(401e) Accurate Continuation of Multi-Dimensional Fem Calculations Involving Drop Breakup beyond the First Singularity
Phenomena involving interface rupture and drop breakup are of common occurrence in nature and technology. The generation of a mist from a waterfall, and the dripping of water from a leaky faucet and the formation of drops in an ink jet printer are just some well recognized examples of such phenomena. Computational analysis of how a drop breaks is therefore essential for advancing the scientific understanding of so-called pinch-off singularities and developing new technological applications. A prototypical example of drop breakup is the formation of drops of an incompressible Newtonian liquid from a tube. As more liquid flows from the tube into the drop, the drop elongates and thins. At the incipience of breakup, a spherical mass ? the precursor of the primary or main drop ? is connected to the rest of the liquid hanging from the tube by a thin thread ? the precursor of one or more satellites. Numerical algorithms designed to capture this phenomenon at finite Reynolds number have been of two types: one type has been based on the finite element method (FEM) and the other has been based on various diffuse interface techniques such as the volume of fluid method (VOF). Numerical solutions must agree with scaling solutions of interface pinch-off, which are exact solutions of the nonlinear Navier-Stokes equations in the vicinity of breakup, and experiments. To date, the diffuse interface type approach has been more popular because it is relatively easy to use and can predict the formation of several drops in sequence. However, predictions made with the diffuse interface type approach are typically too coarse to accord with predictions of analytical scaling theories and at best agree qualitatively with experiments. By contrast, predictions made with FEM algorithms have been shown to be in excellent agreement with both scaling theories and measurements. However, such algorithms have heretofore been able to simulate the dynamics only until the instant of first breakup (Chen, Notz, and Basaran, Phys. Rev. Lett., 2002; Notz and Basaran, J. Fluid Mech. 2004; and Liao, Franses, and Basaran, Phys. Fluids, 2006). Here we describe a new FEM algorithm to predict the dynamics of formation of an arbitrary number of drops in sequence and carry out high-speed visualization experiments to follow the dripping of many drops in succession from a tube. The accuracy of the new algorithm is confirmed by verifying that its predictions accord with both scaling theories and agree with 1% experimental error with the new measurements. The new calculations open the door for the possibility of theoretically predicting the fluid mechanics of leaky faucets which until now has been possible using reduced-order models based on the slender-jet approximation that may be accurate over only selected regions of the space of parameters governing the dynamics (Fuchikami et al., J. Phys. Soc. Jpn., 1999; Ambravaneswaran et al., Phys. Rev. Lett., 2000; 2004; Coullet et al., J. Fluid Mech., 2005 ; Subramani et al., Phys. Fluids, 2006).