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普遍存在，并且在生物医学和环境科学、空气动力学、化学和能源工业以及空间科学中具有重要应用。例如，真空泵在非常低的密度和压力下操纵稀薄气体，其中基于连续介质理论的方法不再有效，连续介质理论是标准温度和压力下气体流动的工程模型。 气体流动的非平衡程度可以用努森数--平均自由程与宏观长度之比--来定性地测量。 非平衡流可以用玻尔兹曼方程来模拟相空间中的单粒子速度分布函数。用于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，它们的计算成本可能非常昂贵，特别是对于时间依赖的问题。该项目旨在为非平衡微流建立有效和高效的模拟工具。对于低速微流，简化的动力学模型，如线性化的Bhatnagar-Gross-Krook-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数范围内产生非常精确的微流结果。 该方程可以转化为宏观变量的线性积分方程组，包括气体的密度，流速和温度，这导致了大幅度的降维，从而大大提高计算效率。 该研究项目的总体目标是开发高效的高阶算法，以解决在整个克努森数范围内的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。 针对微流场模拟中遇到的困难，本文的工作包括以下几个方面：（1）在复平面上计算Abramowitz函数的精确有效的算法，以满足时间相关问题的需要：（2）对积分方程进行理论分析，特别是零空间的分析;（3）二维光滑和非光滑凸域上积分方程的高效高阶算法。 该项目的结果预计将提供使能技术，涉及多尺度多物理场微流的广泛的工程应用。

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based on continuum theory, which is the engineering model for gaseous flows at standard temperature and pressure, is no longer valid.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.