Full similarity solution for the zero-depletion-length regime

Derive and fully verify the similarity solution for the zero-depletion-length case Λ_d=0, including the sharp surfactant-front region and the matching that determines the front prefactor η_f, for the finitely soluble surfactant thin-film equations.

Background

The paper analyzes the singular Λ_d=0 limit, in which the bulk surfactant is advected without adsorption–desorption depletion while the surface concentration remains kinetically coupled through the Biot number. Scaling arguments predict t^{-3} decay of the minimum thickness for finite nonzero Bi, and the authors derive the similarity profiles in the region behind the surfactant front.

However, the complete similarity solution requires resolving and matching the sharp front region. Numerical difficulties prevented the authors from obtaining and fully testing that global solution, so the front prefactor and complete matched structure remain unresolved in this limit.