Numerical simulation of thin-liquid film flow on inclined palne using implicit and explicit finite difference schemes - Jameel, Ahmad Tariq; Hoda, Asif; Ismail, Ahmad Faris (Institute of Applied Mathematics (IAM), Middle East Technical University, Ankara, Turkey, 2012)
The stability and dynamics of thin liquid films has been the subject of extensive study for the past few decades. Especially, thin liquid films subjected to various physico-chemical effects, such as thermocapillarity, solutal-Marangoni and evaporative instabilities at the surface, has been the focus of research since past two decades (Burelbach et al., 1988, Joo et al, 1991; Ali et al, 2005). In addition to these instabilities, an inclined film also experiences the gravity force which may influence the nonlinear dynamics of the film coupled with other forces. van der Waals interactions being ubiquitous in nature play significant role especially at nano-scale thicknesses of such films. Flow of a Newtonian liquid on a solid support and bounded by a passive gas at the free surface is represented by Navier-Stokes equation, equation of continuity and appropriate boundary conditions. The external effects are generally incorporated in the body force term of the Navier-Stokes equation. These governing equations can then be simplified using so called long-wave approximation to arrive at a fourth order nonlinear partial differential equation, henceforth called equation of evolution (EOE) which describes the time evolution of the interfacial instability caused by internal and/or external effects. The details of the derivation of the EOE are available in the literature (Burelbach et al., 1988, Joo et al., 1991; Ali et al, 2005 and others). The linear stability characteristics can be obtained by the solution of the linearized equation of evolution. However, complete characterization of the nonlinear dynamics and surface morphology of thinfilm requires efficient numerical method for the solution of the equation of evolution (EOV). The extent of nonlinearity and the stiffness of the resulting differential equation depend upon the nature of various physico-chemical effects incorporated in the thin-film model. There have been several attempts to solve numerically the EOE for various thin-film models. Burelbach et al. (1988) and Ali et al. (2005) have obtained numerical solution using an implicit finite difference scheme, Joo et al. (1991) has employed Fourier spectral method while Sharma and Jameel (1993) have used Fourier collocations methods. This is certainly not an exhaustive coverage of all such works reported in the literature. Currently, we are working at the numerical simulation of the flow of the inclined thin films subject to evaporative and thermocapillary instabilities as well as instabilities owing to long range van der Waals interactions. A simple explicit finite difference (FD) formulation of the EOE is being attempted probably for the first time to our knowledge, besides an implicit FD scheme. Present work, compares these two numerical schemes - an implicit finite difference scheme called Crank Nicholson mid-point rule and a fully explicit finite difference discretization as described below, and applied to nonlinear equation of evolution for a thin liquid film flowing down an inclined plane under isothermal and non-isothermal conditions