The shape of a fluid drop approaching an interface

The shape of a fluid drop approaching an interface does not change appreciably with time and is very close to the equilibrium dimensions, in spite of the large pressure gradient which is present in the draining film. This is because the net vertical force due to the excess pressure in the draining film above that in the drop is identical with that for an equilibrium film being zero for a plane interface and —2 R σ sin 2 ϕ for a deformable interface. Employing this result in a force balance around the drop which is independent of the bulk interface shows that the area A of the draining film between a fluid drop of volume V and a deformable fluid‐liquid interface is given by\documentclass{article}\pagestyle{empty}\begin{document}$$ \left( {V - v} \right)\Delta \rho g = A\left( {1 + \cos \Phi } \right)\left( {\frac{\sigma }{b} + \frac{{\Delta \rho gh}}{2} - \frac{\sigma }{R}} \right) $$\end{document}where σ is the interfacial tension and Δρ the density difference between the drop and surrounding fluid, 1/ b is the curvature at the top of the drop and h is the distance between this point and the edge of the draining film which is inclined to the horizontal at an angle ϕ. When the interface is a rigid plane the overall curvature 1/ R of the draining film and the volume v enclosed by it, together with the angle ϕ are all zero. The limiting cases of the expression for very small and very large drops agree with those previously established for both deformable and rigid interfaces. An approximate expression which applies when cV 2/3 (where c = Δρ g /σ) is between 0.6 and 13.5 and which gives A to within ± e% is\documentclass{article}\pagestyle{empty}\begin{document}$$ A/V^{2/3} = k\left( {eV^{2/3} } \right)^n $$\end{document}where for a rigid plane interface\documentclass{article}\pagestyle{empty}\begin{document}$$ k = 0.25,\,n = 0.75,\,{\rm and}\,e = 7.5 $$\end{document}and for a deformable interface\documentclass{article}\pagestyle{empty}\begin{document}$$ k = 0.5,\,n = 0.6,\,{\rm and}\,e = 11 $$\end{document}When the densities of the drop and bulk heavy fluids are equal, but their respective interfacial tensions σ 12 and σ 23 with the light fluids are different, the expression becomes\documentclass{article}\pagestyle{empty}\begin{document}$$ A/V^{2/3} = 0.5\left( {eV^{2/3} } \right)^{0.6} \left( {\sigma _{12} /\sigma _{23} } \right)^{0.3}, $$\end{document}which estimates A/V 2/3 to within about ± 25% for σ 12 /σ 23 in the range 0.11 to 9.0 and cV 2/3 (where c = Δρ g /σ 12 ) between 0.6 and 13.5.