Hyperbolicity Analysis of Polydisperse Sedimentation Models via a Secular Equation for the Flux Jacobian

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 Hofler 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 A...

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