A normal form of thin fluid film equations provides initial conditions

Abstract

We use dynamical systems theory to construct the normal form of the Navier--Stokes equations for the flow of a thin layer of fluid upon a solid substrate. The normal form equations illuminate the fluid dynamics by decoupling the long-term flow of interest from the rapid viscous decay of the transient shear modes. The normal form clearly shows the centre manifold of the lubrication model and shows the result that the initial condition for the fluid thickness of the lubrication model is \emph{not} the initial physical fluid thickness, but instead is modified by the initial lateral shear flow. With these initial conditions, better forecasts will be made using the lubrication model. This dynamical systems approach will also enable similar illumination of other complicated models of dynamics.