Well-posedness of a multiscale model for concentrated suspensions

In a previous work [math.AP/0305408] three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and as a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker-Planck type equation with nonhomogeneous, nonlocal and possibly degenerate, coefficients. We show the existence and the uniqueness of the global-in-time weak solution to such a system.Comment: 1 figur

Convergence of equilibria for numerical approximations of a suspension model

In this paper we study the numerical approximations of a non-Newtonian model for concentratedsuspensions. First,weprovethattheapproximativemodelspossessauniquefixedpointandstudy theirconvergencetoastationarypointoftheoriginalequation. Second, we implement an implicit Euler scheme, proving the convergence of these approximationsaswell. Finally,numericalsimulationsareprovided

Macroscopic limit of a one-dimensional model for aging fluids

We study a one-dimensional equation arising in the multiscale modeling of some non-Newtonian fluids. At a given shear rate, the equation provides the instantaneous mesoscopic response of the fluid, allowing to compute the corresponding stress. In a simple setting, we study the well-posedness of the equation and next the long-time behavior of its solution. In the limit of a response of the fluid much faster than the time variations of the ambient shear rate, we derive some equivalent macroscopic differential equations that relate the shear rate and the stress. Our analytical conclusions are confronted to some numerical experiments. The latter quantitatively confirm our derivations