Thermodynamic Theory of Incompressible Hydrodynamics

ETH-Zürich, Institute of Energy Technology, CH-8092 Zürich, Switzerland.
Physical Review Letters (Impact Factor: 7.51). 04/2005; 94(8):080602. DOI: 10.1103/PhysRevLett.94.080602
Source: PubMed


The grand potential for open systems describes thermodynamics of fluid flows at low Mach numbers. A new system of reduced equations for the grand potential and the fluid momentum is derived from the compressible Navier-Stokes equations. The incompressible Navier-Stokes equations are the quasistationary solution to the new system. It is argued that the grand canonical ensemble is the unifying concept for the derivation of models and numerical methods for incompressible fluids, illustrated here with a simulation of a minimal Boltzmann model in a microflow setup.

Download full-text


Available from: Iliya V. Karlin
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Concepts of the lattice Boltzmann method are discussed in detail for the one-dimensional kinetic model. Various techniques of constructing lattice Boltzmann models are discussed, and novel collision integrals are derived. Geometry of the ki- netic space and the role of the thermodynamic projector is elucidated.
    Full-text · Article · Apr 2007 · Communications in Computational Physics
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In flows through microdevices the continuum fluid mechanics description often breaks down and higher order corrections to the Navier-Stokes description arise both at the boundaries and in the bulk. The interaction between the flow geometry, rarefaction and compressibility is not completely understood for such flows. Recent advances in compu- tational kinetic theory, such as the entropic lattice Boltzmann method, provide a simple and realistic framework which enable the systematic study of such interactions. We consider a specific example of entropic lattice Boltzmann model and compare it with Grad's moment system. We show that for the model under consideration, the dispersion relation is closely related to that of Grad's ten-moment system. We perform a parametric study of the flow in a microcavity, which is a prototype problem, where the deviations from incompressible hydrodynamics can be studied conveniently. Simulation results obtained with the entropic lattice Boltzmann method are compared with those of the Direct Simulation Monte-Carlo method. Based on the parametric study, we discuss aspects of the interaction between rarefaction and compressibility.
    Full-text · Article ·
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A new method for the computation of flows at the micrometer scale is presented. It is based on the recently introduced minimal entropic kinetic models. Both the thermal and isothermal families of minimal models are presented, and the simplest isothermal entropic lattice Bhatnagar–Gross–Krook (ELBGK) is studied in detail in order to quantify its relevance for microflow simulations. ELBGK is equipped with boundary conditions which are derived from molecular models (diffusive wall). A map of three-dimensional kinetic equations onto two-dimensional models is established which enables two-dimensional simulations of quasi-two-dimensional flows. The ELBGK model is studied extensively in the simulation of the two-dimensional Poiseuille channel flow. Results are compared to known analytical and numerical studies of this flow in the setting of the Bhatnagar–Gross–Krook model. The ELBGK is in quantitative agreement with analytical results in the domain of weak rarefaction (characterized by Knudsen number Kn, the ratio of mean free path to the hydrodynamic scale), up to Kn∼0.01, which is the domain of many practical microflows. Moreover, the results qualitatively agree throughout the entire Knudsen number range, demonstrating Knudsen's minimum for the mass flow rate at moderate values of Kn, as well as the logarithmic scaling at large Kn. The present results indicate that ELBM can complement or even replace computationally expensive microscopic simulation techniques such as kinetic Monte Carlo and/or molecular dynamics for low Mach and low Knudsen number hydrodynamics pertinent to microflows.
    Full-text · Article · Jan 2005 · Physica A: Statistical Mechanics and its Applications
Show more