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Contact manifolds for bidisperse sedimentation with particle-size-specific hindered-settling factors

Ponente: Pablo Castañeda Rivera
Institución: Instituto Tecnológico Autónomo de México (ITAM)
Cuándo 28/05/2015
de 10:00 a 11:00
Dónde Salón de seminarios Graciela Salicrup
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Bidisperse suspensions consist of two types of small particles differing in size, density or viscosity, which are dispersed in a viscous fluid. During sedimentation, the different particle species segregate and create areas of different composition. Spatially one-dimensional mathematical models of this process can be expressed as strongly coupled, nonlinear systems of first-order conservation laws. The solution of this system is the vector of volume fractions of each species as a function of depth and time, which will in general be discontinuous. The system might fail to be strictly hyperbolic in models for polydisperse suspensions with more than two phases; these can be modeled by the Masliyah-Lockett-Bassoon (MLB) flux vector, where the particles have the same density, and the hindered-settling factor (a multiplicative algebraic expression appearing in the flux vector) depends on the particle size. Even though strict hyperbolicity can be guaranteed for bidisperse suspensions with size-dependent hindered settling factor, for certain parameter choices a contact manifold in the interior of the phase space can be detected. A small perturbation of initial data around this contact manifold leads to a drastic change of the solution structure.

We want to examine the impact of the contact 
manifold on the solution structure of Riemann problems that occur in standard batch settling tests. This examination is carried out along 2x2 systems modeling bidisperse suspensions by the MLB model with particle-size dependent hindered settling factors. A characterization of the contact manifold reveals that it forms part of the Hugoniot locus of the origin. An eigenvalue analysis gives that the contact manifold can be associated to the second characteristic family. The elementary waves that start in the origin of the phase space turn out to be shocks belonging to different classes. Prototypic Riemann problems that connect the origin to any state in the state space are solved semi-analytically. The Riemann problems connecting a state in the state space to the maximum line are solved numerically.

This is a joint work with Stefan Berres from the Universidad Católica de Temuco (Chile).