A Model of a Plug-Chain System Near the Thermodynamic Critical Point: Connection with the Korteweg Theory of Capillarity and Modulation Equations

Abstract

We consider a plug flow of a gas-liquid mixture in a pipe of constant cross-section. We assume that the liquid and gas are distinct and non reacting chemicals, and we consider the case when the gas is near the thermodynamic critical point. Pertinent nonlinear models of the flow pattern are derived: a discrete plug-chain model and its long wave approximation. Explicit solutions to both models are obtained.

A mechanical analogy in the discrete case is derived which represents an infinite chain of mass points moving in the transversal direction only.

It is found that the model for the long -wave approximation of a plug-chain system can be considered as a special example of the Korteweg theory of capillarity.

Modulation equations of the Korteweg theory of capillarity along with “pseudothermodynamics” are obtained. It is found that the modulated quantities are described by Godunov’s system of quasilinear partial differential equations.

The “pseudotemperature” of the plug flow pattern as a function of other “thermodynamic parameters” is obtained in an explicit form.

1

This work was done while S.L.G. was a visitor at Ecole Normale Supérieure de Lyon, FRANCE

2

Member of the Institut Universitaire de France