Spreading of Very Thin Droplets

In the earlier part of this section we showed that pure capillary spreading results in inconsistency of the mathematical treatment in the vicinity of the apparent moving contact line of the spreading droplet. This inconsistency is usually referred to as a singularity at the three-phase contact line. As we already mentioned in the Introduction to this chapter, this inconsistency is the result of 

This is why we consider here the spreading of the miCTodroplet, whose apex is located in the range of action of the surface forces. In the following text, we neglect the effect of capillary pressure. The hydrodynamic pressure in the liquid is described by Equation 3.16. 

the case under consideration refers to the final stage of the spreading of droplets, which completely wets the substrate. The equation describing the droplet profile, h t, r), of a spreading droplet has the following form  

This is obtained by a substitution of Equation 3.16 into the equation of spreading (3.11). The boundary conditions for second-order differential equation (3.37) are as follows  

Let us first consider the simplest form of the isotherm of the disjoining pressure, H(h), in the case of complete wetting  

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