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The mathematics of moving contact lines in thin liquid films. (English) Zbl 0917.35100

The author studies the title problem by using the fourth-order degenerate diffusion equation \(h_t+ \nabla(f(h)\nabla \Delta h)= 0\), where \(h\) is the thickness of the liquid film, and \(f(h)\) is a prescribed function related to the solution-dependent diffusion coefficient, \(f(h)\to 0\) as \(h\to 0\). First, a special discussion is devoted to the finite-time singularities and similarity solutions of a particular one-dimensional equation \(h_t+ (h^n h_{xxx})_x= 0\). Then the author examines weak solutions of the general equation which correspond to various constitutive laws for moving contact lines (Young’s law, Greenspan-McKay law, van der Walls and superdiffusion models). Numerical results obtained by finite difference methods conclude the paper.
Reviewer: O.Titow (Berlin)

MSC:

35Q35 PDEs in connection with fluid mechanics
76D45 Capillarity (surface tension) for incompressible viscous fluids
76D08 Lubrication theory
82B24 Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics
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