Strongly Degenerate Parabolic-Hyperbolic Systems Modeling Polydisperse Sedimentation with Compression

Abstract

Abstract. We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression" or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an N × N coupled system of quasilinear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N = 3 is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convectiondiffusion problems. The numerical scheme exploits the derived bounds on the eigenvalues to keep the numerical diffusion to a minimum.Key words. polydisperse suspensions, sedimentation, systems of conservation laws, strongly degenerate parabolic-hyperbolic systems, central difference approximation AMS subject classifications. 35K65, 35L40, 35L65, 65M06, 76T05 DOI. 10.1137/S00361399024081631. Introduction. Mathematical models for the (controlled) sedimentation of polydisperse suspensions of small particles, which belong to a finite number of species differing in size or density and are suspended in a viscous fluid, are important to many applications such as the chemical engineering, ceramic, pulp and paper, and food industries, mineral processing, wastewater treatment, and medicine [3,50,88,89,101,122]. The characteristic behavior of such mixtures is differential sedimentation, which leads to areas of different composition if an initially homogeneous suspension is allowed to settle. In this paper, we consider the additional property that the solid particles possibly form a compressible sediment layer. A mathematical model for polydisperse suspensions forming compressible sediments is developed, analyzed, and simulated, focusing on three different aspects.First, we show how two existing sedimentation models-one for monodisperse flocculated suspensions, which are described by scalar strongly degenerate parabolichyperbolic equations, and one for polydisperse suspensions of rigid spheres differing

Numerical and experimental analysis of the sedimentation of spherical colloidal suspensions under centrifugal force

“…10 By including the sediment within the model, compression of the sediment can be taken into account, using a compressive yield stress. 11 Such one-dimensional models have also been successfully extended to account for the effects of polydispersity in particle size, 12 and density stratification in the continuous phase. 13 Particles in suspension do not sediment alone, and as a result their motion is hindered by surrounding particles, where the effect of hindrance increases as the particle volume fraction increases.…”

The strong solutions for a class of fluid-particle interaction non-Newtonian models

“…Author details 1 College of Science, Liaoning University of Technology, Jinzhou, Liaoning 121001, P.R. China.…”

The existence of solutions for a shear thinning compressible non-Newtonian models

“…Fluid-particle interaction model arises in many practical applications, and is of primary importance in the sedimentation analysis of disperse suspensions of particles in fluids. This model is one of the commonly used models nowadays in biotechnology, medicine, mineral processing and chemical engineering [27]- [25]. Usually, the fluid flow is governed by the Navier-Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation [4], the system has the form:    ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) + ∇(P (ρ) + η) − µ∆u − λ∇divu = −(η + βρ)∇Φ, η t + div(η(u − ∇Φ)) − ∆η = 0,…”