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Free Boundary Problems

Lubrication approximation in the viscous film theory and the k-szp2_5.gif (935 bytes) 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 - szp2_5.gif (935 bytes) 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, szp2_4.gif (1212 bytes) are positive dimensionless constants.

The free boundary is the geometric boundary of the turbulent domain, where k and e vanish. For different szp2_4.gif (1212 bytes) we 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
Donetsk University
University of Bath, UK

For more information please contact

Róbert Kersner and Zsolt Biró. Room K427
Laboratory of Applied Mathematics
Group of Mathematical Physics
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