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Continuous emission of drops of an incompressible Newtonian liquid from a tube–dripping–is a much studied problem because it is important in applications as diverse as inkjet printing, microarraying, and microencapsulation, and recognized as the prototypical nonlinear dynamical system, viz., the leaky faucet. The faucet’s dynamics are studied in this paper by a combination of experiment, using high-speed imaging, and computation, in which the one-dimensional slender-jet equations are solved numerically by finite element analysis, over ranges of the governing parameters that have heretofore been unexplored. Previous studies when the Bond number G that measures the relative importance of gravitational to surface tension force is moderate, G≈0.5, and the Ohnesorge number Oh that measures the relative importance of viscous to surface tension force is low, Oh≈0.1, have shown that the dynamics changes from (a) simple dripping, i.e., period-1 dripping with or without satellites, to (b) complex dripping, where the system exhibits period doubling bifurcations and hysteresis, to (c) jetting, as the Weber number We that measures the relative importance of inertial to surface tension force increases. New experiments and computations reveal that lowering the Bond number to G≈0.3 while holding Oh fixed results in profound simplification of the behavior of the faucet. At the lower value of G, the faucet exhibits simply period-1 dripping, period-2 dripping, and jetting as We increases. Experimental and computational bifurcation diagrams when G≈0.3 and Oh≈0.1 that depict the variation of drop length or volume at breakup with We are reported and shown to agree well with each other. The range of We over which the faucet exhibits complex dripping when G≈0.3 is shown by both experiment and computation to shrink as Oh increases. Computations are also used to develop a comprehensive phase diagram when G≈0.3 that shows transitions between simple dripping and complex dripping, and those between dripping and jetting in (We,Oh) space. Similar to the case of G≈0.5, dripping faucets of high viscosity (Oh) liquids are shown to transition directly from simple dripping to jetting without exhibiting complex dripping when G≈0.3. When G≈0.3, computed values of We that signal transition from dripping to jetting are further shown to accord well with estimates obtained from scaling analyses. By contrast, new computations in which the Bond number is increased to G≈1, while Oh is held fixed at Oh≈0.1, reveal that the faucet’s response becomes quite complex for large G. In such situations, the computations predict theoretical occurrence of (a) rare period-3 dripping and period-3 intermittence, which have previously been surmised solely by the use of ad hoc spring-mass models of dripping, and (b) chaotic attractors. Therefore, by combining insights from earlier studies and the detailed response of dripping which has been obtained here by varying (i) Oh as 0.01⩽Oh⩽2, a range that is typical of most practical applications, (ii) We from virtually zero to a value just exceeding that at which the system transitions from dripping to jetting, and (iii) G from a small value to a value approaching that beyond which controlled formation of drops is prohibited, this paper provides a comprehensive understanding of the effect of the governing parameters on the nonlinear dynamics of dripping.

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Stresses induced by a spatially nonuniform electric field acting on an initially flat fluid-fluid interface can (i) be exploited beneficially to pattern polymer microstructures without the use of resists, exposure, development, and etching, but (ii) cause undesirable nonuniformity in film thickness in precision coating processes. The equilibrium shape of an interface separating a liquid film from an ambient fluid subjected to a uniform electric field is flat so long as the field strength is below a critical value. A nonuniform electric field, however, results in the deformation of the interface no matter how small its strength, an important difference which previous theoretical studies have not addressed satisfactorily. Hence, whereas under a uniform field loss of stability occurs via a bifurcation from the flat film solution, under a nonuniform field destabilization may occur at a turning point at which the film profile already exhibits a finite-amplitude deformation. This deficiency in understanding is remedied here by analyzing a model problem in which a gas overlying a perfect dielectric liquid film is sandwiched between two electrodes wherein the top electrode is grounded and the electric potential of the bottom electrode varies sinusoidally with distance measured along it. The equilibrium shapes and stability of the liquid-gas interface are determined directly in the present work by simultaneously solving the augmented Young-Laplace equation governing the shape of the free surface and the Laplace equation governing electric potentials theoretically by domain perturbation analysis and numerically by finite element analysis. For small nonuniformities in the electric field, analytical solutions are reported for the profile of the free surface. The computational predictions are shown to be in excellent accord with these small-deformation results. Moreover, computations are used to extend the investigations into the nonlinear regime where nonuniformities in the electric field and deformations of the free surface are large, and loss of stability may occur. The variation of the equilibrium shapes and the limits of stability with the governing dimensionless groups are investigated thoroughly. It is shown that the rich response exhibited by the system can be rationalized by interrogating the computed solutions and scrutinizing the balance of stresses due to the normal component of the electric field, which are destabilizing, and those due to its tangential component, which are stabilizing.