A mixed formulation of the Bingham fluid flow problem: Analysis and numerical solution

In this paper we introduce a mixed formulation of the Bingham fluid flow problem. We consider both the original and a regularized version of the problem, where a parameter ε is introduced, forcing the entire domain to be formally a fluid region. In general, common solvers for the regularized problem experience a performance degradation when the parameter ε gets smaller. The method studied here introduces an auxiliary tensor variable and shows enhanced numerical properties for small values of ε. A good performance is also observed for the non-regularized case. The well posedness for the regularized problem and the equivalence – at the continuous level – between the original (primitive variables) and the mixed formulation are demonstrated. We analyze properties of linearized problems that are relevant for the convergence of numerical solvers. A finite element method for the mixed formulation is discussed. Numerical results confirm the predicted better performances of the mixed formulation when compared to the primitive variables formulation. A comparison to a non-regularized solver based on the augmented Duvaut–Lions–Glowinski formulation of the problem is carried out as well.