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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 ${h}_{t}+\nabla \left(f\left(h\right)\nabla {\Delta }h\right)=0$, where $h$ is the thickness of the liquid film, and $f\left(h\right)$ is a prescribed function related to the solution-dependent diffusion coefficient, $f\left(h\right)\to 0$ as $h\to 0$. First, a special discussion is devoted to the finite-time singularities and similarity solutions of a particular one-dimensional equation ${h}_{t}+{\left({h}^{n}{h}_{xxx}\right)}_{x}=0$. 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) 76D08 Lubrication theory 82B24 Interface problems; diffusion-limited aggregation (equilibrium statistical mechanics)