The motions of liquid films in a gas

The motions in a gas of thin films of a viscous incompressible liquid acted upon by capillary forces are considered. The surface tension depends on the impurity concentration of a surface-active material, soluble or insoluble in the liquid, and the liquid is non-volatile. The inertia of the liquid, viscous stresses, the Laplace pressure and the surface-tension gradients, impurity transfer and also the particular properties of super-thin films are taken into account. The motions of the films are described using the model of quasi-steady viscous flow. Systems of equations are obtained in the approximation of an ideal compressible medium and for small Mach numbers. The conditions for the incompressible film surface approximation to hold are obtained. The severe limitations of the gas-dynamic approximation in the case of a soluble impurity due to attenuation of the waves related to diffusion are investigated. A continuum model of the film as a compressible medium with a non-equilibrium pressure is constructed. The asymptotic form of the solutions of unsteady problems of impurity transfer in the limit of weak non-equilibrium is obtained. Integrals of the equations of motion of the films in steady one-dimensional problems are derived. Integral forms of the equations of momentum and its moment for an arbitrary contour of the film are presented, which hold for steady flows in a film and in quasi-statics. The boundary conditions for the solutions of the system of equations of motion of films are given.