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 lea...
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Published in | Physics of fluids (1994) Vol. 18; no. 3; pp. 032106 - 032106-13 |
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Main Authors | , , , , , |
Format | Journal Article |
Language | English |
Published |
Melville, NY
American Institute of Physics
01.03.2006
|
Subjects | |
Online Access | Get full text |
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Summary: | 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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ISSN: | 1070-6631 1089-7666 |
DOI: | 10.1063/1.2185111 |