Two-dimensional Riemann problem involving three contact discontinuities for 2×2 hyperbolic conservation laws in anisotropic media

Abstract

In this study, we consider the two-dimensional Riemann problem for a system of conservation laws, which models polymer flooding in an anisotropic oil reservoir. The initial data are constants in three sectors centered at the origin, which only involve three contact discontinuities. By the generalized characteristic analysis method and the phase plane analysis method, we find that the solutions are not unique for certain values of the initial data. We propose an additional stability condition for the interaction of the contact discontinuities. By this stability condition, all of the exact solutions and their corresponding criteria can be obtained. The solutions exhibit various geometrical structures. In particular, envelope rarefaction waves develop in some solutions.

Non‐selfsimilar global solutions to a two‐dimensional system of conservation laws

“…Besides, it is noticed that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non-selfsimilar global solutions and are essentially different from those of the self-similar global solutions [22][23][24]. Moreover, we believe that the technique used in this context is also available to the study of this type of Riemann problem for other multi-dimensional systems of conservation laws.…”

“…Meanwhile, in [23], they dealt with the case where A ¤ B and the initial data took two different constant values separated by x-negative and y-positive axes, where they presented all of the solutions explicitly. Pang and Yang [24] studied the case where A ¤ B and the initial data took three different constant values involving three contact discontinuities. They proposed a stability condition for the interaction of the contact discontinuities and obtained all of the exact solutions and their corresponding criteria.…”

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

“…2 Many two-dimensional hyperbolic conservation laws that are applicable to polymer flooding of an oil reservoir, elastic theory and magneto-hydrodynamics, and so forth have been discussed extensively in recent years. We refer the reader to previous works [13][14][15] for more details.…”