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Article Dans Une Revue SIAM Journal on Numerical Analysis Année : 2018

A bi-projection method for incompressible Bingham flows with variable density, viscosity and yield stress

Un schéma de bi-projection pour des écoulements incompressibles visco-plastiques avec densité, viscosité et seuil de plasticité variables

Rénald Chalayer
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Laurent Chupin
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Thierry Dubois

Résumé

A new numerical scheme for solving incompressible Bingham flows with variable density, plastic viscosity and yield stress is proposed. The mathematical and computational difficulties due to the non-differentiable definition of the stress tensor in the plug regions, i.e. where the strain-rate tensor vanishes, is overcome by using a projection formulation as in the Uzawa-like method for viscoplastic flows. This projection definition of the plastic tensor is coupled with a fractional time-stepping scheme designed for Newtonian incompressible flows with variable density. The plastic tensor is treated implicitly in the first sub-step of the fractional time-stepping scheme and a fixed-point iterative procedure is used for its computation. A pseudo-time relaxation term is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed-point algorithm. This is a key feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor. Stability and error analyses of the bi-projection scheme are provided. The use of the discrete divergence-free velocity to convect the density in the mass conservation equation allows us to derive lower and upper bounds for the discrete density. The error induced by the pseudo-time relaxation term is controlled by a prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field and the density, as well as the dependent parameters that are the plastic viscosity and the yield stress.
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Dates et versions

hal-01803711 , version 1 (30-05-2018)

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Rénald Chalayer, Laurent Chupin, Thierry Dubois. A bi-projection method for incompressible Bingham flows with variable density, viscosity and yield stress. SIAM Journal on Numerical Analysis, 2018, ⟨10.1137/17M113993X⟩. ⟨hal-01803711⟩
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