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：本文致力于研究变密度宾汉流动。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。随后，我们的计算方法采用二阶散度相容离散化Huber正则化宾汉本构方程，并结合不连续伽辽金格式处理质量密度。我们利用散度相容和不连续伽辽金公式的特性引入迎风离散化以稳定格式。分析了连续问题及全离散格式的稳定性。此外，提出了一种半光滑牛顿方法，用于在每个时间步求解所获得的完全离散方程组。最后，给出了若干数值算例，以展示问题的主要特征和数值格式的特性。