2009/02/28 by Stephen C. Anco, Amanullah Dar
Engineering · Mathematics · Physics and Astronomy · #Barotropic fluid #Classical mechanics #Compressibility #Computational Fluid Dynamics and Aerodynamics #Conservation law #Conservative vector field #Conserved quantity #Equation of state #Euler equations #Eulerian path #Lagrangian and Eulerian specification of the flow field #Mathematical analysis #Mathematical physics #Mathematics #Mechanics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #Physics #Polytropic process #Quantum mechanics #Vortex #Vorticity #math-ph #math.MP #physics.flu-dyn
paper · pdf · doi:10.1098/rspa.2009.0072
published as Proc. Roy. Soc. A 465 (2009), 2461-2488 · 24 pages; published version with misprints corrected
openalex publication_date 2009/06/01 · arxiv created 2009/11/03 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
For the Euler equations governing compressible isentropic fluid flow with a barotropic equation of state (where pressure is a function only of the density), local conservation laws in n >1 spatial dimensions are fully classified in two primary cases of physical and analytical interest: (i) kinematic conserved densities that depend only on the fluid density and velocity, in addition to the time and space coordinates, and (ii) vorticity conserved densities that have an essential dependence on the curl of the fluid velocity. A main result of the classification in the kinematic case is that the only equation of state found to be distinguished by admitting extra n -dimensional conserved integrals, apart from mass, momentum, energy, angular momentum and Galilean momentum (which are admitted for all equations of state), is the well-known polytropic equation of state with a dimension-dependent exponent, γ=1+2/ n . In the vorticity case, no distinguished equations of state are found to arise, and here the main result of the classification is that, in all even dimensions n ≥2, a generalized version of Kelvin’s two-dimensional circulation theorem is obtained for a general equation of state.