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The Boltzmann Equation

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Mathematical Modeling for Complex Fluids and Flows

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Abstract

For about two decades, the lattice Boltzmann method (LBM) has made a major breakthrough in the numerical solution of fluid flow problems and has become a real and efficient alternative with respect to the traditional route of CFD tools and software. Seen from afar, the method looks like a toy algorithm that is easily written as a one page program to solve, for example, the square cavity problem. However, if a more careful and thorough analysis is undertaken, the method reveals itself as a very acute and deep instrument to simulate the behavior of complex fluid motion. LBM represents the motion of virtual particles that collide according to some simple rules and then stream and move across a computational grid named the lattice. These particles mimic the fluid flow and yield excellent results for many difficult problems. The design of the underlying model is made at the mesoscopic scale, that is, at an intermediate scale between the continuous representation and the nanoscale. Nevertheless a multi-scale analysis using the Chapman-Enskog expansion produces the well known (weakly compressible) Navier-Stokes equations.

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Authors and Affiliations

  1. Institute of Mechanical Engineering, Swiss Federal Institute of Technology, EPFL, Lausanne, Switzerland

    Michel O. Deville

  2. Institute PPRIME, CNRS-Université de Poitiers-ENSMA, Futuroscope Chasseneuil, France

    Thomas B. Gatski

  3. Center for Coastal Physical Oceanography and Ocean, Earth and Atmospheric Sciences, Old Dominion University, Norfolk, USA

    Thomas B. Gatski

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  1. Michel O. Deville
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  2. Thomas B. Gatski
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Correspondence to Michel O. Deville .

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Deville, M.O., Gatski, T.B. (2012). The Boltzmann Equation. In: Mathematical Modeling for Complex Fluids and Flows. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25295-2_7

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