Simplicity and complexity in a dripping faucet

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 approximate to 0.5, and the Ohnesorge number Oh that measures the relative importance of viscous to surface tension force is low, Oh approximate to 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 approximate to 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 approximate to 0.3 and Oh approximate to 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 approximate to 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 approximate to 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 approximate to 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 approximate to 0.3. When G approximate to 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 approximate to 1, while Oh is held fixed at Oh approximate to 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. (C) 2006 American Institute of Physics.