Raimund Bürger, Rosa Donat, Pep Mulet, Carlos A. Vega:
On the hyperbolicity of certain models of polydisperse sedimentation
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 since 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 J. Appl. Math. 70:2186-2213, 2010] 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, Lin. Alg. Appl. 246:49-70, 1996], 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.
This preprint gave rise to the following definitive publication(s):
Raimund BüRGER, Rosa DONAT, Pep MULET, Carlos A. VEGA: On the hyperbolicity of certain models of polydisperse sedimentation. Mathematical Methods in the Applied Sciences, vol. 35, 6, pp. 723-744, (2012).