Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows

Lei Wu, Jason M. Reese, Yonghao Zhang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 linearized 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 linearized Boltzmann equation. Then, a number of two-dimensional microflows 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 linearized 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 linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized 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 languageEnglish
Pages (from-to)53-84
Number of pages32
JournalJournal of Fluid Mechanics
Early online date28 Mar 2014
Publication statusPublished - 10 May 2014


  • computational methods
  • micro-/nano-fluid dynamics
  • rarefied gas flow
  • diffusive transport
  • Boltzmann
  • molecular models
  • channel flow
  • numerical methods
  • spectral methods
  • rarefaction
  • collision operator
  • micro gas flow
  • Knudsen number

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