A Riemann problem for poroelastic materials with the balance equation for porosity. Part I
- Radkevich, Evgeniy V.
- Wilmański, Krzysztof
2010 Mathematics Subject Classification
- 41A60 35Q51 35L45 76S05
- Porous and granular media, solitons, Riemann Problem, asymptotic expansions
The paper is devoted to the asymptotic analysis of the multicomponent model of poroelastic materials in which the porosity is described by its own field equation. The model is weakly nonlinear due to the kinematic contributions and a nonlinear dependence of material parameters on an equilibrium porosity. It is shown that the model contains two small parameters. The first one describes the coupling of the skeleton (solid component) with the fluid and its contribution through partial stresses is similar to a dynamical pressure of extended thermodynamics. Mathematically it leads to a dispertion effect similar to this appearing in the Korteweg - de Vries equation. The second small parameter describes a relaxation of porosity. We consider two cases. In the first one the order of magnitude of both small parameters is the same. This seems to correspond to usual porous materials. In the second case the dimensionless relaxation time is propotional to the square of the other parameter. We call such materials granular-like porous. They seem to correspond to compact granular materials with hard and smooth particles. We prove the existence of soliton-like solutions for the porosity and kink-like solutions for the partial velocities under a natural entropy-like selection condition which is also presented in the paper. The proof is based on the asymptotic analysis in which two steps of approximations were investigated. We show that the diffusive interaction force of components - a kind of an internal friction - yields decaying amplitudes of discontinuities. We show as well that in one class of Riemann problems a Saffman-Taylor instability appears. The paper is divided into two parts solely for technical reasons. Therefore the references appear after the second part. In the Appendix we show a few examples of a numerical simulation of a two-dimensional Riemann problem. These were obtained by dr. O. A. Vassilieva (Moscow University). The full numerical analysis shall be presented separately.