The well-known kinematic sedimentation model by Kynch states that the settling velocity of small equal-sized particles in a viscous fluid is a function of the local solids volume fraction. This assumption converts the one-dimensional solids continuity equation into a scalar, nonlinear conservation law with a nonconvex and local flux. This work deals with a modification of this model, and is based on the assumption that either the solids phase velocity or the solid-fluid relative velocity at a given position and time depends on the concentration in a neighbourhood via convolution with a symmetric kernel function with finite support. This assumption is justified by theoretical arguments arising from stochastic sedimentation models, and leads to a conservation law with a nonlocal flux. The alternatives of velocities for which the nonlocality assumption can be stated lead to different algebraic expressions for the factor that multiplies the nonlocal flux term. In all cases, solutions are in general discontinuous and need to be defined as entropy solutions. An entropy solution concept is introduced, jump conditions are derived and uniqueness of entropy solutions is shown. Existence of entropy solutions is established by proving convergence of a difference-quadrature scheme. It turns out that only for the assumption of nonlocality for the relative velocity it is ensured that solutions of the nonlocal equation assume physically relevant solution values between zero and one. Numerical examples illustrate the behaviour of entropy solutions of the nonlocal equation.

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We construct global weak solutions to the different modes of sedimentation appearing in the theory of Kynch and show that, with constant initial concentration, only five modes of sedimentation exkt. Wc also generalize the method of construction to the case of a monotonically decreasing initial concentration.

We develop a general phenomenological theory of sedimentation‐consolidation processes of flocculated suspensions, which are considered as mixtures of two superimposed continuous media. Following the standard approach of continuum mechanics, we derive a mathematical model for these processes by applying constitutive assumptions and a subsequent dimensional analysis to the mass and linear momentum balance equations of the solid and liquid component. The resulting mathematical model can be viewed as a system of Navier‐Stokes type coupled to a degenerating convection‐diffusion equation by singular perturbation terms. In two or three space dimensions, solvability of these equations depends on the choice of phase and mixture viscosities. In one space dimension, however, this model reduces to a quasilinear strongly degenerate parabolic equation, for which analytical and numerical solutions are available. The theory is applied to a batch sedimentation‐consolidation process.

This paper provides a concise review of the contributions to research in sedimentation and