A new operator splitting algorithm for elastoviscoplastic flow problems

This paper presents an efficient time-dependent decoupled approach for the numerical resolution of the highly nonlinear set of coupled partial differential equations appearing in elastoviscoplastic fluid flow problems. The two main nonlinear difficulties, the viscoplasticity and the viscoelasticity, are then solved separately. Numerical simulations suggest an optimal convergence rate with respect to the space discretization. Finally, numerical results compare well with experimental measurements on liquid foams in a complex geometry. Future works will explore flows of liquid foams for tridimensional geometries where experimental data are available and also compare to flows of others soft glassy materials such as carbopol solutions.

[1]  G. G. Stokes.  On the Effect of the Internal Friction of Fluids on the Motion of Pendulums , 2009 .

[2]  A. Putz,et al.  Settling of an isolated spherical particle in a yield stress shear thinning fluid , 2008 .

[3]  Jacques Baranger,et al.  Finite element approximation of viscoelastic fluid flow: Existence of approximate solutions and error bounds , 1992 .

[4]  P. Saramito,et al.  Understanding and predicting viscous, elastic, plastic flows , 2011, The European physical journal. E, Soft matter.

[6]  P. Coussot.  Rheometry of Pastes, Suspensions, and Granular Materials: Applications in Industry and Environment , 2005 .

[7]  M. Fortin,et al.  A new approach for the FEM simulation of viscoelastic flows , 1989 .

[8]  F Graner,et al.  Discrete rearranging disordered patterns, part II: 2D plasticity, elasticity and flow of a foam , 2008, The European physical journal. E, Soft matter.

[9]  K. Gardner.  Desh-Bidesh - Sylheti images of home and away , 1993 .

[10]  P. Saramito,et al.  Numerical modelling of foam Couette flows , 2008, The European physical journal. E, Soft matter.

[11]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[12]  R. Glowinski,et al.  Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics , 1987 .

[13]  Pierre Saramito,et al.  A new constitutive equation for elastoviscoplastic fluid flows , 2007 .

[14]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[15]  Bo Dong,et al.  Optimal Convergence of the Original DG Method on Special Meshes for Variable Transport Velocity , 2010, SIAM J. Numer. Anal..

[16]  François Graner,et al.  The role of fluctuations and stress on the effective viscosity of cell aggregates , 2009, Proceedings of the National Academy of Sciences.

[17]  G. Richter.  An Optimal-Order Error Estimate for the Discontinuous Galerkin Method , 1988 .

[18]  P. Lesaint,et al.  Finite element collocation methods for first-order systems , 1979 .

[19]  R. Verfürth.  Finite element approximation on incompressible Navier-Stokes equations with slip boundary condition , 1987 .

[20]  P. Saramito,et al.  Steady Couette flows of elastoviscoplastic fluids are nonunique , 2012 .

[21]  P. Saramito,et al.  Efficient simulation of nonlinear viscoelastic fluid flows , 1995 .

[22]  J. Oldroyd.  On the formulation of rheological equations of state , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[23]  M. Möbius,et al.  Rate dependence and role of disorder in linearly sheared two-dimensional foams. , 2007, Physical review letters.

[24]  S. Frey,et al.  Flow of elasto-viscoplastic thixotropic liquids past a confined cylinder , 2013 .

[25]  Todd E. Peterson,et al.  A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation , 1991 .

[26]  Francois Graner,et al.  Two-dimensional flow of foam around a circular obstacle: local measurements of elasticity, plasticity and flow , 2006, Journal of Fluid Mechanics.

[27]  Pierre Saramito,et al.  An adaptive finite element method for Bingham fluid flows around a cylinder , 2003 .

[28]  K. Deckelnick,et al.  Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition , 1999 .

[30]  R. Verfürth.  Finite element approximation of steady Navier-Stokes equations with mixed boundary conditions , 1985 .

[31]  P. Saramito.  A new elastoviscoplastic model based on the Herschel–Bulkley viscoplastic model , 2009 .

[32]  L Preziosi,et al.  An elasto-visco-plastic model of cell aggregates. , 2010, Journal of theoretical biology.

[33]  A. Colin,et al.  Spatial cooperativity in soft glassy flows , 2008, Nature.

[34]  J. M. Piau,et al.  Flow characteristics of viscoelastic fluids in an abrupt contraction by using numerical modeling , 1994 .

[35]  T. Colin,et al.  An elasto-visco-plastic model for immortal foams or emulsions , 2007, The European physical journal. E, Soft matter.

[36]  J. Oldroyd.  A rational formulation of the equations of plastic flow for a Bingham solid , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[37]  Juhani Pitkäranta,et al.  An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation , 1986 .

[38]  Two-dimensional foam rheology with viscous drag. , 2006, Physical review letters.

[39]  Roland Glowinski,et al.  On the Numerical Simulation of Viscoplastic Fluid Flow , 2011 .

[40]  E. C. Bingham.  Fluidity And Plasticity , 1922 .

[41]  Yoël Forterre,et al.  Drop impact of yield-stress fluids , 2009, Journal of Fluid Mechanics.