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Zbl 0917.35100
Bertozzi, Andrea L.
The mathematics of moving contact lines in thin liquid films. (English)
[J] Notices Am. Math. Soc. 45, No.6, 689-697 (1998). ISSN 0002-9920; ISSN 1088-9477/e

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.
[O.Titow (Berlin)]

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
76D45 Capillarity
76D08 Lubrication theory
82B24 Interface problems (equilibrium)

Keywords: van der Waals model; superdiffusion model; Young's law; fourth-order degenerate diffusion equation; solution-dependent diffusion coefficient; finite-time singularities; similarity solutions; weak solutions; Greenspan-McKay law; finite difference methods

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Zentralblatt MATH Berlin [Germany]