Thermal Lattice Boltzmann Method for Dilute Fluids of Bosons and Fermions
R. C. V. Coelho1; A. S. Ilha2; M. M. Doria1
1 UFRJ; 2 Inmetro
Resumo
Nearly twenty years ago a numerical method was developed to solve the Boltzmann equation with the BGK (Bhatnagar, Gross and Krook) collision. This method, based on the discretization of the phase space, was very successful in solving various problems of fluid mechanics, including problems with complex geometry, interfacial phenomena and multicomponent fluids. Known as LBM - Lattice Bolztmann Method - it describes the evolution of a set of statistical distributions of particles defined on a regular space lattice in which each site has a finite number of velocities directed to neighboring sites. The advantage over other methods lies in the simplicity of its dynamics and especially the flexibility for implementation in parallel computing. Although the LBM breaks the translational invariance in the mesoscopic scale, it leads to the expected hydrodynamic equations in the continuous macroscopic scale, such as conservation of mass and momentum (Navier Stokes). In recent years there has been a great deal for the construction of an LBM able to describe compressible and thermal fluids, which also carries the description of energy conservation. In this work we develop the LBM for the treatment of quantum fluids, i.e., those such that the particle distributions describe bosons or fermions. We show that the LBM for thermal and compressible classical fluid, described by the Maxwell-Boltzmann statistics, and the LBM for quantum fluids, described by the Bose-Einstein and Fermi-Dirac statistics, are based on the same mathematical structure. Both cases require the expansion to fourth order of the equilibrium distribution functions in Hermite polynomials, in such a way to reproduce a consistent thermodynamics. Once done this fourth order expansion we obtain the correct macroscopic equations describing the quantum fluid, i.e., the equation of conservation of mass, momentum (Navier-Sokes) and energy. From these equations we retrieve the well-known coefficients of viscosity and thermal conductivity. As an application, we do numerical
Palavras-chave: Fluidodynamics, Plasma and Turbulence Modeling, Numerical Simulation and Optimization