Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 401-439, January 2021.
In this article, we show how to derive a multiphase model of Baer and Nunziato type with a simple stochastic model. Baer and Nunziato models are known to be unclosed, namely, they depend on modeling parameters, as interfacial velocity and pressure, and relaxation terms, whose exact expression is still an open question. We prove that with a simple stochastic model, interfacial and relaxation terms are equivalent to the evaluation of an integral, which cannot be explicitly computed in general. However, in different particular cases matching with a large range of applications (topology of the bubbles/droplets, or special flow regime conditions), the interfacial and relaxation parameters can be explicitly computed, leading to different models that are either nonlinear versions or slight modifications of previously proposed models. The validity domains of previously proposed models are clarified, and some modeling parameters of the averaged system are linked with the local topology of the flow. Last, we prove that usual properties like entropy dissipation are ensured with the new closures found.

中文翻译：

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Baer 和 Nunziato 型多相模型的推导和闭合通过对简单随机模型求平均

多尺度建模与仿真，第 19 卷，第 1 期，第 401-439 页，2021 年 1 月。
在本文中，我们展示了如何使用简单的随机模型推导出 Baer 和 Nunziato 类型的多相模型。众所周知，Baer 和 Nunziato 模型是不封闭的，即它们依赖于建模参数，如界面速度和压力以及松弛项，其确切表达仍然是一个悬而未决的问题。我们证明，使用简单的随机模型，界面和松弛项等效于积分的评估，通常无法明确计算。然而，在与大范围应用（气泡/液滴的拓扑结构或特殊流态条件）匹配的不同特定情况下，可以明确计算界面和松弛参数，从而产生非线性版本或轻微修改的不同模型先前提出的模型。阐明了先前提出的模型的有效性域，并且平均系统的一些建模参数与流的局部拓扑相关联。最后，我们证明了发现的新闭包确保了熵耗散等常用属性。