Rana Anirudh Singh, Gupta Vinay Kumar
Department of Mathematics, Indian Institute of Technology Indore, Indore, Madhya Pradesh 453552, India.
Department of Mathematics, Birla Institute of Technology and Science Pilani, Rajasthan 333031, India.
Phys Rev E. 2023 Jul;108(1-2):015306. doi: 10.1103/PhysRevE.108.015306.
The inability of the Navier-Stokes-Fourier equations to capture rarefaction effects motivates us to adopt the extended hydrodynamic equations. In the present work, a hydrodynamic model, which consists of the conservation laws closed with the recently propounded coupled constitutive relations (CCR), is utilized. This model is referred to as the CCR model and is adequate for describing moderately rarefied gas flows. A numerical framework based on the method of fundamental solutions is developed to solve the CCR model for rarefied gas flow problems in quasi two dimensions. To this end, the fundamental solutions of the linearized CCR model are derived in two dimensions. The significance of deriving the two-dimensional fundamental solutions is that they cannot be deduced from their three-dimensional counterparts that do exist in literature. As applications, the developed numerical framework based on the derived fundamental solutions is used to simulate (i) a rarefied gas flow between two coaxial cylinders with evaporating walls and (ii) a temperature-driven rarefied gas flow between two noncoaxial cylinders. The results for both problems have been validated against those obtained with the other classical approaches. Through this, it is shown that the method of fundamental solutions is an efficient tool for addressing quasi-two-dimensional multiphase microscale gas flow problems at a low computational cost. Moreover, the findings also show that the CCR model solved with the method of fundamental solutions is able to describe rarefaction effects, like transpiration flows and thermal stress, generally well.
纳维-斯托克斯-傅里叶方程无法捕捉稀薄效应，这促使我们采用扩展的流体动力学方程。在当前工作中，我们使用了一个流体动力学模型，该模型由通过最近提出的耦合本构关系（CCR）封闭的守恒定律组成。这个模型被称为CCR模型，适用于描述中等稀薄气体流动。我们开发了一个基于基本解方法的数值框架，用于求解CCR模型，以解决准二维稀薄气体流动问题。为此，我们推导了二维线性化CCR模型的基本解。推导二维基本解的意义在于，它们无法从文献中已有的三维对应解推导得出。作为应用，基于推导得到的基本解开发的数值框架被用于模拟：（i）两个具有蒸发壁的同轴圆柱之间的稀薄气体流动，以及（ii）两个非同轴圆柱之间由温度驱动的稀薄气体流动。这两个问题的结果已与其他经典方法得到的结果进行了验证。由此表明，基本解方法是一种以低计算成本解决准二维多相微尺度气体流动问题的有效工具。此外，研究结果还表明，用基本解方法求解的CCR模型通常能够很好地描述稀薄效应，如蒸发流和热应力。
