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

非平衡微流场普遍存在于传感器、微流体和微电子机械系统中，在生物医学和环境科学、空气动力学、化工和能源工业以及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作稀薄气体，而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。气体流动的不平衡程度由克努森数--平均自由程与宏观长度之比--来定性地测量。非平衡流可以用相空间中单粒子速度分布函数的Boltzmann方程来描述。标准的数值解方法可能不够精确，而且由于相空间的高维性，特别是对于时间相关的问题，它们的计算成本可能高得令人望而却步。该项目旨在为非平衡微流创造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，能够在整个克努森数范围内产生非常精确的微流结果。该方程可转化为包括气体密度、流速和温度在内的宏观变量的线性积分方程组，从而大大降低了维数，从而大大提高了计算效率。该研究项目的主要目标是开发有效的高阶算法来求解与整个Knudsen数范围内各种几何形状的非平衡气体流动有关的积分方程组，以应用于微流动。为了克服微流模拟中遇到的挑战，本文的工作包括以下几个方面：(1)精确和高效地计算复杂平面上的Abramowitz函数，如时间相关问题所需；(2)积分方程组的理论分析，特别是在零空间上；(3)二维光滑和非光滑凸域上积分方程组的高效和高阶算法。该项目的成果预计将为涉及多尺度、多物理微流的广泛工程应用提供使能技术。

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based on continuum theory, which is the engineering model for gaseous flows at standard temperature and pressure, is no longer valid.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