# Interdisciplinary Applied Mathematics

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where p is the viscosity. Here, fx and fz denote external interfacial forces, and Ф^ is a potential evaluated at the liquid-air interface, i.e., at z = h. Physically, the terms in the above equation represent the action: external forces, surface tension and its nonuniformity, gravity, and viscous damping, respectivbely starting from the second term. Specifically, the potential Ф^ may include any conservative force, e.g., gravity, centrifugal forces, or van der Waals forces. In the case of an isothermal film with constant surface tension, the above equation reduces to (Oron, 2001)

hx)x Y -y(/i hxxx)x g    (8*12)

which expresses a balance of viscous damping, gravity, and capillary forces. This is a nonlinear equation for diffusion enhanced by a dissipation term due to surface tension. It is stable, which implies that any disturbance imposed at the interface will decay very fast and the film will return ito its original shape. On the other hand, a change in the sign of the gravity term leads to a very unstable system, the well-known Rayleigh-Taylor instability, which will eventually lead to rupture of the thin film.

The van der Waals forces between an apolar liquid and a solid can be modeled by a standard Lennard-Jones potential of the form

Ф = a3h-3 — a9h-9,

where aj are the positive Hamaker coefficients. A different potential appropriate also for rough solid substrates, derived in (Oron and Bankoff, 1999), is

Ф = a3h-3 — a4h-4.

A simpler potential for apolar liquids is

Ф

-3

where positive a corresponds to an attractive force, while negative a corresponds to a repulsive force driving the interface toward a flat profile. In the case of constant surface tension and neglecting gravity, the above potential leads to the following film evolution equation:

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