Variational principles and stability of stationary flows of barotropic ideal fluid
Abstract
Some wellknown results of Arnold are generalized and it is proved that the stationary threedimensional flow of an ideal (inviscid) barotropic fluid yields an extremum of the total mechanical energy with respect to variations of the hydrodynamic fields that possess the same vorticity. In the twodimensional case some functionals are constructed using integrals of the energy and vortex conservation laws. These functionals are also integrals of the equations of motion and their extremum corresponds to the stationary flows under consideration. The formulae for the second variations of these functionals are then derived, and in this way some sufficient conditions for the stability of the corresponding stationary flows are found. In particular, it is proved that plane stationary flows of a barotropic ideal fluid are stable in the Lyapunov sense if two conditions hold: the first one is a natural generalization of the RayleighArnold criterion, the latter a requirement of the subsonic character of the flow.
 Publication:

Geophysical and Astrophysical Fluid Dynamics
 Pub Date:
 February 1984
 DOI:
 10.1080/03091928408210133
 Bibcode:
 1984GApFD..28...31G
 Keywords:

 Barotropic Flow;
 Flow Stability;
 Ideal Fluids;
 Inviscid Flow;
 Steady Flow;
 Three Dimensional Flow;
 Variational Principles;
 Equations Of Motion;
 Hydrodynamic Equations;
 Liapunov Functions;
 Two Dimensional Flow;
 Vortices