Abstract

We describe twin-rainbow metrology, a new optical technique used to measure the thickness of thin films in a cylindrical geometry. We also present an application of the technique: measurement of the thickness of a Newtonian fluid draining under gravity. We compare these measurements with fluid mechanics models.

© 2003 Optical Society of America

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References

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  15. A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
    [CrossRef]
  16. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1836).
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    [CrossRef]
  20. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 240–246.
  21. M. Abromowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 478.
  22. H. Jeffreys, “The drainage of a vertical plate,” Proc. Cambridge Philos. Soc. 26, 204–205 (1930).
    [CrossRef]
  23. C. Gutfinger, J. A. Tallmadge, “Some remarks on the problem of drainage of fluids on vertical surfaces,” AIChE. J. 10, 774–780 (1964).
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  24. L. H. Tanner, “The form and motion of draining oil drops,” J. Phys. D 18, 1311–1326 (1985).
    [CrossRef]
  25. D. Ramakrishna, N. Ch. Pattabhiramacharyulu, “Dusty viscous drainage on a vertical cylinder,” Proc. Indian Acad. Sci. Sect. A 53, 257–266 (1987).
  26. P. C. Hiemenz, Principles of Colloid and Surface Chemistry, 2nd. ed. (Marcel Dekker, New York, 1986), pp. 287–299.
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  28. C. L. Adler, J. A. Lock, B. R. Stone, C. J. Garcia, “High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1305–1315 (1997).
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2002 (1)

2001 (2)

2000 (1)

T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
[CrossRef]

1999 (4)

T. Nousiainen, K. Muinonen, “Light scattering by Gaussian randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
[CrossRef]

P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transfer 63, 341–351 (1999).
[CrossRef]

H. Hattori, “Simulation study on refractometry by the rainbow method,” Appl. Opt. 38, 4037–4046 (1999).
[CrossRef]

1998 (2)

1997 (2)

1996 (1)

1994 (1)

1987 (1)

D. Ramakrishna, N. Ch. Pattabhiramacharyulu, “Dusty viscous drainage on a vertical cylinder,” Proc. Indian Acad. Sci. Sect. A 53, 257–266 (1987).

1985 (1)

L. H. Tanner, “The form and motion of draining oil drops,” J. Phys. D 18, 1311–1326 (1985).
[CrossRef]

1979 (1)

1977 (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

1964 (1)

C. Gutfinger, J. A. Tallmadge, “Some remarks on the problem of drainage of fluids on vertical surfaces,” AIChE. J. 10, 774–780 (1964).
[CrossRef]

1951 (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1950 (1)

1930 (1)

H. Jeffreys, “The drainage of a vertical plate,” Proc. Cambridge Philos. Soc. 26, 204–205 (1930).
[CrossRef]

1836 (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1836).

Aden, A. L.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Adler, C. L.

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1836).

Chan, C. W.

Chigier, N.

Garcia, C. J.

Giannoulis, D.

Greenler, R.

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980), pp. 8–10.

Gutfinger, C.

C. Gutfinger, J. A. Tallmadge, “Some remarks on the problem of drainage of fluids on vertical surfaces,” AIChE. J. 10, 774–780 (1964).
[CrossRef]

Hattori, H.

Hiemenz, P. C.

P. C. Hiemenz, Principles of Colloid and Surface Chemistry, 2nd. ed. (Marcel Dekker, New York, 1986), pp. 287–299.

Hom, J.

Jamison, J. M.

Jeffreys, H.

H. Jeffreys, “The drainage of a vertical plate,” Proc. Cambridge Philos. Soc. 26, 204–205 (1930).
[CrossRef]

Kagawa, K.

Kakui, H.

Keller, H. B.

Keller, J. B.

Kerker, M.

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kurniawan, H.

Lee, W. K.

Lin, C.-Y.

Lock, J. A.

Marston, P. L.

P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transfer 63, 341–351 (1999).
[CrossRef]

Minnaert, M.

M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 195–197.

