Collaborative Research: Efficient High-Order Algorithms for Nonequilibrium Microflows Over the Entire Range of Knudsen Number

非平衡微流在传感器、微流体和微机电系统中无处不在，在生物医学和环境科学、空气动力学、化学、能源工业和空间科学中有着重要的应用。例如，真空泵在非常低的密度和压力下操纵稀薄气体，在这种情况下，基于连续介质理论（即标准温度和压力下气体流动的工程模型）的方法不再有效。气体流动的非平衡程度是用克努森数——平均自由程与宏观长度的比值——来定性地测量的。非平衡流动可以用相空间中单粒子速度分布函数的玻尔兹曼方程来模拟。数值解的标准方法可能不够精确，并且由于相空间的高维性，特别是对于时变问题，其计算成本可能过高。本项目旨在为非平衡微流创造有效的模拟工具。研究生也参与了这项研究。对于低速微流，简化动力学模型，如线性化的bhatnagar - gross - krookk - welander （BGKW）方程，结合漫反射边界条件，是可靠的，并且能够在整个Knudsen数范围内对微流产生非常精确的结果。该方程可以转化为包含气体密度、流速、温度等宏观变量的线性积分方程组，大大减少了维数，从而大大提高了计算效率。本研究项目的首要目标是开发高效的高阶算法，以解决在整个Knudsen数范围内各种几何形状的非平衡气体流动的积分方程系统，用于微流。为了克服微流模拟中遇到的挑战，本工作包括以下技术成分：(1)在复杂平面上精确高效地评估Abramowitz函数的算法，以满足时间相关问题的要求；(2)积分方程的理论分析，特别是关于零空间的理论分析；(3)二维光滑和非光滑凸域上积分方程的高效高阶算法。该项目的成果预计将为涉及多尺度多物理场微流的广泛工程应用提供支持技术。

Nonequilibrium microflows are ubiquitous in sensors, microfluidics, and microelectromechanical systems and have important applications in bio-medical and environmental sciences, aerodynamic, chemical, and energy industries, and space science. For example, vacuum pumps manipulate rarefied gases at very low density and pressure, where approaches based on continuum theory, which is the engineering model for gaseous flows at standard temperature and pressure, is no longer valid. The extent of nonequilibrium of a gaseous flow is qualitatively measured by the Knudsen number -- the ratio of mean free path to a macroscopic length. Nonequilibrium flows may be modeled by the Boltzmann equation for the single-particle velocity distribution function in phase space. Standard methods for numerical solution may not be accurate enough, and their computational cost may be prohibitively expensive due to the high-dimensionality of phase space, especially for time-dependent problems. This project aims to create effective and efficient simulation tools for nonequilibrium microflows. Graduate students are involved in the research.For low-speed microflows, reduced kinetic models, such as the linearized Bhatnagar-Gross-Krook-Welander (BGKW) equation, coupled with the diffuse reflection boundary condition, are reliable and capable of producing very accurate results for microflows in the whole range of the Knudsen number. The equation can be transformed into a system of linear integral equations for macroscopic variables including the density of the gas, the flow velocity, and the temperature, which leads to great dimension reduction, consequently drastic enhancement of computational efficiency. The overarching goal of this research project is to develop efficient high-order algorithms to solve a system of integral equations pertaining to nonequilibrium gaseous flows in various geometries over the entire range of Knudsen number, for applications to microflows. The work consists of the following technical ingredients to overcome the challenges encountered in simulation of microflows: (1) An accurate and efficient algorithm to evaluate the Abramowitz function on the complex plane, as required for time-dependent problems; (2) Theoretical analysis, especially on the nullspace, of the integral equations; (3) Efficient and high-order algorithms for the integral equations on smooth and nonsmooth convex domains in two dimensions. The results of the project are anticipated to provide enabling technologies for a broad range of engineering applications involving multiscale multi-physics microflows.