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 nonlinear first-order conservation laws for the vector $\Phi$ 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 $\Phi$, so its Jacobian is a rank-m perturbation of a diagonal matrix. This allows us to identify its eigenvalues as the zeros of a particular rational function $R(\lambda)$, which in turn is related to the determinant of a certain $m\times m$ matrix. The coefficients of $R(\lambda)$ follow from a representation formula due to Anderson [Linear Algebra Appl., 246 (1996), pp. 49–70]. It is demonstrated that the secular equation $R(\lambda)=0$ 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 aim of this paper is to derive a separable entropy for a one-dimensional reduced blood flow model, which will be used to treat the symmetrizability of the model in full generality and for constructing entropy conservative fluxes, which are one of the essential building blocks of designing entropy-stable schemes. Time discretization is conducted by implicit–explicit (IMEX) Runge–Kutta schemes, but solutions for nonlinear systems will not be required due to the particular form of the source term. To validate the numerical schemes obtained, some numerical tests are presented.