Liquid-vapour travelling waves by a kinetic model of van der Waals fluids
by
Kazimierz Piechór
Pedagogical University of Bydgoszcz and Polish Academy of Sciences
Coauthors: Bogdan Kazmierczak

Our aim is to study a flow of a liquid-vapour system from the point of view of a kinetic theory. This is because in the transition zone separating the liquid and vapour phases the gradients of the flow parameters are very large and the zone itself is very narrow. So the flow is very non-uniform with a strong heat and mass transfer. In order to have a kinetic equation suitable to liquid dynamics and phase change Grmela [1], [2] proposed so-called Enskog-Vlasov equation. In this model, the intermolecular potential is split into a hard-core and an attractive tail. The hard-core is treated as in the standard or revised Enskog equation, whereas the tail enters the equation only linearly in a mean-field term. Unfortunately, it turns then that the Enskog-Vlasov equation, despite its merits, is too complicated for a purpose like that. That is why we turned to the so-called discrete kinetic theory developed for the Boltzmann kinetic equation, which considers such mathematical models of it that the molecular velocity space is not all Rd (where d = 1, 2, 3) but a finite, fixed in advance set of d-dimensional vectors and extended its ideas to our present needs. In this paper we confine ourselves to an analysis of shock waves using to this end the simplest 4- velocity model of the Enskog-Vlasov equation. Unfortunately, it turns then that the Enskog-Vlasov equation, despite its merits, is too complicated for a purpose like that. That is why we turned to the so-called discrete kinetic theory developed for the Boltzmann kinetic equation, which considers such mathematical models of it that the molecular velocity space is not all Rd (where d = 1, 2, 3) but a finite, fixed in advance set of d-dimensional vectors and extended its ideas to our present needs. In this paper we confine ourselves to an analysis of shock waves using to this end the simplest 4- velocity model of the Enskog-Vlasov equation. Although, from the physical point of view this model is very simple, mathematically it is quite complicated. Due to this complicity we performed its various simplifications, which will presented and discussed. We look for travelling wave solutions to these simplified versions.

We pay some attention to the monotonicity of the density component of the travelling wave.

Finally, we introduce a simplified model and present some numerical results concerning the hydrodynamic and kinetic shock wave structures paying special attention to the impending shock splitting. The new feature is that kinetic effects alone are unable to kill the artificial phenomenon of impending shock splitting. [1] Grmela M. (1971) Kinetic approach to phase transitions. J. Statistical Physics 3: 347-364 [2] Grmela M. (1974) On the approach to equilibrium in kinetic theory. J. Math. Physics 1:

35-40

Date received: February 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caco-10.