Investigation of the Conservative System of Equations for a Vertically Flowing Liquid Film

A new system of equations for modeling the dynamics of long-wave perturbations on the surface of a viscous liquid film flowing down a vertical plane was investigated. It is shown that the system obtained agrees with a known system derived by the variable changing method but unlikely has a conservative form that is perfect for designing of numerical efficient conservative difference schemes. The system is reduced to a single conservative equation for the function analogous to the hydrodynamic stream function with the appropriate boundary conditions. For the case of moderate Reynolds number it is noted that the equation with boundary conditions coincide with the well known Shkadov’s model under the assumption of self-similar profile of longitudinal velocity. At small Reynolds numbers typical to the condition of microgravity it was proved that the system is reduces to one evolution equation for the film thickness.