Mathematics & Physics 2015, 8(1), 3–6 УДК 532.51 The 2D Motion of Perfect Fluid with a Free Surface Victor K.Andreev∗ Institute of Computational Modelling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.10.2014, received in revised form 10.11.2014, accepted 20.12.2014 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. <...> Governing flow equations and main results The Euler equations for 2D motions of a perfect fluid are recorded by ut +uux +vuy + 1 ρ px = 0, ux +vy = 0, vt +uvx +vvy + 1 ρ py = 0, (1) where 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. <...> Let us introduce the Lagrangian coordinates (η, t) by the solving Cauchy problem dy dt = v(y, t), y ∗andr@icm.krasn.ru Siberian Federal University. <...> So, all unknowns can be determined in analytical form. <...> Now we show that this solution can be interpreted as an unsteady motion in a strip with one free boundary, see Fig. 1. (5) (6) Fig. 1 Geometry of the motion rigid wall. <...> Initial velocity field has the form u0(x, y) = u10(y)x+u20(y), v0(y) = − Really, at the initial time liquid fills the strip of thickness y = h0 = const. <...> The upper line y = h0 is a free boundary and at the initial time the pressure p(h0, 0) coincides with outer pressure pout = p10 + p00x2/2. <...> For all t > 0 the strip motion is described – 4 – Victor <...>