Traveling Waves Solution of Convection–Diffusion Systems Whose Convection Terms are Weakly Nonconservative: Application to the Modeling of Two-Phase Fluid Flows

The paper is concerned with the derivation of explicit approximate jump conditions for the shock waves solutions of hyperbolic systems in nonconservation form extracted from a known second-order convection–diffusion system; the shock waves are defined as the limit, as the diffusion tends to zero, of the traveling waves solution of the second-order system. Although no explicit jump conditions may be found for general hyperbolic systems in nonconservation form, we derive an asymptotic expansion of the nonexplicit jump relations in the case of two-phase fluid flows. We indeed observe that the terms in nonconservation form arise from pressure gradient forces which act on droplets and are small in the sense that the a small number appears in front of them. We obtain an asymptotic expansion of the nonexplicit jump relations whose first two terms can be computed explicitly.