A Bi-Projection Method for Incompressible Bingham Flows with Variable Density, Viscosity, and Yield Stress

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.

[1]  T. Papanastasiou.  Flows of Materials with Yield , 1987 .

[2]  M. Fuchs,et al.  Regularity results for the quasi–static Bingham variational inequality in dimensions two and three , 1998 .

[3]  Roland Glowinski,et al.  On the numerical simulation of Bingham visco-plastic flow: Old and new results , 2007 .

[4]  P. Colella,et al.  A Conservative Adaptive Projection Method for the Variable Density Incompressible Navier-Stokes Equations , 1998 .

[5]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[6]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[7]  Thierry Dubois,et al.  A bi-projection method for Bingham type flows , 2016, Comput. Math. Appl..

[8]  Pierre Fabrie,et al.  Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles , 2006 .

[9]  Michel Bercovier,et al.  A finite-element method for incompressible non-Newtonian flows , 1980 .

[10]  Jie Shen.  On error estimates of projection methods for Navier-Stokes equations: first-order schemes , 1992 .

[11]  J. Bell,et al.  A Second-Order Projection Method for Variable- Density Flows* , 1992 .

[12]  Chris J. Price,et al.  An accelerated dual proximal gradient method for applications in viscoplasticity , 2015, 1509.05084.

[13]  Jie Shen,et al.  Gauge-Uzawa methods for incompressible flows with variable density , 2007, J. Comput. Phys..

[14]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[15]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (I) , 1969 .

[16]  Pierre Saramito,et al.  Progress in numerical simulation of yield stress fluid flows , 2017, Rheologica Acta.

[17]  J.-L. GUERMOND,et al.  Error Analysis of a Fractional Time-Stepping Technique for Incompressible Flows with Variable Density , 2011, SIAM J. Numer. Anal..

[18]  P. Vigneaux,et al.  Augmented Lagrangian simulations study of yield-stress fluid flows in expansion-contraction and comparisons with physical experiments , 2017 .

[19]  S. Zagatti.  On viscosity solutions of Hamilton-Jacobi equations , 2008 .

[20]  I. Frigaard,et al.  On the usage of viscosity regularisation methods for visco-plastic fluid flow computation , 2005 .

[21]  R. Rannacher,et al.  Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization , 1990 .

[22]  Jean-Luc Guermond,et al.  On the approximation of the unsteady Navier–Stokes equations by finite element projection methods , 1998, Numerische Mathematik.

[23]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[24]  Gregory Seregin,et al.  Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids , 2001 .

[25]  Xiangdi Huang,et al.  Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity , 2015 .

[26]  L. Evans.  On solving certain nonlinear partial differential equations by accretive operator methods , 1980 .

[27]  P. Saramito.  A damped Newton algorithm for computing viscoplastic fluid flows , 2016 .

[28]  L. Muravleva.  Uzawa-like methods for numerical modeling of unsteady viscoplastic Bingham medium flows , 2015 .

[29]  Neil J. Balmforth,et al.  Yielding to Stress: Recent Developments in Viscoplastic Fluid Mechanics , 2014 .

[30]  M. Lewicka,et al.  On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions , 2015, 1501.01917.

[31]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[32]  Abner J. Salgado,et al.  A splitting method for incompressible flows with variable density based on a pressure Poisson equation , 2009, J. Comput. Phys..

[33]  L. Quartapelle,et al.  A projection FEM for variable density incompressible flows , 2000 .