Shape relaxation of an elongated viscous drop

The shape relaxation of a distorted viscous drop suspended in a quiescent immiscible liquid is analyzed in the creeping flow limit. The shape of the drop is axisymmetric, but otherwise arbitrary. The relaxation process is assumed to be driven by a constant interfacial tension and rate-limited by the Newtonian viscosities of the dispersed and continuous phases. For analysis, a least squares technique is developed which, compared to the more common boundary integral methods, is simpler to implement and especially suited for systems where one liquid is much more viscous than the other (i.e., when the viscosity ratio lambda, defined as the ratio of the dispersed to continuous phase viscosities, approaches either zero or infinity). To demonstrate the validity of the proposed least squares technique, its results are shown to agree well with boundary integral calculations for moderate values of lambda, and with experimental data when lambda is much larger than unity (approximately 10(6)). Predictions at infinite viscosity ratio--the regime in which the least squares technique is most useful--are then used to evaluate interfacial tensions associated with a system of practical importance, namely, the dispersion of heavy crude oil in an aqueous environment. This amounts to a novel and accurate technique for determining interfacial tensions--especially those of low values (1 mN/m or less)--between density-matched liquids where at least one of the phases is highly viscous. The experimental part of this study involves the use of suction pipettes to manipulate the shapes of individual micrometer-sized droplets, thus avoiding the need for complex flow-generating devices to create drop deformations.