Efficient numerical computations of yield stress fluid flows using second-order cone programming

This work addresses the numerical computation of the two-dimensional flow of yield stress fluids (with Bingham and Herschel-Bulkley models) based on a variational approach and a finite element discretization. The main goal of this paper is to propose an alternative optimization method to existing procedures such as penalization and augmented Lagrangian techniques. It is shown that the minimum principle for Bingham and Herschel-Bulkley yield stress fluid steady flows can, indeed, be formulated as a second-order cone programming (SOCP) problem, for which very efficient primal-dual interior point solvers are available. In particular, the formulation does not require any regularization of the visco-plastic model as is usually the case for existing techniques, avoiding therefore the difficult choice of the regularization parameter. Besides, it is also unnecessary to adopt a mixed stress-velocity approach or discretize explicitly auxiliary variables as frequently proposed in existing methods. Finally, the performance of dedicated SOCP solvers, like the Mosek software package, enables us to solve large-scale problems on a personal computer within seconds only. The proposed method will be validated on classical benchmark examples and used to simulate the flow generated around a plate during its withdrawal from a bath of yield stress fluid.