Simulation of the flow of a two-layer liquid film on a cylindrical surface

Abstract

The problem of a stationary joint flow of a two-layer liquid film and gas along the outer (or inner) surface of a circular cylinder of radius r0 is considered. It is assumed that the films are insoluble in one another, and there are no chemical reactions. The axis of the body is located vertically, and the films flow down from its top. The film is affected by gravity, as well as a gas stream directed upwards or downwards. A cylindrical coordinate system (r, θ, z) is introduced: the z coordinate is measured along the axis of the cylinder, r and θ are the polar coordinates in a plane perpendicular to the axis of the body. To describe the flow of a liquid film, a viscous incompressible fluid model is used, which is based on the equations of continuity and Navier-Stokes. The following boundary conditions are set on the interface surfaces: on the solid surface - draw off “sticking”, on the “liquid-liquid” and “liquid-gas” interfacial surfaces - the conditions of equilibrium of forces and continuity of speeds. To simplify these differential equations, the method of a small parameter, for which the relative thickness of the films is selected, is applied. Solutions of simplified equations (in a zero approximation) are obtained in analytical form. Functional dependences are obtained for calculating the optimal effect of the gas flow on the "working" film. In accordance with the described method, calculations of the flow of a two-layer film on the outer and inner surfaces of the circular cylinder are performed in cases where the gas stream is directed upwards, downwards , and also when the gas flow is absent. An increase in the relative thicknesses of the films δ1 and δ2 (with decreasing radius of the cylinder r0) leads to an increase in deviations from the case of a plane surface that corresponds to the limiting case δ1 = δ2 = 0. The results of calculations of the flow of a two-layer liquid film on the surface of a circular cylinder are presented. The analysis of the influence of physical parameters on the speed profiles is carried out. The results of calculations for determining the optimal effect of gas flow on a liquid film are presented, when the profile of the speed of the "working" film is the most uniform