Conservation Laws for Multiphase Flow

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Multiphase flow is a branch of fluid dynamics of considerable difficulty, importance, and interest. The defining equations result from averages of some primitive equation, such as the Navier-Stokes or Euler equations. Nonlinear terms in the primitive equations lead to new unknowns in the averaged equation and the need for closure relations. These are difficult to determine, or even to measure experimentally. Closure is nonunique, as different closures apply to different flow regimes, and different closures may define competing equations to describe the same flow regime. The different equations may be complementary in that they describe the same situation in differing levels of detail. They also may reflect unresolved differences of scientific opinion. For these multiple reasons, the analysis of the equations is of considerable importance to science. The proposal here is primarily concerned with methods of analysis that delimit or shed light on the closures that are valid descriptions of fluid flow.Multiphase flow, and more generally the study of turbulence, is one of the major unsolved problems of importance to physics and to engineering. Flow of oil, gas, and water mixtures in a pipeline or in the rocks of a petroleum reservoir provide examples of such flows. Thermal mixing layers in meteorology leading to formation of thunderstorms provide another example. The formation of salt domes in geological formations, the study of controlled fusion to provide ample energy sources, and the study of late stage supernovae, or stellar explosions are further examples. In all cases, the phenomena is too complex, detailed and varied to be described usefully at a fine level of detail. Just as with the process of addition of milk to coffee, the initial swirls of milk in the coffee are artistic and complex, but the coffee-milk mixture after stirring is better described by averages of coffee and milk. Such a study of averages of mixtures, and the appropriate equations is the purpose of this proposal. Because of the importance of the problem, many methods are used for the study of mixtures: experiment, theory and numerical simulation. The mathematical aspects of the equations describing the mixture also give an important window into this subject and will be the primary focus of this investigation.