Analysis of thin liquid film flow using new evolution equations based on an integral boundary layer approach

Abstract

New evolution equations are derived, based on application of the K??rm??n-Polhausen integral method and the imposition of a velocity profile accounting for inertial effects, rather than the usual parabolic profile, for the analysis of thin liquid film flows with/without condensation. First, the validity and accuracy of the equations without condensation are demonstrated via both a linear and nonlinear stability analysis for the case of flow down an inclined plane, and the solutions obtained are seen to be an improvement over earlier results. In addition, the evolution equations are used to model the case of a thin liquid film flowing down an inclined plane containing a well defined trench topography, with the predicted film profile for Reynolds number greater than zero shown to be in excellent agreement with a recent corresponding finite element solution of the full Navier-Stokes equations taken from the literature. Using a similar procedure, a new form of the condensation rate is derived from the energy equation using a temperature profile accounting for energy convection effects. From this condensation rate as well as the evolution equations including condensation effect, convenient formulae for the film thickness and the heat transfer rate as a function of Jacob and Prandtl numbers are derived for the laminar condensate film condensation on a vertical plate. The results are in good agreement with the previous similarity solutions of the boundary layer equations. The validity and accuracy of the present approach is also demonstrated via the linear stability analysis of vertical condensate film flow. Finally, the evolution equations are used to model the condensate film flow over topography. The result is seen to follow the analytic solution derived in the present analysis when the inflow and outflow boundaries are approached and has a wave profile similar to that of an isothermal flow case near the topographic region.