# Interdisciplinary Applied Mathematics

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The process of dewetting of a solid surface is of great interest in microfluidic applications and has been the subject of many fundamental computational and experimental studies. Theoretical work supports the idea that the dominant mechanism driving film evolution is the so-called spinodal dewetting, and molecular dynamics simulations support this thesis (Koplik and Banavar, 2000); however, some experiments point to nucleation of holes from defects of the surface. The governing equations in this case should include both gravity and capillary forces but also intermolecular repulsive and attractive forces. The general equation (8.11) reduces in this case to

pht ^ [h3(pgh — jhxx + Ф)ж]ж = 0.    (8.15)

Small disturbances usually decay fast, but when the van der Waals forces overcome the stabilizing effect of gravity, rupture may take place.

The above modeling assumes isothermal conditions. However, surface tension is a strong function of temperature. Assuming a constant ambient temperature TTO,    then if the temperature    at    the    bottom    of    the    film    T0    is

greater than TTO, then the surface tension (difference) Ду is positive. On the other hand, if Тж > T0, then Ду > 0. The equation for the evolution of the film thickness should now include the thermocapillary stress, so equation (8.11) becomes

phtpg{h3K)x + yy(h3hxxx)x + ат^ (h2hx)x = 0,    (8.16)

3    3    2k

where k, aT are the thermal conductivity and diffusivity, respectively. In general, the effect of the thermocappilary stress is to produce an unstable interface. However, thermocapillary stresses due to internal heat generation have a stabilizing effect (Oron and Peles, 1998).

##### 8.4 Dynamics of Capillary Spreading

In this section we consider isothermal conditions and examine in more detail the spreading of liquids on homogeneous substrates with roughness as well as heterogeneous substrates consisting of hydrophobic and hydrophilic stripes. Of interest is the evolution of the spreading front, i.e., its location and speed as a function of time. A classical result due to (Tanner, 1979) concerns the spreading of a Newtonian liquid droplet on a homogeneous smooth substrate. The radial advance r(t) of the liquid of volume V is

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