This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of the divergence conforming and discontinuous Galerkin formulations to incorporate upwind discretizations to stabilize the formulation. The stability of the continuous problem and the full-discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully-discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.
翻译:本文研究了具有变密度的Bingham流。我们基于Huber平滑步骤提出了一种应力张量的局部双粘度正则化方法。接下来,我们的计算方法基于对Huber正则化Bingham本构方程的二阶散散耦合离散化,并结合质量密度的不连续Galerkin方案。我们利用散散耦合和不连续Galerkin方案的性质,采用上风离散化来稳定公式。研究了连续问题和完全离散方案的稳定性。进一步,提出了一种半平滑牛顿方法,用于在每个时间步骤中求解所得到的完全离散化方程组。最后,提供了几个数值实例,以说明问题的主要特征和数值方案的性质。