Critical Eotvos numbers for buoyancy-induced oil drop detachment based on shape analysis

Critical values of the Eotvos number, which is half the Bond number, above which buoyancy induced drop detachment occurs, are estimated based on force balance equations available in the literature Colloids Surf. A: Physicochem. Eng. Aspects 178 (2001) 249]. Since there are two significantly different expressions of the capillary retention force responsible for holding oil drops on a solid substrate in an aqueous phase, the critical dimensionless number is estimated with these two distinct equations. The differential equation defining the drop shape, with the constraints of the drop volume and the 'pinned' or 'receding' contact line, is numerically solved. The equilibrium drop shapes predicted are shown to match the experimentally observed variations in drop shape. From the numerical solution, it is observed that for interfacial tension (IFT) values lower than a certain limit for a given drop size, no numerically estimated drop shape can fulfil the drop volume constraint. Similarly, for the dimensionless number above a critical value, no shape can meet all the constraints. These critical Eotvos numbers are estimated, based on the above numerical approach, for initial contact angles measured in oil varying from 20 degrees to 90 degrees. It is found that the critical Eotvos numbers estimated from the numerical shape analysis are between the critical values estimated from the two force-balance equations. Near 90 degrees, the critical values estimated from the drop shape analysis matches the values from one of the force balance estimates, but merges with the critical values of the dimensionless number, estimated from the other force balance model near 10 degrees. From this analysis, it appears that a combination of the two equations for the capillary retention force is required, with one dominating when the contact angles are high, while the other applies for low values of the contact angle.