Mathematical models of incompressible viscoelastic media

Start: 01/26/2015 - 3:30pm
End  : 01/26/2015 - 4:30pm

Applied Math Seminar

V. V. Pukhnachev (Lavrentyev Institute of Hydrodynamics, Siberian Division of RAS, and Novosibirsk State University)


This research was stimulated by experiments carried out in Yekaterinburg (Apakashev and Pavlov, 1997) and in Chelyabinsk (Korenchenko and Beskachko, 2008). Authors found that at small strain rates ordinary water and similar liquids demonstrate not only viscous properties but also elastic ones. At the same time, we can neglect liquid compressibility. As a result, we lose the important property of hyperbolicity for the governing PDE system, which is typical for many mathematical models in continuum mechanics. There is no general theory of initial boundary value problems for these systems. However we can obtain a lot of valuable information analyzing exact solutions of corresponding systems in one and two dimensions. In particular, we study problem of filling in a spherical cavity in Maxwell and Kelvin-Voigt viscoelastic media and the analog of the classical Couette problem in plane and cylindrical geometries for both of them. We found a deep analogy between Kelvin-Voigt model and model of acoustics of viscous gas. In contrast, equations of incompressible Maxwell model are similar to equations of inviscid gas with non-convex constitutive law. It means that existence of weak or strong discontinuities is possible in the motion of incompressible Maxwell medium. We give examples of this phenomenon for motion with plane waves.

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