A normal form of thin fluid film equations resolves the transient paradox

Abstract

Imagine two constant thickness, thin films of fluid colliding together: the transient flow forms a hump where they collide; thereafter they slowly relax. But, apparently reliable lubrication models expressed only in the thickness of the fluid forecast that precisely nothing happens. How can we resolve this paradox? Dynamical systems theory constructs a normal form of the Navier-Stokes equations for the flow of a thin layer of fluid upon a solid substrate. These normal form equations illuminate the fluid dynamics by decoupling the interesting long-term 'lubrication' flow from the rapid viscous decay of transient shear modes. The normal form clearly shows the slow manifold of the lubrication model and demonstrates that the initial condition for the fluid thickness of the lubrication model is not the initial physical fluid thickness, but instead is modified by any initial lateral shear flow. With these initial conditions, the lubrication model makes better forecasts. This dynamical systems approach could similarly illuminate other models of complicated dynamics. © 2006 Elsevier Ltd. All rights reserved.