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

我们打算在微/纳米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，其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）的多尺度多物理场模拟方法，用于工程兴趣的微和纳米尺度流动。为了避免直接求解玻尔兹曼方程的计算挑战，我们将利用晶格玻尔兹曼方程（LBE）作为一个简化的动力学系统，并允许系统地增加其复杂性和建模更高Knudsen数（Kn）流的能力，通过增加离散速度的数量和增加碰撞模型的复杂性，增加多个松弛时间。为了克服纳米尺度气体流动动力学理论的不足，即无法包括气体和表面分子之间的范德华相互作用以及其他表面物理效应，我们提出了一种气体流动的分子动力学（MD）方法，该方法将使计算成本降低约三个数量级。MD方法还将用于提取气-表面相互作用的平均场势，以包含在微流动的LBE模型中。为了实现我们的目标，我们提出了基于动力学理论的介观模型，其中包括从MD提取的气体表面相互作用的一致平均场势。我们不相信这是一种可行的方法，以一种特别的方式，将在不同尺度的不同物理制度中有效的模型，如宏观连续体的Navier-Stokes方程和微观系统的分子动力学，结合在一起。我们希望所提出的方法能够正确地模拟非零Kn引起的非平衡效应和由于气体表面相互作用引起的表面效应，这在微纳米流动中是至关重要的。本研究的成功将对微、纳米流动的建模与仿真产生广泛而重要的影响，在生物医学等微流体领域具有重要而广泛的应用前景。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们国家快速发展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.