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 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.