Conservation Laws for Multiphase Flow

多相流是流体力学的一个分支，具有相当大的难度、重要性和趣味性。定义方程由一些原始方程的平均值产生，例如纳维斯托克斯方程或欧拉方程。原始方程中的非线性项导致了平均方程中的新未知数，并需要闭合关系。这些都很难确定，甚至很难通过实验来衡量。闭合不是唯一的，因为不同的闭合适用于不同的流型，并且不同的闭合可能定义相互竞争的方程来描述相同的流型。不同的方程式可能是互补的，因为它们以不同的详细程度描述了相同的情况。它们还可能反映出尚未解决的科学观点分歧。由于这些多重原因，对这些方程的分析对科学具有相当重要的意义。这里的建议主要涉及分析方法，这些方法界定或阐明了有效描述流体流动的闭合。多相流，更广泛地说，对湍流的研究，是对物理和工程具有重要意义的主要未解决问题之一。石油、天然气和水的混合物在管道或储油层岩石中的流动就是这种流动的例子。气象学中的热混合层导致雷暴的形成是另一个例子。地质构造中盐丘的形成，提供充足能源的受控聚变的研究，以及晚期超新星或恒星爆炸的研究，都是进一步的例子。在所有情况下，这些现象都过于复杂、详细和多样，无法以精细的细节进行有用的描述。就像在咖啡中加入牛奶的过程一样，咖啡中最初的牛奶漩涡是艺术和复杂的，但搅拌后的咖啡和牛奶混合物更好地描述了咖啡和牛奶的平均值。这项建议的目的是研究混合物的平均值和适当的方程式。由于这一问题的重要性，人们采用了多种方法来研究混合物：实验、理论和数值模拟。描述混合物的方程的数学方面也为这一主题提供了一个重要的窗口，也将是本研究的主要焦点。

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.