Incompressible limit and non isentropic fluids

Didier Bresch

Description

Incompressible limit and non-isentropic fluids Didier Bresch Laboratoire de Mathematiques, UMR5127 CNRS, Universite de Savoie 73000 Le Bourget du lac cedex, FRANCE Joint works with : B. DESJARDINS (Momagroup, ENS Paris), E. GRENIER (ENSL). Oscillatory limits with changing eigenvalues, in preparation (2008). Keywords: Transversality, crossing eigenvalues, resonant set, oscillatory limit, singular flows, low mach (low Froude) number limits. March 2008 CIRM, 2008 – p.1/3

• field small


• compressible isentropic


• sound velocity


• anisotropic compressible


• incompressible limit


• velocity motions


• rotating fluids


• euler equations

Sujets

• Speed of sound
• List of topics named after Leonhard Euler

Informations

• Publié par: pefav
• Nombre de lectures: 13
• Langue: English

Extrait

Incompressible limit and non-isentropic uids

Didier Bresch

LaboratoiredeMath ´ ematiques,UMR5127CNRS,Universit ´ edeSavoie 73000 Le Bourget du lac cedex, FRANCE [e-mail]

Joint works with : B. D ESJARDINS (Momagroup, ENS Paris), E. G RENIER (ENSL).

Oscillatory limits with changing eigenvalues, in preparation (2008).

Keywords: Transversality, crossing eigenvalues, resonant set, oscillatory limit, singular ows, low mach (low Froude) number limits.

March 2008

ICRM,0280–p.1/3

Introduction

Generalities:Incompressible ows equations justication:

From compressible Navier–Stokes or Euler equations (shallow-water system) Flow velocity eld small compared to sound velocity

Limit = incompressible equations. Correction = acoustic waves, gravity waves.....

Small parameter = Mach number, Froude number For instanceε= Mach = uid velocity / sound velocity

Car: 50 km/h / 120 km/h = 1/20 Plane: 800 km/h / 1200 km /h = 0.66

velocity motions<150 km are essentially incompressible

Difference = Noise (waves..)

ICRM,0280–.p/23

Introduction

Compressible isentropic Euler equations:

Scaling:

gives

Then limitε→0provides

∂ t ρ+ div(ρu) = 0

∂ t (ρu) + div(ρu⊗u) +∇p(ρ) = 0

u(t x) =εU(ε t x)

∂ t ρ+ div(ρU) = 0

∂ t (ρU) + div(ρU⊗U) + ∇p( 2 ρ) =0 ε

∇p(ρ) = 0

Thus, using the mass equation,ρis a constantρ= 1=⇒divergence free condition divU= 0(Udenoteduin the sequel).

CIRM,0208–.p/23