The 2D Motion of Perfect Fluid with a Free Surface

Abstract

The 3D continuous subalgebra is used to searching new partially invariant solution of incompressible perfect fluid equations. It can be interpreted as a non-stationary motion of a plane layer with one free surface. The velocity field and pressure are determined in analytical form by using Lagrangian coordinates.Keywords: perfect fluid, partially invariant solution, non-stationary motion, free surfaces. Governing flow equations and main resultsThe Euler equations for 2D motions of a perfect fluid are recorded bywhere ρ is the constant fluid density, u and v are the velocity components in the x and y directions, respectively, p is the pressure. The group of point transformations admitted by the system (1) is computed in [1]. Corresponding this group basic continuous Lie algebra includes the three parametrical subalgebra ∂ x , t∂ u + ∂ x , ∂ p . It has the invariants t, y, v and partly invariant solution of (1) rang two and defect two necessary to seek in the form u = u(x, y, t), v = v(y, t), p = p(x, y, t). From continuity equation u x + v y = 0 we obtain the relationsImpulse equations (1) are equivalent to the followingwith arbitrary function f (t). Let us introduce the Lagrangian coordinates (η, t) by the solving Cauchy problem dy dt = v(y, t), y t=0 = η. (4) * andr@icm.krasn.ru c Siberian Federal University. All rights reserved -3 -