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

我们打算在微/纳米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）的多尺度多物理模拟方法，用于工程感兴趣的微和纳米尺度流动。 为了规避直接求解玻尔兹曼方程的计算挑战，我们将利用格子玻尔兹曼方程（LBE）作为一个简化的动力学系统，并允许系统增加其复杂性和能力，通过增加离散速度的数量和增加复杂性的碰撞模型与多个弛豫时间模型更高的努森数（Kn）的流量。 为了克服纳米尺度约束中气体流动的动力学理论的不足，即，由于该方法不能考虑气体和表面分子之间的货车德瓦耳斯相互作用以及其它表面物理效应，我们提出发展一种用于气体流动的分子动力学（MD）方法，该方法将减少大约三个数量级的计算成本。MD方法也将被用来提取气体-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，其中包括一致的平均场电位提取的气体表面相互作用的MD。我们不认为这是一个可行的方法缝合在一起，在一个特设的方式，模型是有效的，在不同的物理制度，不同的尺度，如纳维尔-斯托克斯方程的宏观连续和分子动力学的微观系统。 我们期望所提出的方法，以正确的模型由于非零Kn的非平衡效应和由于气体-表面相互作用，这是至关重要的微和纳米flow.The成功的研究建议将有广泛的和显着的影响建模和模拟的微和纳米流，这在微流体生物医学和其他应用的重要和广泛的应用。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.