A BGK model for small Prandtl number in the Navier-Stokes approximation

1993 ◽  
Vol 71 (1-2) ◽  
pp. 191-207 ◽  
Author(s):  
Fran�ois Bouchut ◽  
Beno�t Perthame
Keyword(s):  
Prandtl Number ◽  
Navier Stokes ◽  
Bgk Model ◽  
Stokes Approximation

scholarly journals Hydrodynamic limits of kinetic equations for polyatomic and reactive gases

2017 ◽  
Vol 8 (1) ◽  
pp. 23-42 ◽  
Author(s):  
M. Bisi ◽  
G. Spiga
Keyword(s):  
Degrees Of Freedom ◽  
Continuum Limit ◽  
Fluid Dynamic ◽  
Energy Levels ◽  
Kinetic Equations ◽  
Navier Stokes ◽  
Bgk Model ◽  
Hydrodynamic Limits ◽  
Stress Diffusion ◽  
Reactive Gases

Abstract Starting from a kinetic BGK-model for a rarefied polyatomic gas, based on a molecular structure of discrete internal energy levels, an asymptotic Chapman-Enskog procedure is developed in the asymptotic continuum limit in order to derive consistent fluid-dynamic equations for macroscopic fields at Navier-Stokes level. In this way, the model allows to treat the gas as a mixture of mono-atomic species. Explicit expressions are given not only for dynamical pressure, but also for shear stress, diffusion velocities, and heat flux. The analysis is shown to deal properly also with a mixture of reactive gases, endowed for simplicity with translational degrees of freedom only, in which frame analogous results can be achieved.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

ON THE ASYMPTOTIC ANALYSIS OF THE BGK MODEL TOWARD THE INCOMPRESSIBLE LINEAR NAVIER–STOKES EQUATION

2010 ◽  
Vol 20 (08) ◽  
pp. 1299-1318 ◽  
Author(s):  
A. BELLOUQUID
Keyword(s):  
Asymptotic Analysis ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Uniqueness Theorems ◽  
Bgk Model ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Systems ◽  
Global Strong Solutions

This paper deals with the analysis of the asymptotic limit for BGK model to the linearized Navier–Stokes equations when the Knudsen number ε tends to zero. The uniform (in ε) existence of global strong solutions and uniqueness theorems are proved for regular initial fluctuations. As ε tends to zero, the solution of BGK model converges strongly to the solution of the linearized Navier–Stokes systems. The validity of the BGK model is critically analyzed.

Initial Stage of Natural Convection Over a Hot Aerosol Sphere

2006 ◽  
Vol 129 (6) ◽  
pp. 695-701
Author(s):  
Elaad Mograbi ◽  
Ezra Bar-Ziv
Keyword(s):  
Prandtl Number ◽  
Linear Process ◽  
Navier Stokes ◽  
Transient Temperature ◽  
Square Root ◽  
Analytical Treatment ◽  
Initial Development ◽  
Initial Flow ◽  
Linear Solution ◽  
Initial Stage

Background: Analytical study is presented on the transient problem of buoyancy-induced motion due to the presence of a hot aerosol sphere in unbounded quiescent fluid. Method of Approach: Because the initial flow field is identically zero, the initial stage of the process is governed by viscous and buoyancy forces alone where the convective inertial terms in the momentum and energy balances are negligible, i.e., the initial development of the field is a linear process. The previous statement is examined by analyzing the scales of the various terms in the Navier-Stokes and energy equations. This scale analysis gives qualitative limitations on the validity of the linear approximation. A formal integral solution is obtained for arbitrary Prandtl number and for transient temperature field. Results: We consider, in detail, the idealized case of vanishing Prandtl number for which the thermal field is developed much faster than momentum. In this case, analytical treatment is feasible and explicit expressions for the field variables and the drag acting on the particle are derived. Detailed quantitative analysis of the spatial and temporal validity of the solution is also presented. Conclusions: The linear solution is valid throughout space for t<10 diffusion times. For t>10, an island in space appears in which inertial effects become dominant. The transient process is characterized by two different time scales: for short times, the development of the field is linear, while for small distances from the sphere and finite times, it is proportional to the square root of time. The resultant drag force acting on the sphere is proportional to the square root of time throughout the process.

scholarly journals The Gaussian-BGK model of Boltzmann equation with small Prandtl number

2000 ◽  
Vol 19 (6) ◽  
pp. 813-830 ◽  
Author(s):  
Pierre Andries ◽  
Patrick Le Tallec ◽  
Jean-Philippe Perlat ◽  
Benoı̂t Perthame
Keyword(s):  
Boltzmann Equation ◽  
Prandtl Number ◽  
Bgk Model
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