary wall meets the lid is presented and is shown to simulate the real physics when the local approximations are in higher order spaces and when h d → 0. For both model problems shear rate γ ˙ ˆ is examined in the flow domain to establish validity of the UCMM constitutive model.