Quantification of thermally-driven flows in microsystems using Boltzmann equation in deterministic and stochastic contexts

Abstract

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow-an observation attributed to the thermo-stress convection effects at microscale. The dynamics of the overall thermo-stress convection process is governed by the Boltzmann equation-an integro-differential equation describing the evolution of the molecular distribution function in six-dimensional phase space-which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general inter-molecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method; and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models-necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method; Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK, and Shakhov kinetic models; and the widely-used stochastic DSMC method, are utilized to assess the thermo-stress convection process in MIKRA-Micro In-Plane Knudsen Radiometric Actuator-a microscale compact low-power pressure sensor utilizing the Knudsen forces. BGK model under-predicts the heatflux, shear-stress, and flow speed; S-model over-predicts; whereas ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems.

An efficient discrete velocity method with inner iteration for steady flows in all flow regimes

“…[1][2][3][4] Besides, the microelectromechanical systems (MEMS) will also face rarefied flow problems due to the very small characteristic scale. [5][6][7] Moreover, the local Knudsen number in practical engineering problems may vary significantly over several orders of magnitude, which results in the flows covering all flow regimes. Taking the hypersonic flow over a flying vehicle at Reynolds number of 59 373 and Mach number of 4 as an example, the minimum and maximum local Knudsen numbers are 7:61 Â 10 À5 and 19.9, respectively.…”

Local Discrete Velocity Grids for Multi-Species Rarefied Flow Simulations

“…In [37], [36], [33], [34], [28], the Discrete Unified Gas Kinetic Scheme (DUGKS) is extended to multi-species flows described by several multi-species models. Then, a Discontinuous Galerkin (DG) approach has been applied to the Boltzmann equations in [23] and [22].…”