Model Equations and Instability Regions for the Sedimentation of Polydisperse Suspensions of Spheres

Abstract

The one‐dimensional kinematical sedimentation theory for suspensions of small spheres of equal size and density is generalized to polydisperse suspensions and several space dimensions. The resulting mathematical model, obtained by introducing constitutive assumptions and performing a dimensional analysis, is a system of first‐order conservation laws for the concentrations of the solids species coupled to a variant of the Stokes system for incompressible flow describing the mixture. Various flux density vectors for the first‐order system have been proposed in the literature. Some of them cause the first‐order system of conservation laws to be non‐hyperbolic, or to be of mixed hyperbolic‐elliptic type in the bidisperse case. The criterion for ellipticity is equivalent to a well‐known instability criterion predicting phenomena like blobs and viscous fingering in bidisperse sedimentation. We show that loss of hyperbolicity, that is the occurrence of complex eigenvalues of the Jacobian of the first‐order system, can be viewed as an instability criterion for arbitrary polydisperse suspensions, and that for tridisperse mixtures this criterion can be evaluated by a convenient calculation of a discriminant. We determine instability regions (or alternatively prove stability) for three different choices of the flux vector of the first‐order system of conservation laws. Consequently, mixed or non‐hyperbolic, rather than hyperbolic, systems of conservation laws are the appropriate general mathematical framework for polydisperse sedimentation. The stability analysis examines a first‐order system of conservation laws, but its predictions are applicable to the full multidimensional system of model equations. The findings are consistent with experimental evidence and are appropriately embedded into the current state of knowledge of non‐hyperbolic systems of conservation laws.

On Riemann problems and front tracking for a model of sedimentation of polydisperse suspensions

“…The polydisperse sedimentation model goes back to Masliyah [41] and Lockett and Bassoon [38] ("MLB model"); further details are given in [4,5,11,53]. This model is similar to other multi-species kinematic models, for example for traffic flow [3,12,51], granular flow [19], and the settling of oil-in-water emulsions [46].…”

Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

“…Thus, our model is new and therefore derived completely in section 2. On the other hand, we essentially combine arguments that have been discussed extensively in previous works that focus on modeling either flocculated monodisperse [26,27,35] or nonflocculated polydisperse suspensions [14,19,22]. Thus the presentation in section 2 is fairly concise, and we refer to the cited papers for additional details and justification.…”

Modelling of Sedimentation for Polydisperse Mixture

“…Polydisperse suspensions consist of particles differing in size or density that are dispersed in a viscous fluid. During sedimentation, the different particle species segregate and create areas of different composition [1], [2].…”

“…As a result of these two forces, the particles falling down direction is a resultant function between v p and v 0 . The movement of solid particles in a perfect horizontal settler shown in eqn (1). If the settling velocity of particles in the way H is denoted in eqn (1):…”

“…The suspended particles vary considerably in source, composition charge, particle size, shape, and density. Correct application of coagulation and flocculation processes and selection of the coagulants depend upon understanding the interaction between these factors [1]. In practice, the coagulation is carried out by coagulants, where the hydrolysis products cause electrolyte coagulation and the coagulation by the colloid with opposite sign.…”