The Boltzmann equation provides a rigorous description of gas flows at all degrees of gas rarefaction. Asymptotic analyses of this equation yields valuable insight into the physical mechanisms underlying gas flows. In this article, we report an asymptotic analysis of the Boltzmann-BGK equation for a slightly rarefied gas when the acoustic wavelength is comparable to the macroscopic characteristic length scale of the flow. This is performed using a three-way matched asymptotic expansion, which accounts for the Knudsen layer, the viscous layer and the outer Hilbert region—these are separated by asymptotically disparate length scales. Transport equations and boundary conditions for these regions are derived. The utility of this theory is demonstrated by application to three problems: (1) flow generated by uniformly heating two plates, (2) oscillatory thermal creep induced between two plates, and (3) the flow generated by an oscillating sphere. Comparisons to numerical simulations of the Boltzmann-BGK equation and previous asymptotic theories (for long wavelength) are performed. The present theory is distinct from previous asymptotic analyses that implicitly assume long or short acoustic wavelength. This theory is expected to find application in the design and characterisation of nanoelectromechanical devices, which often generate acoustic oscillatory flows of a rarefied nature.

中文翻译：

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声音在稀有气体中流动

玻耳兹曼方程式对气体稀薄度的所有角度下的气体流量进行了严格的描述。对该方程进行渐近分析，可以深入了解气流背后的物理机理。在本文中，当声波波长与流动的宏观特征长度尺度相当时，我们报告了一种稍微稀薄的气体的Boltzmann-BGK方程的渐近分析。这是使用三向匹配渐近展开来执行的，该渐进展开考虑了Knudsen层，粘性层和外部Hilbert区域-它们通过渐近不同的长度标度分开。推导了这些区域的输运方程和边界条件。该理论在以下三个问题上的应用证明了其实用性：（1）均匀加热两块板而产生的流动，（2）在两个板之间引起的振荡热蠕变，以及（3）振荡球产生的流动。对Boltzmann-BGK方程和以前的渐近理论（对于长波长）的数值模拟进行了比较。本理论不同于以前的渐进分析，后者隐含了长或短的声波波长。预期该理论将在纳米机电装置的设计和表征中找到应用，该装置通常会产生稀有性质的声振荡流。本理论不同于以前的渐进分析，后者隐含了长或短的声波波长。预期该理论将在纳米机电装置的设计和表征中找到应用，该装置通常会产生稀有性质的声振荡流。本理论不同于以前的渐进分析，后者隐含了长或短的声波波长。预期该理论将在纳米机电装置的设计和表征中找到应用，该装置通常会产生稀有性质的声振荡流。