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One method to determine the solution values at the boundary for the vertical two dimensional equation system

In this paper, the boundary conditions for modified Navier-Stokes equations system are presented, and the complementary equations on the boundaries are established. | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 3 (24 - 36) ONE METHOD TO DETERMINE THE SOLUTION VALUES AT THE BOUNDARY FOR THE VERTICAL TWO-DIMENSIONAL EQUATION SYSTEM TRAN G IA Lrcm Institute of Mathematics, Hanoi, Vietnam ABSTRACT. In this paper, the boundary conditions for modified Navier-Stokes equations system are presented, and the complementary equations on the boundaries are established. Keywords: Partial differential equation, the equation of mathematical physics, linear algebra. 1. Navier-Stokes equation system The N a vier-Stokes equation system describing vertical two-dimensional unsteady flow for viscous incompressible fluid has the following form (see [1]-[5]) au au au .1 ap = vf.u, at ax az pax aw aw aw 1 ap - + u - +w- + - - = vf.w, at ax az paz au aw ax+ az = o, ·- + u-·- + w- + - - (1.1) where-x, z are coordinate axes, (u,w) is the velocity vector, p- the pressure, p0 the density, v - the kinematic viscosity, t. = + 02 • 2 ax oz It is well known that, the equation system (1.1) with the initial condition U1 (x,z,O) = Uf(x,z), where 24 and the boundary conditions on the boundary aG of the region in consideration G: .U1 (x,z,t)j 8G = 0 has the unique solution in the space of generalised functions (see [1], [2)}. The determination of pressure p from equation (1.1} is very difficult. To avoid it, the artificial compression component is added to the continuity equation of (1.1). Then we obtain the following equation system (see [3), [4)}. (Let us take p = 1) au au au· ap -at +u+wax az +ax = vl:l.u, aw aw aw ap -at + uaz- + waz +az = vl:l.w, ap au aw- 0 at+ ax+ az- . (1.2} In actual fact, it is difficult to give all 2 boundary conditions at any part of the boundary. To overcome this insufficiency of given boundary conditions, at the neighborhood of the part of boundary, where all the 2 boundary conditions can not be given, we consider the following modified Navier-Stokes equation system for determining the solution