University of Southern Queensland

Abstract:

There exist numerous models describing the long term dynamics of thin fluid films over a solid substrate, such as the lubrication model. Frequently, the film evolution is described by the spatio-temporal variation of the local fluid layer thickness. These models do no account for any quickly decaying transients such as, for example, the decay of the initial fluid velocity field due to viscosity. On the other hand the details of the initial conditions may have long term effect. For example, the initial velocity field could lead to a finite displacement of the surfactant before the flow is stopped by the action of viscosity. thus correct predictions cannot be made by the dynamical models unless correct initial conditions for the models are supplied. However, the choice of proper initial conditions for nonlinear models is a nontrivial task. In particular we show that the initial fluid thickness generally is \emph (not) the correct initial condition for the lubrication model. Theory recently developed by Roberts\footnote (A.J. Roberts. Appropriate initial conditions for asymptotic descriptions of the long term evolution of dynamical systems. J.Austral. Math. Soc. B31:48--75, 1989 and Cox and Roberts\footnote and Cox and Roberts\footnote S.M. Cox and A.J. Roberts. Initial conditions for models of dynamical systems. Physica D., 85:126--141, 1995 describes the correct projection of initial conditions onto a model of dynamical system. This projection is determined by requiring that the model solution exponentially quickly approaches the actual fluid dynamics. For the lubrication model, we show that although the initial free surface shape contributes the most to the initial conditions for the model, the initial velocity field is also of influence.