A new measure for drop deformation in extensional flows at low Reynolds number

Abstract

The critical capillary number of a drop, which represents the state where the interfacial tension force of the drop cannot overcome the viscous force exerted by a surrounding flow, is usually determined in low Reynolds number [<O(1)] extensional flows by progressively elongating the drop in stagnant extensional flows. Below the critical capillary number, all the elongated states of the drop are steady. The unsteady drop states beyond the critical capillary number, usually seen in practically relevant non-stagnant extensional flows which breakup the drop, provide no information about critical capillary number and are usually studied separately. In this study, we present a new measure—called the semi-minor capillary number—which uniquely describes the drop deformation process at both steady and unsteady states. The measure uses the instantaneous semi-minor dimension of the deforming drop as the length scale in calculating the capillary number. Our experiments at small initial capillary numbers, compared to the critical capillary number, yielded steady drops with a constant value of semi-minor capillary number. For large initial capillary numbers and unsteady states, the drops elongated continuously, and the same constant represented an asymptotic limit of the self-similar deformation. The new measure of semi-minor capillary number rationalized drop behavior at both small and large initial capillary numbers compared to the critical capillary number. More importantly, it provided significance to drop behavior at large initial capillary numbers, which is an unstudied parametric space in the context of determining the critical capillary number. Finally, we discuss the significance of the new measure by presenting the critical semi-minor capillary number at different viscosity ratios.