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Hamiltonian Dynamic Equations for Fluid Films
Author(s) -
Grinfeld P.
Publication year - 2010
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2010.00485.x
Subject(s) - surface tension , material derivative , euler equations , hamiltonian (control theory) , laplace transform , mathematical analysis , functional derivative , covariant derivative , nonlinear system , curvature , classical mechanics , hamiltonian system , covariant transformation , mathematics , physics , mathematical physics , geometry , thermodynamics , quantum mechanics , mathematical optimization
Two‐dimensional models for hydrodynamic systems, such as soap films, have been studied for over two centuries. Yet there has not existed a fully nonlinear system of dynamic equations analogous to the classical Euler equations. We propose the following exact system for the dynamics of a fluid filmHere δ/δ t is the invariant time derivative, ρ is the two‐dimensional density of the film, C is the normal component of the velocity field, V α are the tangential components, B αβ is the curvature tensor, and ∇ α is the covariant surface derivative. The surface energy density e (ρ) is a generalization of the common surface tension and e ρ is its derivative. The Laplace model corresponds to e (ρ) =σ/ρ , where σ is the surface tension density. The proper choice of e (ρ) in paramount in capturing particular effects displayed by fluid films. The proposed system is exact in the sense that neither velocities nor deviation from the equilibrium are assumed small. The system is derived in the classical Hamiltonian framework. The assumption that e is a function of ρ alone can be relaxed in practical physical and biological applications. This leads to more complicated systems, briefly discussed in the text.