Thermodynamic theory of incompressible hydrodynamics

ETH-Zürich, Institute of Energy Technology, CH-8092 Zürich, Switzerland.
Physical Review Letters (Impact Factor: 7.73). 04/2005; 94(8):080602. DOI: 10.1103/PhysRevLett.94.080602
Source: arXiv

ABSTRACT 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.

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    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.
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    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.