A bi-projection method for Bingham type flows

Laurent Chupin (1) , Thierry Dubois (1)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal

We propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic Bingham medium. The main difficulty, for both theoretical and numerical approaches, is due to a lack of definition of the plastic stress tensor in aeras where the deformation tensor vanishes. This is handled by introducing a projection formulation for the yield stress tensor. A new time scheme based on the classical incremental projection method for the Newtonian Navier-Stokes equations is proposed. The plastic tensor is treated implicitely in the first sub-step of the projection scheme and is computed by using a fixed point procedure, which is shown to converge geometrically. This key feature of our method ensures its numerical efficiency. Stability and error analyses of the numerical scheme are provided. A first-order estimate of the time error is obtained for the velocity field. A second-order cell-centered finite volume scheme on a staggered grid is applied for the spatial discretization. The scheme is assessed against previous plubished benchmark results for both Newtonian and Bingham flows in a lid-driven cavity at Reynolds number equals $10^3$. The proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplastic medium in the absence of any energy source term in the equations.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Laurent Chupin, Thierry Dubois. A bi-projection method for Bingham type flows. 2015. ⟨hal-01166406v1⟩