Interdisciplinary Applied Mathematics

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pht Ф v o.(h hx)x T ~^/{h hxxx)x = 0.    (8.13)


This equation generates solutions with very small wavelength at least initially, due to van der Waals forces. However, asymptotically, surface tension acts to cut off the smaller scales and stabilize the interface (Williams and Davis, 1982). The attractive van der Waals force included in the equation establishes naturally a precursor film, which removes the stress singularity encountered in problems with moving contact lines. Using different contact line models to relieve the stress singularity leads to some differences in the initial evolution of the thin film, but its asymptotic stability does not seem to depend on the particular model. Thefore, using a flat precursor film or a slip boundary condition or employing van der Waals forces will not change the basic dynamics obtained in long-time integration (Diez et al., 2000; Davis and Troian, 2003).

A more general equation regarding the upward thermocapillary spreading of a Newtonian liquid film in an inclined plane with angle ф was derived in (Oron et al., 1997), and also studied in (Davis and Troian, 2003), and it is given by

ht +



pg sin

in фИ3



-I X


V(—pghcos ф + jV2h + ah 3)


where т is the shear stress and x indicates the flow direction. The dynamic viscosity is assumed variable, since in some applications, such as thermocapillary pumping, the viscosity may be changing due to temperature variation. We will discuss this topic in some detail in Section 8.5.

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