3:00 pm Friday, October 10, 2014
Math - ICES Center of Numerical Analysis Seminar: Boltzmann equation and hydrodynamics at the Burnett level by Alexander Bobylev (Keldysh Institute of Applied Mathematics, Moscow) in POB 6.304
We present a review of some results on Burnett-type hydrodynamic equations derived from the Boltzmann equation. The well-known problem here is connected with regularization of classical (ill-posed) Burnett equations[1-5]. There are several ways to deal with this problem. We discuss in detail one of the approaches, proposed in [1] and further developed in [2-4]. Our approach is based on infinitesimal changes of variables, it shows that the way of truncation of the Chapman-Enskog series is not unique. It is the only approach which does not use any information beyond the classical Burnett equations. We show how to derive a two-parameter family of stable Generalized Burnett Equations (GBEs) [2] and discuss the optimal choice of the parameters. Surprisingly the resulting well-posed equations are simpler than the original Burnett equations. The equations are derived for arbitrary intermolecular forces. Some special properties of (a) stationary problems and (b) linear non-stationary problems are discussed in more detail. Finally we present some recent results on the shock-wave structure [4], which show that GBEs yield certain improvement of the Navier-Stokes results for moderate Mach numbers. Some open questions are also discussed. References: [1] A.V.Bobylev, Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations, J.Stat.Phys. 124, 371 (2006). [2] A.V. Bobylev, Generalized Burnett hydrodynamics, J.Stat.Phys. 132, 569 (2008). [3] M.Bisi, M.P.Cassinari and M.Groppi, Qualitative analysis of the Generalized Burnett Equations and applications to half-space problems, Kinet. Relat. Models 1, 295 (2008). [4] A.V.Bobylev, M.Bisi, M.P.Cassinari and G.Spiga, Shock wave structure for generalized Burnett equations, Phys. of Fluids 23, 1 (2011). [5] A.V.Bobylev and A.Windfall, Boltzmann equation and hydrodynamics at the Burnett level, Kinet. Relat. Models 5, No.2 (2012). Submitted by
|
|