Viscous Flow of a Liquid Over a Rotating Surface With Gas Drag

Abstract

Differential equations are set up to describe the viscous flow of a liquid on a rotating surface with a shear stress exerted by a flowing gas stream along the surface and normal to the radial direction. Such a problem is met in the low-pressure stages of a steam turbine. The equations are solved first for a uniform flux of liquid onto the surface along a radial boundary R > R0 , where R is the radial position and R0 is constant. It is then shown that absence of liquid flowing in along the radial boundary for R < R0 casts what is called a “flow shadow” and that two solutions of the equations are required, one for the “shadow flow” and one for the “main flow” region. The solutions are extended to the case where the shear stress exerted by the gas varies in a manner described by a simple power function of the distance along the surface, x, normal to the radial direction. All these solutions can be represented on a single graph showing streamlines and the shadow boundary. In conclusion, it is demonstrated that the two solutions for the main and shadow flows are special cases of a more general solution in which flow in along the radial boundary at x = 0 varies as 1 − R0R1/32