# Interdisciplinary Applied Mathematics

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If gravity is important,    then    the    total    energy    of    the    system    consists    of

the interfacial as well as the gravitational potential energy. The latter is given by Eg = R4ag(9), where ag(9) is a shape function. This expression was derived in (Shapiro et al., 2003a), using scaling arguments, where the shape function was also calculated. The total energy in this case is

E = Ei + Eg = R2 [2nYgl(1 — cos 9) + n sin2 9(yis — Ysg)]    (8.10)

2n

+RA — pg{3 + cos9) sin6(0/2).

By minimizing this energy, as before, we obtain the corresponding Young equation with gravity

Ygi    Ygi

cos 0/2, — cos(20) 12

1

4

0.

We see that in this case the contact angle depends on the radius R of the droplet, unlike the case without gravity.

##### 8.3 Governing Equations for Thin Films

We have already presented some basic concepts for droplet formation and equilibrium equations involving surface tension as the dominant mechanism. We will present more modeling details in the following sections, but in the current section we deal with thin films. The evolution equation of a liquid film on a solid surface is derived from the incompressible Navier-Stokes equations. The fundamental assumption is that the mean thickness is much smaller than the characteristic length of the interfacial disturbance, the so-called long-wave approximation. Let us denote by z the direction along the film thickness. Then the equation for the film thickness z = h(x, t) in two dimensions is

pht + 2 [h2(fx + Y®)]x + g {G[(/z§h)x + «fhxxx]}x =0,    (8.11)

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