The drop weight method is an accurate yet simple technique for determining surface tension σ. It relies on dripping a liquid of density ρ at a low flow rate Q̃ from a capillary of radius R into air and measuring the combined volumes of the primary and satellite drops that are formed. The method’s origin can be traced to Tate, who postulated that the volume Ṽideal of the drop that falls from the capillary should be given by ρgṼideal=2πRσ, where g is the gravitational acceleration. Since Tate’s law is only an approximation and the actual drop volume Ṽf<Ṽideal, in practice the surface tension of the liquid-air interface is determined from the experimental master curve due to Harkins and Brown (HB). The master curve is a plot of the fraction of the ideal drop volume, Ψ≡Ṽf∕Ṽideal, as a function of the dimensionless tube radius, Φ≡R∕Ṽf1∕3. Thus, once the actual drop volume Ṽf, and hence Φ, is known, σ is readily calculated upon determining the value of Ψ from the master curve and that Ψ=ρgṼf∕2πRσ. Although HB proposed their master curve more than 80 years ago, a sound theoretical foundation for the drop weight method has heretofore been lacking. This weakness is remedied here by determining the dynamics of formation of many drops and their satellites in sequence by solving numerically the recently popularized one-dimensional (1–d) slender-jet equations. Computed solutions of the 1-d equations are shown to be in excellent agreement with HB’s master curve when Q̃ is low. Moreover, a new theory of the drop weight method is developed using the computations and dimensional analysis. The latter reveals that there must exist a functional relationship between the parameter Φ, where Φ−3 is the dimensionless drop volume, and the gravitational Bond number G≡ρgR2∕σ, the Ohnesorge number Oh≡μ∕(ρRσ)1∕2, where μ is the viscosity, and the Weber number We≡ρQ̃2∕π2R3σ. When We→0, the computed results show that Φ depends solely on G. In this limit, a new correlation is deduced which has a simple functional form, G=3.60Φ2.81, and is more convenient to use than that of HB. The computed results are also used to show how the original drop weight method can be extended to situations where We is finite and resulting drop volumes are not independent of Oh.

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.

Motivated by the desire to improve the theoretical understanding of drop-on-demand (DOD) ink-jet printing, a computational analysis is carried out to simulate the formation of liquid drops of incompressible Newtonian fluids from a simple capillary tube by imposing a transient flow rate upstream of the nozzle exit. Since the flow in a typical ink-jet nozzle is toward the nozzle outlet during part of the time and away from the nozzle outlet at other times, an inflow rate is adopted here that captures the essential physics and is given in dimensionless form by Q=(πWe∕2)sinΩt, where We is the Weber number (inertial/surface tension force), Ω is the frequency, and t is time. The dynamics are studied as functions of We, Ω, and the Ohnesorge number Oh (viscous/surface tension force). For a common ink forming from a nozzle of 10μm radius, Oh=0.1. For this typical case, a phase or operability diagram in (We,Ω)-space is developed that shows that three regimes of operation are possible. In the first regime, where We is low, breakup does not occur, and drops remain pendant from the nozzle and undergo time periodic oscillations. Thus, the simulations show that fluid inertia, and hence We, must be large enough if a DOD drop is to form, in accord with intuition. A sufficiently large We causes both drop elongation and onset of drop necking, but flow reversal is also necessary for the complete evacuation of the neck and capillary pinching. In the other two regimes, at a given Ω, We is large enough to cause drop breakup. In the first of these two regimes, where Wec1<We<Wec2, DOD drops do form but have negative velocities, i.e., they would move toward the nozzle upon breakup, which is undesirable. In the second breakup regime, where We>Wec2, not only are DOD drops formed, but they do so with positive velocities.

A series of potential biomarkers including tyrosine, creatinine, linoleic acid, β-hydroxybutyric acid and ornithine have been identified by metabolomic profiling, which may be used to identify the metabolic changes during hyperlipidemia progression.

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