Drop formation in viscous flows at a vertical capillary tube

Drop formation at the tip of a vertical, circular capillary tube immersed in a second immiscible fluid is studied numerically for low-Reynolds-number flows using the boundary integral method. The evolution and breakup of the drop fluid is considered to assess the influences of the viscosity ratio $\lambda$ , the Bond number B, and the capillary number C for ${10}^{\text{−2}}\text{⩽λ⩽10}$ , ${10}^{\text{−2}}⩽\mathcal{C}\text{⩽1}$ , and $\text{0.1⩽}\mathcal{B}\text{⩽5}$ . For very small $\lambda$ , breakup occurs at shorter times, there is no detectable thread between the detaching drop and the remaining pendant fluid column, and thus no large satellite drops are formed. The distance to detachment increases monotonically with $\lambda$ and changes substantially for $\text{λ>1}$ , but the volume of the primary drop varies only slightly with $\lambda$ . An additional application of the numerical investigation is to consider the effect of imposing a uniform flow in the ambient fluid [e.g., Oguz and Prosperetti, J. Fluid Mech. 257, 111 (1993)], which is shown to lead to a smaller primary drop volume and a longer detachment length, as has been previously demonstrated primarily for high-Reynolds-number flows.