Abstract
Incompressible, inviscid, free boundary fluid flows are Hamiltonian systems, i.e., the evolution of the flow is determined by a Poisson bracket and a Hamiltonian function. The Poisson structure for such flows generalizes a previous structure of Zakharov for irrotational free boundary flow and a structure of Arnold for flow of an incompressible fluid in a region with fixed boundary. The Poisson bracket is determined by reduction from canonical variables in the Lagrangian (material) description.
The Hamiltonian structure of the flow is used in the “Energy-Casimir” method to demonstrate formal stability of a rigidly rotating circular drop. That method involves the determination of an appropriate combination of the Hamiltonian and conserved quantities associated to the symmetries of the system. This combination is chosen such as to make the equilibrium in question a critical point of the combined function whose second variation is then tested for positive (or negative) definiteness at that point.
One of our main results is that a formally stable relative equilibrium of the equations of motion of a planar drop with a free boundary with surface tension and self-gravitation or charge is (conditionally) nonlinearly stable. Here conditional simply means that the result is conditional on the assumption of existence of solutions in the function spaces we construct for the stability analysis. The proof of this stability theorem is based on techniques from the classical calculus of varialions. A careful Taylor series analysis shows that an equilibrium with positive-definite second variation is a strong minimum of the functional. This conclusion about the minimum, which is applicable to a large class of variational problems, is used in this paper to obtain bounds on the variation of the Hamiltonian for the rotating liquid drop. The existence of these bounds implies that the equilibrium is conditionally nonlinearly stable.
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Lewis, D. Nonlinear stability of a rotating planar liquid drop. Arch. Rational Mech. Anal. 106, 287–333 (1989). https://doi.org/10.1007/BF00281351
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DOI: https://doi.org/10.1007/BF00281351