Abstract. Polydisperse suspensions consist of small particles which are dispersed in a viscous fluid, and which belong to a finite number N of species that differ in size or density. Spatially one-dimensional kinematic models for the sedimentation of such mixtures are given by systems of N non-linear first-order conservation laws for the vector Φ of the N local solids volume fractions of each species. The problem of hyperbolicity of this system is considered here for the models due to Masliyah, Lockett and Bassoon, Batchelor and Wen and Höfler and Schwarzer. In each of these models, the flux vector depends only on a small number m < N of independent scalar functions of Φ, so its Jacobian is a rank-m perturbation of a diagonal matrix. This allows to identify its eigenvalues as the zeros of a particular rational function R(λ), which in turn is the determinant of a certain m × m matrix. The coefficients of R(λ) follow from a representation formula due to Anderson [Lin. Alg. Appl. 246:49-70, 1996]. It is demonstrated that the secular equation R(λ) = 0 can be employed to efficiently localize the eigenvalues of the flux Jacobian, and thereby to identify parameter regions of guaranteed hyperbolicity for each model. Moreover, it provides the characteristic information required by certain numerical schemes to solve the respective systems of conservation laws.

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The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first‐order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Bürger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the models by Batchelor and Wen, Höfler and Schwarzer, and Davis and Gecol, for which the Jacobian of the flux is a rank‐3 or rank‐4 perturbation of a diagonal matrix. Explicit estimates of the regions of hyperbolicity of these models are derived via the approach of the so‐called secular equation (J. Anderson. Linear Algebra and Applications 1996; 246:49–70), which identifies the eigenvalues of the Jacobian with the poles of a particular rational function. Hyperbolicity of the system is guaranteed if the coefficients of this function have the same sign. Sufficient conditions for this condition to be satisfied are established for each of the models considered. Some numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.

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