Abstract

Falling water drops from a dripping faucet, illuminated from above, exhibit a row of bright strips of light, a few centimeters apart at a fixed distance below the faucet. Flash photographs of the drops show that they are oblate in shape when the flashes occur, and the bright flashes of light originate from the edge of the drop that is on the opposite of the overhead light source. Here we show that the spots result from the same internal reflection that gives rise to the rainbow in a cloud of spherical drops. The periodic flashes reflect the capillary oscillations of the liquid drop between alternating prolate and oblate shapes, and the dramatic enhancement in the oblate phase results from a combination of several optical effects. Ray tracing analysis shows that the flashes occur when the rainbow angle is 42° in spherical drops but sweeps over a wide range between 35° and 65° for typical ellipsoidal drops, and the intensity of the caustic is strongly enhanced in the oblate phase. This phenomenon can be seen in all brightly lit water sprays with millimeter size drops and is responsible for their white color.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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2007

1991

1990

D. H. Peregrine, G. Shoker, and A. Symon, “The bifurcation of liquid bridges,” J. Fluid Mech. 212, 25-39 (1990).
[CrossRef]

1980

1959

S. Chandrasekhar, “The oscillations of a viscous liquid globe,” Proc. London Math. Soc. s3-9, 141-149 (1959).
[CrossRef]

1892

J. W. S. Rayleigh, “On the stability of cylindrical fluid surfaces,” Phil. Mag. 34, 177 -181 (1892).

1877

Ph. Lenard, “Über die Schwingungen fallender Tropfen,” Ann. Phys. Chem. 30, 209 (1877).

Rayleigh, J. W. S.

J. W. S. Rayleigh, “On the stability of cylindrical fluid surfaces,” Phil. Mag. 34, 177 -181 (1892).

Bohren, C. E.

C. E. Bohren and D. R. Huffman, Absorption and Scattering or Light by Small Particles (Wiley-Interscience, 1983).

Brochard-Wyart, F.

P.-G. de Gennes, F. Brochard-Wyart, D. Qur, and A. Reisinger, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004).

Chandrasekhar, S.

S. Chandrasekhar, “The oscillations of a viscous liquid globe,” Proc. London Math. Soc. s3-9, 141-149 (1959).
[CrossRef]

Clift, R.

R. Clift, J. R. Grace, and M. E. Weber, Bubbles Drops and Particles (Academic, 1978), p. 185.

de Gennes, P.-G.

P.-G. de Gennes, F. Brochard-Wyart, D. Qur, and A. Reisinger, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004).

Descartes, René

René Descartes, Discourse on Method (Cambridge University Press, 1637).

Grace, J. R.

R. Clift, J. R. Grace, and M. E. Weber, Bubbles Drops and Particles (Academic, 1978), p. 185.

Han, X.

Huffman, D. R.

C. E. Bohren and D. R. Huffman, Absorption and Scattering or Light by Small Particles (Wiley-Interscience, 1983).

Jiang, Huifen

Lenard, Ph.

Ph. Lenard, “Über die Schwingungen fallender Tropfen,” Ann. Phys. Chem. 30, 209 (1877).

Li, R.

Marston, P. L.

Peregrine, D. H.

D. H. Peregrine, G. Shoker, and A. Symon, “The bifurcation of liquid bridges,” J. Fluid Mech. 212, 25-39 (1990).
[CrossRef]

Qur, D.

P.-G. de Gennes, F. Brochard-Wyart, D. Qur, and A. Reisinger, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004).

Reisinger, A.

P.-G. de Gennes, F. Brochard-Wyart, D. Qur, and A. Reisinger, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004).

Ren, K. F.

Shi, L.

Shoker, G.

D. H. Peregrine, G. Shoker, and A. Symon, “The bifurcation of liquid bridges,” J. Fluid Mech. 212, 25-39 (1990).
[CrossRef]

Symon, A.

D. H. Peregrine, G. Shoker, and A. Symon, “The bifurcation of liquid bridges,” J. Fluid Mech. 212, 25-39 (1990).
[CrossRef]

van de Hulst, H. C.

Volz, F. E.

F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” Physics of Precipitation, H. Weickmann, ed. (American Geophysical Union, 1960), Monograph 5, pp. 280-286.

Wang, R. T.

Weber, M. E.

R. Clift, J. R. Grace, and M. E. Weber, Bubbles Drops and Particles (Academic, 1978), p. 185.

Ann. Phys. Chem.

Ph. Lenard, “Über die Schwingungen fallender Tropfen,” Ann. Phys. Chem. 30, 209 (1877).