Muinonen, K.

T. Nousiainen, K. Muinonen, “Light scattering by Gaussian randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

Nash, J.

Nash, J. K.

Nousiainen, T.

T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
[CrossRef]

T. Nousiainen, K. Muinonen, “Light scattering by Gaussian randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

Nussenzveig, H. M.

Pattabhiramacharyulu, N. Ch.

D. Ramakrishna, N. Ch. Pattabhiramacharyulu, “Dusty viscous drainage on a vertical cylinder,” Proc. Indian Acad. Sci. Sect. A 53, 257–266 (1987).

Phipps, D.

Ramakrishna, D.

D. Ramakrishna, N. Ch. Pattabhiramacharyulu, “Dusty viscous drainage on a vertical cylinder,” Proc. Indian Acad. Sci. Sect. A 53, 257–266 (1987).

Riethmuller, M. L.

Saunders, K.

Saunders, K. W.

Stone, B. R.

Tallmadge, J. A.

C. Gutfinger, J. A. Tallmadge, “Some remarks on the problem of drainage of fluids on vertical surfaces,” AIChE. J. 10, 774–780 (1964).
[CrossRef]

Tanner, L. H.

L. H. Tanner, “The form and motion of draining oil drops,” J. Phys. D 18, 1311–1326 (1985).
[CrossRef]

van Beeck, J. P. A. J.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 240–246.

Yamanaka, H.

Yokoi, S.

Zimmer, L.

AIChE. J. (1)

C. Gutfinger, J. A. Tallmadge, “Some remarks on the problem of drainage of fluids on vertical surfaces,” AIChE. J. 10, 774–780 (1964).
[CrossRef]

Appl. Opt. (8)

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

J. Atmos. Sci. (1)

T. Nousiainen, “Scattering of light by raindrops with single-mode oscillations,” J. Atmos. Sci. 57, 789–802 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

L. H. Tanner, “The form and motion of draining oil drops,” J. Phys. D 18, 1311–1326 (1985).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

P. L. Marston, “Catastrophe optics of spheroidal drops and generalized rainbows,” J. Quant. Spectrosc. Radiat. Transfer 63, 341–351 (1999).
[CrossRef]

T. Nousiainen, K. Muinonen, “Light scattering by Gaussian randomly oscillating raindrops,” J. Quant. Spectrosc. Radiat. Transfer 63, 643–666 (1999).
[CrossRef]

Opt. Lett. (1)

Proc. Cambridge Philos. Soc. (1)

H. Jeffreys, “The drainage of a vertical plate,” Proc. Cambridge Philos. Soc. 26, 204–205 (1930).
[CrossRef]

Proc. Indian Acad. Sci. Sect. A (1)

D. Ramakrishna, N. Ch. Pattabhiramacharyulu, “Dusty viscous drainage on a vertical cylinder,” Proc. Indian Acad. Sci. Sect. A 53, 257–266 (1987).

Sci. Am. (1)

H. M. Nussenzveig, “The theory of the rainbow,” Sci. Am. 236, 116–127 (1977).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 (1836).

Other (5)

R. Greenler, Rainbows, Halos and Glories (Cambridge U. Press, Cambridge, UK, 1980), pp. 8–10.

M. Minnaert, Light and Color in the Outdoors (Springer-Verlag, New York, 1993), pp. 195–197.

P. C. Hiemenz, Principles of Colloid and Surface Chemistry, 2nd. ed. (Marcel Dekker, New York, 1986), pp. 287–299.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 240–246.

M. Abromowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970), p. 478.

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Figures (11)

Fig. 1
Fig. 1

Coated cylinder cross section and the trajectory of the twin-rainbow rays.

Fig. 2
Fig. 2

Experimental setup for twin-rainbow metrology.

Fig. 3
Fig. 3

Theoretical profile of a draining thin film as a function of time for μ = 12 cm2/s. Note the different scales for the r and z axes.

