Stability and rupture of liquid film flowing down an inclined plane - Atieh, Muataz Ali (2001)
Liquid film flowing down inclined or vertical planes find applications in thin film heat and mass transfer, wetted wall columns, liquid drainage in packed columns, surface coating operations, and the like. The film is modeled as a two-dimensional Newtonian liquid of constant density p and viscosity u flowing down an inclined plane. The liquid film of mean thickness ho is bounded above by a passive gas and laterally extends to infinity (two-dimensional model). Then such a flow can be represented by a two-dimensional Navier-Stokes equation coupled with continuity equation and associated boundary conditions. The body force term in the Navier-Stokes equation is modified by the inclusion of excess intermolecular interactions between fluid film and the solid surface owing to long-range van der Waals force, in addition to gravity force. The modified Navier-Stokes equation with associated boundary conditions is solved under long wave approximation method to obtain a nonlinear equation of evolution of the film interface. A nonlinear theory based upon the condition of infinitesimal perturbation on the film surface is derived to obtain the growth coefficient, dominant wavelength (i.e., wavelength corresponding to maximum growth coefficient of the surface instability) and the film rupture time. The nonlinear equation of evolution is solved numerically in conservative form as part of an initial-value problem for spatially periodic boundary condition on the fixed domain 0< x< 21t/k, where k is a wavenumber. Centered difference in space and the midpoint (Crank-Nicholson) rule in time are employed. The mesh size is taken sufficiently small so that space and time errors are negligible. The nonlinear algebraic equations obtained as a result of finite difference discretization are solved using efficient-numerical technique employing IMSL subroutine DNEQNJ. The nonlinear simulation shows that the dominant wavelengths (corresponding to minimum time) for film rupture are very close to the prediction of the linear theory for all types of films. There seems to be no influence of surface inclination on the instability of thin films. Inclination dose influence the growth of instability in thick films. The film rupture time increases with increasing film thickness for inclined planes. Increase in the amplitude of perturbation results into decreased time of rupture. The deviations between the predictions of nonlinear and linear theory results are minimum around dominant wavelength. The linear theory may overestimate or underestimates the time of rupture by several orders of magnitude depending upon thin film parameters. Hence linear theory is inadequate to describe the stability characteristics of inclined films and therefore, the need of a nonlinear approach to the study of inclined film dynamics.