Abstract

We review some recent advances in the mathematical modelling and numerical simulation of polydisperse suspensions of small particles dispersed in a viscous fluid. Such a mixture contains small particles of multiple species that differ in size or density, and that segregate and form areas of different composition. In one space dimension, common models that describe the settling as such mixtures, and that generalize the well-known kinematic theory of sedimentation [38], can be formulated as systems of first-order nonlinear conservation laws for the unknown volume fractions as functions of height and time. These models differ in the algebraic form of the flux vector. One of the models that has received major acceptance through extensive support by experimental data was proposed by Masliyah [41] and Lockett and Bassoon [40] (``MLB model''). Several properties of these models are discussed. Of particular theoretical and practical interest is the property of hyperbolicity, that is, the existence of pairwise distinct real eigenvalues and a full set of corresponding eigenvectors of the flux Jacobian. This property is related to the stability of the sedimentation process and allows the implementation of high-resolution numerical schemes for its simulation. The spectral decomposition required to identify conditions for hyperbolicity is not available in closed form but can be characterized by the approach of the so-called secular equation [3,17]. The MLB and related models have been extended to describe suspensions forming compressible sediments. In that case, the governing equation is a system of nonlinear degenerate convection-diffusion type. The numerical solution of such models by explicit schemes is usually very inefficient due to the severe time step restriction. A more efficient, and still easy-to-implement alternative are so-called implicit-explicit numerical schemes (``IMEX schemes'') that treat the convective part by an explicit scheme, and the nonlinear diffusive part by an implicit scheme [13,14]. Finally, an outlook on some current developments, and in particular on spatially multi-dimensional models will be given [22].