Free Boundary Problems
Lubrication approximation in the viscous film theory and the k- system in turbulence theory
Free boundary problems (FBP) constitute (mathematical, biological, physical, etc.) research subject characterized by the occurence of frontiers whose locations are a priori unknown. These boundaries separate geometric regions with different properties. FBP arise mainly in boundary value problems of partial differencial equations.1./ The so-called k - system can be derived from the exact transport equation for the Reynolds stresses by using some closing approximations. The simplest version modeling the evolution of turbulent bursts is
where k(x, t) in the turbulent energy density, e (x, t) is dissipation rate of turbulent energy,are positive dimensionless constants.
The free boundary is the geometric boundary of the turbulent domain, where k and e vanish. For differentwe have very different kinds of evolution.
2./ The lubrication approximation for a thin film of liquid on a solid surface yields a fourth order degenerate diffusion equation for the film height h.
Numerical calculations and matched asymptotics lead to results which suggest a very rich structure of solutions. We study the existence of free boundaries, their behaviour and provide different energy estimates h, and their dependence on the parameter n.
University of Rome II
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