Appl. Opt.

J. Fluid Mech.

D. H. Peregrine, G. Shoker, and A. Symon, “The bifurcation of liquid bridges,” J. Fluid Mech. 212, 25-39 (1990).
[CrossRef]

Phil. Mag.

J. W. S. Rayleigh, “On the stability of cylindrical fluid surfaces,” Phil. Mag. 34, 177 -181 (1892).

Proc. London Math. Soc.

S. Chandrasekhar, “The oscillations of a viscous liquid globe,” Proc. London Math. Soc. s3-9, 141-149 (1959).
[CrossRef]

Other

F. E. Volz, “Some aspects of the optics of the rainbow and the physics of rain,” Physics of Precipitation, H. Weickmann, ed. (American Geophysical Union, 1960), Monograph 5, pp. 280-286.

P.-G. de Gennes, F. Brochard-Wyart, D. Qur, and A. Reisinger, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2004).

René Descartes, Discourse on Method (Cambridge University Press, 1637).

C. E. Bohren and D. R. Huffman, Absorption and Scattering or Light by Small Particles (Wiley-Interscience, 1983).

R. Clift, J. R. Grace, and M. E. Weber, Bubbles Drops and Particles (Academic, 1978), p. 185.

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Figures (6)

Fig. 1
Fig. 1

Geometry used to photograph falling drops. (a) Ellipsoidal drop at the origin with its symmetry axis vertical. The light source is located at L at an angle θ above the horizontal. The camera at C is at an angle ϕ above the horizontal plane. The scattering angle α is the angle between the light source and the observer at C. (b) Rays propagating inside the drop.

Fig. 2
Fig. 2

Stroboscopic photo of two water drops from a dripping faucet. Images are 10 ms apart. (a) Sequence of photos that starts at the time of release of the drop, and (b) a sequence that starts somewhat later. We show that the bright spots seen on the lower left of certain drops are the result of total internal reflection of the light source (a camera flash in the stroboscopic mode) incident from the upper right. The water drops that show the bright spots are oblate in shape. The intermediate, dark drops are either spherical or prolate. The elapsed time between the oblate phases is 30 ms , consistent with the calculated time for capillary shape oscillations of the drop.

Fig. 3
Fig. 3

Time exposure of a water drop on the left, and a stroboscopic image of a drop under the same conditions of flow rate on the right.

Fig. 4
Fig. 4

“Rainbow” images of backscattering from ellipsoidal water drops. Light is incident at 45 ° from the vertical symmetry direction of the drop. Various eccentricities are shown. The dashed circle is where the rainbow from spherical drops is located, 42 ° from the incident direction marked by a star at the center of the figure. Ellipsoidal drops give rise to the elliptically shaped rainbows as shown. However a rapidly oscillating drop would show elliptical patterns that change in size and shape as seen in the figure. The oblate ellipsoid gives rise to a particularly well defined ellipsoid. As the shape approaches this form, the scattered efficiency increases dramatically in strength and angular coverage, up to 65 ° from the incident direction. The dashed vertical box shows the region of angles that are included in Fig. 5.

Fig. 5
Fig. 5

Differential scattering efficiency of light Q scattered from ellipsoidal drops, plotted as a function of the angle from the incident direction ( 45 ° above the horizontal plane) in a vertical direction. From left to right curves range in eccentricity ϵ from ϵ = 0.06 (prolate ellipsoid) to sphere ϵ = 0.00 to oblate ellipsoid ϵ = + 0.06 . As the shape approaches the oblate ellipsoid form, the scattered efficiency increases dramatically in strength and angular size.

Fig. 6
Fig. 6

(a) Flash photograph of a water spray from a collection of 1 mm diameter nozzles delivering drops with an initial velocity of 300 cm / s . The flash is held 45 ° from the camera’s optic axis. Note the irregular shape of the larger drops. Some of the drops show bright flashes of light near the edge of the drop. (b) Photograph of the same spray as in (a) but using an incandescent source and a time exposure. Strips of light can be seen. Their oval shape suggests they arise from the internal reflection of light where two virtual images of the source merge at the “rainbow” angle of maximum deviation. These strips are a few centimeters apart consistent with the notion that they related to capillary oscillations of the mm size drops seen in (a).

Equations (3)

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ω n 2 = n [ n 1 ( n + 2 ) ] σ ρ a 3 ,
ω 2 2 = 8 σ ρ a 3 .
1 / τ = ν ( n 1 ) ( 2 n + 1 ) / a 2 ;

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