Fig. 4
Fig. 4

Video image of twin rainbows for μ = 0.015 cm2/s and t = 10 s.

Fig. 5
Fig. 5

(a) α rainbow gap width Z α 0(t) as a function of time for several different viscosity silicone oils. (b) β rainbow gap width Z β 0(t) as a function of time for several different viscosity silicone oils.

Fig. 6
Fig. 6

Comparison of inverted TRM data (solid curves) to predictions made by the Jeffreys theory (filled circles). Data were taken from the α rainbow for μ = 10 cm2/s nominal viscosity. The best-fit viscosity to the data is μ = 12 cm2/s.

Fig. 7
Fig. 7

Comparison of inverted TRM data for the α (filled squares) and β rainbows (filled triangles) at t = 120 s. The nominal viscosity for the fluid is μ = 10 cm2/s. The best-fit viscosity to the data is μ = 12 cm2/s.

Fig. 8
Fig. 8

Video image of the twin-rainbow interference pattern for μ = 0.015 cm2/s and t = 45 s.

Fig. 9
Fig. 9

Computer simulation of the twin-rainbow interference pattern of Fig. 8.

Fig. 10
Fig. 10

(a) Theoretical profile of the liquid thin film including effects of surface tension; (b) predicted shape of the α rainbow including effects of surface tension for μ = 0.015 cm2/s and t = 20 s.

Fig. 11
Fig. 11

Contact line position as a function of time for μ = 0.015 cm2/s.

Tables (1)

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Table 1 Power-Law Scaling Coefficients Aα,β for the Rainbow Gapa

Equations (33)

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cos θiR=n2-1p2-11/2,
ΘR=2θiR+p-1π-2pθtR,
sin θtR=sin θiRn=p/n2-1p2-11/2.
θαR=θR+2ra4-n123n22+n12-41/2-4-n12n12-11/2,
θβR=θR+2ra24-n123n22+n12-41/2-4-n12n12-11/2.
Ai-w=31/3π0cost3-31/3wtdt,
Eθ  Ai- X2/3h1/3θ-θR,
h=944-n121/2n12-13/2
Iθ=I1Ai- X2/3h1/3θ-θαR2+I2Ai- X2/3h1/3θ-θβR2+2I1I21/2Ai- X2/3h1/3θ-θαR×Ai- X2/3h1/3θ-θβR×cos4πrλ3n22+n12-431/2,
z=gμ r2t,
Zα,β=z1+RLf+γα,βRdz/dr,
γα=231/23n22+n12-41/2-n12-11/2,
γβ=231/2 23n22+n12-41/2-n12-11/2.
Zα,β0t=32γα,β2R21+RLf1/3μ2gt1/3.
Zα,β0t=Aα,βt-1/3+Bα,β.
dzdr=γα,βRZα,βr-1+R/Lfz.
limrzr=Zα,βr1+R/Lf,
rint  a23n22+n12-44-n121/2h1/3X2/3  3 μm.
Δrb=λ233n22+n12-41/2 =0.277 μm.
z=gμ r2t+kσrρg1/3.
ΔZα,β0=231+RLf8/9 kLσ2/3γα,βRμ24g21/9 t- 2/9.
rz=μzgt1/2,
n0 = cosγcosψ0ux - cosγsinψ0uy + sinγuz,
tan γ=drdz0.
cos ψ0=n12-131/2, sin ψ0=4-n1231/2
ξn22-sin2 ψ01/2-cos ψ0γn2.
n1 sin η=n2 sin ξ.
ξ3ξ+2γn2n22-sin2 ψ01/2
n1 sin η3=n2 sin ξ3,
ξ6n2ξ3-γcos ψ0-n22-sin2 ψ01/2=2γ31/223n22+n12-41/2-n12-11/2,
ξ4n2ξ-γcos ψ0-n22-sin2 ψ01/2]=2γ31/23n22+n12-41/2-n12-11/2
Zα,β=z+d+R-αξα,β,
Zα,β=z+R-az-zcf+R-aξα,β,

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