A unified modeling approach for micro- and nano-scale gas flows

我们打算在微纳尺度现象学和计量学的数学建模方面建立一个全面的研究计划，总体目标是开发一种基于玻尔兹曼方程(BE)和分子动力学(MD)的多尺度多物理模拟方法，用于工程上感兴趣的微纳尺度流动。为了避免直接求解Boltzmann方程的计算挑战，我们将利用格子Boltzmann方程(LBE)作为一个简化的动力学系统，并通过增加离散速度的数量和增加碰撞模型的复杂性来系统地增加其复杂性和模拟更高的克努森数(Kn)流的能力。为了克服纳米尺度约束下气体流动的动力学理论的不足，即不能考虑气体与表面分子之间的van der Waals相互作用以及其他表面物理效应，我们提出了一种用于气体流动的分子动力学(MD)方法，该方法将计算成本降低约三个数量级。MD方法还将被用来提取用于气体-表面相互作用的平均场势，以包括在微流的LBE模型中。为了达到目标，我们提出了基于动力学理论的介观模型，该模型包含了从分子动力学中提取的用于气体-表面相互作用的一致平均场。我们认为，以一种特殊的方式将在不同尺度的不同物理体系中有效的模型拼接在一起是不可行的，例如宏观连续介质的Navier-Stokes方程和微观系统的分子动力学。我们期望所提出的方法能够正确地模拟由于非零Kn引起的非平衡效应和由于气体-表面相互作用而产生的表面效应，这是微米和纳米流动中的关键。所提出的研究的成功将对微米和纳米流动的建模和模拟产生广泛而重要的影响，这些流动在生物医学和其他应用的微流体中具有重要而广泛的应用。实际的例子是微惯性传感器或微镜阵列。我们的研究还将培养研究生成为我国快速增长的工程和科学领域的合格劳动力，这些领域对我们的国家经济和安全至关重要。

We intend to build a comprehensive research program in the mathematical modeling of micro/nanoscale phenomenology and metrology, with the overall objective to develop a multi-scale multi-physics simulation methodology based on the Boltzmann equation (BE) and molecular dynamics (MD) for micro- and nano-scale flows of engineering interest. To circumvent the computational challenge for directly solving the Boltzmann equation, we will utilize the lattice Boltzmann equation (LBE) as a reduced kinetic system, and allow systematic increase in its complexity and ability to model higher Knudsen number (Kn) flows by increasing the number of discrete velocities and augmenting the complexity of the collision model with multiple relaxation times. To overcome a deficiency in kinetic theory for gas flows in nano-scale confinements, i.e., its inability to include the van der Waals interactions between the gas and surface molecules, and other surface physics effects, we propose to develop a molecular dynamics (MD) method for gas flows that will reduce the computational cost by about three orders of magnitude. The MD method will also be used to extract mean-field potentials for gas-surface interactions to be included in the LBE models for micro-flows. To achieve ourobjective, we propose kinetic-theory based mesoscopic models which include consistent mean-field potentials extracted from MD for gas-surface interactions. We do not believe it is a viable approach to stitch together, in an ad hoc fashion, models which are valid in different physical regimes of disparately different scales, such as the Navier-Stokes equations for macroscopic continua and molecular dynamics for microscopic systems. We expect the proposed method to properly model both the nonequilibrium effects due to non-zero Kn and the surface effects due to gas-surface interactions, which are crucial in micro- and nano-flows.The success of the proposed research will have broad and significant impact on modeling and simulation of micro- and nano-flows, which have important and wide range of applications in micro-fluidics for bio-medical and other applications. Practical examples are micro-inertial sensors or micro-mirror arrays. Our research will also train graduate students to become qualified workforce for our nation in fast growing areas of engineering and science, which are crucial to our national economy and security.