Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation

Abstract

We present in detail a theoretical framework for representing hydrodynamic systems through a systematic discretization of the Boltzmann kinetic equation. The work is an extension of a previously proposed formulation. Conventional lattice Boltzmann models can be shown to be directly derivable from this systematic approach. Furthermore, we provide here a clear and rigorous procedure for obtaining higher-order approximations to the continuum Boltzmann equation. The resulting macroscopic moment equations at each level of the systematic discretization give rise to the Navier–Stokes hydrodynamics and those beyond. In addition, theoretical indications to the order of accuracy requirements are given for each discrete approximation, for thermohydrodynamic systems, and for fluid systems involving long-range interactions. All these are important for complex and micro-scale flows and are missing in the conventional Navier–Stokes order descriptions. The resulting discrete Boltzmann models are based on a kinetic representation of the fluid dynamics, hence the drawbacks in conventional higher-order hydrodynamic formulations can be avoided.

Predicting different adhesive regimens of circulating particles at blood capillary walls

“…Here, we adopt a discretization in the velocity space of the MB distribution based on the Hermite polynomial expansion of this distribution (Shan et al 2006). An effective forcing term accounting for the boundary presence,  i , can be included as an additional factor on the righthand side of Eq.…”

Adaptive kinetic-fluid solvers for heterogeneous computing architectures

“…Later, it has been extended beyond the level of the Navier Stokes hydrodynamics, and capable to describe some kinetic effects. 34 LBM operates with a Velocity Distribution Function (VDF) defined on a minimal set of discrete velocities to obtain governing equations for the fluid dynamics alternative to conservation equations based on VDF moments. Most LBM works are devoted to low speed isothermal flows close to equilibrium.…”

Thermodynamic consistency in deriving lattice Boltzmann models for describing segregation in non-ideal mixtures

“…In fact, in its early development, the LBM was considered to be applicable only to flows with small Mach number and constant temperature. A key breakthrough in this area was the identification of the correspondence between the LBE and the kinetic continuous equation, and the invention of highorder LB schemes [16][17][18], leading to velocity sets that, when used in a discrete velocity kinetic scheme, ensure accurate recovery of the high-order hydrodynamic moments.…”

“…In this attempt, the thermodynamic consistency of the derived kinetic equations is thoroughly investigated in this paper. Owing to space limitations, only the kinetic equation in continuous space is presented in this paper, the LBEs themselves being considered as discrete forms of the continuous kinetic model, after using an appropriate velocity discretizaton scheme [16][17][18]. In this respect, it is important to distinguish errors due to the kinetic model used for non-isothermal segregation from errors due to the discretization procedure.…”