NavierStokes Hamiltonian for the Similarity Renormalization Group
Abstract
The NavierStokes Hamiltonian is derived from first principles. Its Hamilton equations are shown to be equivalent to the continuity, NavierStokes, and energy conservation equations of a compressible viscous fluid. The derivations of the Euler and NavierStokes Hamiltonians are compared, with the former having identical dynamics to the Euler equation with the viscosity terms of the NavierStokes equation dropped from the beginning. The two Hamiltonians have the same number of degrees of freedom in three spatial and one temporal dimension: six independent scalar potentials, but their dynamical fields are necessarily different due to a theory with dissipation not mapping smoothly onto one without. The dynamical coordinate field of a dissipative fluid is a vector field that stores the initial position of all of its fluid particles. Thus these appear to be natural coordinates for studying arbitrary separations of fluid particles over time. The classical similarity renormalization group is introduced and the first steps are carried out deriving a flow equation for the NavierStokes Hamiltonian. Finally, the symmetries of a nonrelativistic viscous fluid are discussed through its galilean algebra with dissipative canonical Poisson brackets.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.1035
 Bibcode:
 2014arXiv1407.1035J
 Keywords:

 Physics  Fluid Dynamics
 EPrint:
 51 pages, 1 figure. v4 changes: new title plus minor polishing edits. v3 changes: 1. Added two appendices. 2. Went back to original A(x,t) notation because everyone seems to use that notation, so it should be clearer