Similarity Solutions for Film Rupture

The instability in the presence of London-van der Waals attractive forces that was studied in the preceding subsection is indicative of the fact that small perturbations in the film thickness 

An interesting question is whether we can demonstrate that such a self-similar behavior should exist as the film-rupture singularity is approached. We consider the combined equations (6-83) and (6-85) with the gravitational term neglected, as already explained. Using the notation developed earlier in this section, we can write this equation in the form 

Starting from this equation, we can utilize what we have learned from the numerics and seek a transformation to a self-similar form. The numerical result suggests that we seek a solution of the form 

However, in case we had not carried out the numerical analysis, we might hope that the form of the proposed self-similar solution might be derivable directly from usual process of reducing the governing equation, (6-107), to a self-similar form, Hence we seek a solution of the same type but with the coefficients unknown, i.e., 

By substituting (6-108) into (6-107), it is straightforward to show that we can reduce the problem to an ODE for the function II (ij) and that this process leads to the unique set of values p = 1 /5 and a = 2/5 that was suggested by the numerical study by Zhang and Lister of the solutions of (6-107). 

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