Collaborative Research: Efficient High-Order Algorithms For Nonequilibrium Microflows Over The Entire Range Of Knudsen Number

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based on continuum theory, which is the engineering model for gaseous flows at standard temperature and pressure, is no longer valid.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