ELSEVIER Physica D 74 (1994) 372-385 Nondissipative shock waves in two-phase flows T. Elperin a'l, N. Kleeorin a, A. Krylov b aThe Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, P.O. Box 653, Israel bInstitute of Physics of Earth, Russian Academy of Sciences, Moscow, Russia Received 17 August 1993; revised 19 November 1993; accepted 15 February 1994 Communicated by F.H. Busse Abstract It is shown that the Korteweg-de Vries equation which describes dissipationless processes can occur also in the system of equations of multiphase hydrodynamics when dissipation is compensated by external supply of energy. Therefore all the phenomena which are characteristic for the Korteweg-de Vries equation (solitons, periodic waves, nondissipative shock waves) can occur also in multiphase hydrodynamics. The study analyzes in particular the phenomenon of formation of nondissipative shock waves. It is shown that multiphase filtration is accompanied by formation of a continuously expanding region with small scale undamping oscillations of phase composition and velocity. The analysis uses the Korteweg-de Vries equation which is derived from the system of conservation laws describing multiphase hydrodynamics in porous media. Obtained results are relevant for the analysis of multiphase filtration (viscous fingering in the hydrocarbon recovery process) and in the hydrodynamics of fluidized bed (formation of bubbles). 1. Introduction The flows of fluids through porous media are widespread in nature and encountered in various engineering applications. The typical examples include filtration of water and oil through the ground, reactant flows in catalytic packed bed reactors, flows in fluidized bed reactors etc. In this investigation we consider only some general properties of these flows. Obviously, the here analyzed processes of pattern formation in multi- phase flows do not exhaust all the variety of the phenomena occurring in such flows. 1E-mail address: elperin@menix.bgu.ac.il It is generally believed that filtration through porous media can be adequately described by the empirical Darcy law [1]: vP= (1) where k is a permeability of porous medium, /z is the dynamic viscosity of filtrating fluid, VP is the pressure gradient and V is the flow rate velocity. In the case of multiphase filtration the consequence of this equation is the formation of self-steepening kinematic shock waves [2]. Such behavior arises due to nonlinear dependence of the interphase friction upon the phase composi- tion. Another important field of application of 0167-2789/94/$07.00 (~ 1994 Elsevier Science B.V. All rights reserved SSD1 0167-2789(94)00027-N