The Evacuation Wave Interaction And Vacuum

The present Ph.D. dissertation is concerned with the propagation of perfect com-pressible steady plane flows in a jet steam and a convex duct respectively. The existence of solutions to the interaction of rarefaction waves and the reflection of rarefaction waves on a free boundary and a convex wall is established. Meanwhile, the appearance of vacuum is carefully considered.Jet stream and pipe flow are widely used in our daily life and national defense. We first present a mathematical analysis of a supersonic jet stream out of an orifice into the atmosphere. The existence of classical solution in the interaction region arising from the interaction of rarefaction waves is established. For small pressure difference between the oncoming flow and the atmosphere, the existence of classical solution in the interaction region formed by the reflection of rarefaction waves on a free boundary is also obtained. After that we study the interaction of waves in a two-dimensional convex duct. We obtain the existence of classical solution of Euler system in the whole interaction region. Meanwhile, we discuss the problem of the appearance of vacuum in detail. We find that vacuum is always adjacent to the walls, and the appearance of vacuum depends on the slope of the walls and the relative location of the curved bends. The main methods are hodograph transformation and characteristics analysis.The whole contents are organized as follows.First of all, we present the relative background, historical development, research status as well as the main results of this thesis, including the idea, methods and feature, in the hope of describing a full view to the readers.The interaction of rarefaction waves in a jet stream is discussed in detail in Chapter 2. Including the existence of solution to the interaction of rarefaction waves of general strength, we consider in addition the existence of solution to the interaction formed by the reflection of rarefaction waves on the boundary of the jet stream. Due to the compression of the boundary, singularity may arise in this region. We prove that when the pressure difference between the oncoming flow and the atmosphere is not large, no singularity appears in that region. Moreover, we prove that there is no vacuum in the interaction region formed by the interaction of two strong centered waves.The propagation of the supersonic flow in a convex duct is discussed in Chapter 3. We first study the reflection of a centered wave on a convex wall by hodograph transformation and get the global existence of the classical solution in the interaction region formed by the reflection. Then we return to the interaction of waves in the duct and obtain the global solution in the whole interaction region. Next, we consider where vacuum may appear if it does. By contradiction we prove that vacuum is always adjacent to the walls and will never be totally surrounded by the flow. Furthermore, we prove that for a given oncoming flow, when the slope of the walls is larger than a constant depending only on the Mach number and the adiabatic exponent of the flow, vacuum will appear in the duct.