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### Abstract

Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearised Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager-Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearised Boltzmann equation. Then, a number of two-dimensional micro flows in the transition and free molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearised Boltzmann equation. For thermally driven flows in the free molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed

rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearised Poiseuille and thermal creep

flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearised Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearised Poiseuille and thermal creep

flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearised Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

Original language | English |
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Pages (from-to) | 53-84 |

Number of pages | 32 |

Journal | Journal of Fluid Mechanics |

Volume | 746 |

Early online date | 28 Mar 2014 |

DOIs | |

Publication status | Published - 10 May 2014 |

### Keywords

- computational methods
- micro- and nano-scale flows
- fluid dynamics
- rarefied gas flows

## Fingerprint Dive into the research topics of 'Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows'. Together they form a unique fingerprint.

## Projects

- 1 Finished

## Non-Equilibrium Fluid Dynamics for Micro/Nano Engineering Systems

Reese, J.

EPSRC (Engineering and Physical Sciences Research Council)

1/01/11 → 16/02/16

Project: Research

## Cite this

Wu, L., Reese, J. M., & Zhang, Y. (2014). Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows.

*Journal of Fluid Mechanics*,*746*, 53-84. https://doi.org/10.1017/jfm.2014.79