In fluid dynamics, Landau–Squire jet or Submerged Landau jet describes a round submerged jet issued from a point source of momentum into an infinite fluid medium of the same kind. This is an exact solution to the Navier-Stokes equations, which was first discovered by Lev Landau in 1944 and later by Herbert Squire in 1951. The self-similar equation was in fact first derived by N. A. Slezkin in 1934, but never applied to the jet. Following Landau's work, V. I. Yatseyev obtained the general solution of the equation in 1950.
Mathematical description
The problem is described in spherical coordinates with velocity components. The flow is axisymmetric, i.e., independent of. Then the continuity equation and the incompressible Navier–Stokes equations reduce to where A self-similar description is available for the solution in the following form, Substituting the above self-similar form into the governing equations and using the boundary conditions at infinity, one finds the form for pressure as where is a constant. Using this pressure, we find again from the momentum equation, Replacing by as independent variable, the velocities become and the equation becomes After two integrations, the equation reduces to where and are constants of integration. The above equation is a Riccati equation. After some calculation, the general solution can be shown to be where are constants. The physically relevant solution to the jet corresponds to the case . Therefore, The function is related to the stream function as, thus contours of for different values of provides the streamlines. The constant describes the force at the origin acting in the direction of the jet, the exact relation between the force and the constant is given by The solution describes a jet of fluid moving away from the origin rapidly and entraining the slowly moving fluid outside of the jet. The edge of the jet can be defined as the location where the streamlines are at minimum distance from the axis, i.e.,e the edge is given by Therefore, the force can be expressed alternatively using this semi-angle of the conical-boundary of the jet, When the force becomes large, the semi-angle of the jet becomes small, in which case, and the solution inside and outside of the jet become The jet in this limiting case is called the Schlichting jet. On the other extreme, when the force is small, the semi-angle approaches 90 degree, the solution itself goes to
