Mathematics & Physics 2015, 8(2), 140–147 УДК 532.51 Unsteady 2DMotions a Viscous Fluid Described by Partially Invariant Solutions to the Navier–Stokes Equations Victor K.Andreev∗ Institute of Computational Modelling RAS SB Akademgorodok, 50/44, Krasnoyarsk, 660036 Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.02.2015, received in revised form 03.03.2015, accepted 30.03.2015 3D continuous subalgebra is used to searching partially invariant solution of viscous incompressible fluid equations. <...> It can be interpreted as a 2D motion of one or two immiscible fluids in plane channel. <...> The arising initial boundary value problem for factor-system is an inverse one. <...> Unsteady problem for creeping motions is solved by separating of variables method for one fluid or Laplace transformation method for two fluids. <...> Introduction The Navier–Stokes equations for 2D motions of a viscous fluid are recorded by ut +uux +vuy + 1 ρ px = ν(uxx +uyy), vt +uvx +vvy + 1 ρ py = ν(vxx +vyy)−g, ux +vy = 0, 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 and g is the gravity acceleration, ν is the fluid viscosity. <...> The group of point transformations admitted by the system (0.1) is computed in [1, 2]. <...> Corresponding this group basic continuous Lie algebra includes the three parametrical subalgebra ∂x, ∂u+t∂x, ∂p. <...> Flow in layer with two rigid walls In this section the solution (0.2) under consideration shall be interpreted as 2Dmotion viscous liquid fills the layer 0 < y < h with a rigid walls y = 0, y = h = const. <...> Thus, the function w(y, t) is the solution of integro-differential equation wt −wy w(z, t) dz +w2 = νwyy +f(t) with initial and boundary conditions (1.1), (1.2). <...> Hence, we deduce the so-called loaded equation wt = νwyy +wy(1, t)−wy(0, t). <...> Flow in layer with one rigid wall and free boundary In the same assumptions like section 1 the function w <...>