Incompressible Fluid Flows with Free Boundary and the Methods for their Research

Abstract

Thomson W. and Tait P. were the first who applied the Hamilton principle to the fluid [1,2]

$$\int\limits_{{{{t}_{1}}}}^{{{{t}_{2}}}} \delta Ldt = \int\limits_{{{{t}_{1}}}}^{{{{t}_{2}}}} {dt} \int\limits_{{\partial \Omega }} {p{{\delta }_{n}}} \bullet ,dS, {{\delta }_{n}}({{t}_{1}}) = {{\delta }_{n}}({{t}_{2}}) = 0, $$

(1.1)

Where δ_n is the virtual displacement of fluid flow boundary ∂Ω, the integral represents the work of the force of fluid pressure p on the displacement δ_n, L is the functional of the boundary and the normal velocity (the Lagrange function).