The numerical simulation of non-isothermal thin liquid film flow on inclined plane

Abstract

The classical problem of the stability and dynamics of thin liquid films on solid surfaces has been studied extensively. Particularly, thin liquid films subjected to various physico-chemical effects such as thermocapillarity, solutal-Marangoni and evaporative instabilities at the film surface has been the focus of research for more than two decades. Various flow configurations of thin film such as thin film on plane, inclined, and wavy surfaces had been the subject of recent investigations.. An inclined film compared to a horizontal film, also experiences the gravity force which may significantly influence the nonlinear dynamics of the film coupled with other forces. In this research, an attempt is made to characterise qualitatively the stability and dynamics of a thin liquid film on an inclined plane which is subjected to instabilities owing to thermocapillarity and evaporative effects as well as van der Waals attractive intermolecular forces, and to compare the results to those in the standard literature. For a Newtonian liquid, flow in thin liquid film on a planar support and bounded by a passive gas, is represented by Navier-Stokes equation, equation of continuity and appropriate boundary conditions. The external effects are incorporated in the body force term of the Navier-Stokes equation. These governing equations are simplified using the so called long-wave approximation to arrive at a nonlinear partial differential equation, henceforth called equation of evolution (EOE), which describes the time evolution of the interfacial instability in the film caused by internal and/or external effects. Efficient numerical method is required for the solution of the equation of evolution (EOE) in order to comprehend the nonlinear dynamics of the thin film. Here we present the results of our numerical simulations using Crank-Nicholson implicit finite difference scheme applied to the thin film model incorporating instabilities owing to gravity, evaporation and thermo-capillarity. Comparison of our results with those obtained from Spectral method, show remarkable agreement for most of the cases investigated. The film rupture times obtained using our implicit finite difference method closely conform to those of Fourier spectral results of Joo et al. (1991) within a deviation of 1.5% for films going to rupture. For cases not resulting into rupture, the film interface profiles at various stages of deformation are almost identically similar to the profiles obtained in our numerical simulations. Thus implicit Crank-Nicholson mid-point rule proves to be an efficient and reliable method for numerical simulation of the nonlinear dynamics of thin film flows. Keywords: Thin liquid film flow; Nonlinear Dynamics; Implicit Finite Difference; Spectral Method.