Scaling Laws for Transition from Varicose to Whipping Instabilities in Electrohydrodynamic Jetting

Whether an electrified jet breaks up while the jet maintains a straight line (varicose), or alternatively, the jet trajectory becomes chaotic (whipping), depends on the competition of the interfacial forces. Starting from the competition of normal stresses on the jet surface, we derive scaling laws in different electrohydrodynamic operating regimes as a function of fluid properties and the flow rate. The onset of whipping occurs when this scaling function reaches a threshold, which is independent of the electric field strength. However, experimental evidence indicates that this onset condition applies only when the viscosity and electrical conductivity of a liquid are small enough. As a result, we further introduce a general parameter to incorporate viscous effects into the scaling law. A unified threshold value for this parameter is found through a substantial number of experiments for liquids having a wide range of properties and under a wide variety of operational conditions of flow rate.