Abstract

The heating of a laser-irradiated droplet is analyzed theoretically and numerically by solving the heat transport equation. Two regimes of droplet heating are considered, slow and fast heating. In the slow heating regime, the thermal diffusion term plays an important role and the droplet may not experience explosive vaporization during the lifetime of the laser pulse. In the fast heating regime, the vaporization term plays the dominant role and the temperature profile inside the droplet is similar to the heat production profile except for a thin shell near the surface. Numerical results are presented for the case of water droplets irradiated by 10.6 μm laser radiation.

© 1989 Optical Society of America

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  1. R. L. Armstrong, S. A. W. Gerstl, A. Zardecki, “Nonlinear Pulse Propagation in the Presence of Evaporating Aerosols,” J. Opt. Soc. Am. A 2, 1739 (1985).
    [CrossRef]
  2. K. D. Egorov, V. P. Kandidov, S. S. Chesnokov, “Numerical Analysis of the Propagation of High-Intensity Laser Radiation in the Atmosphere,” Sov. Phys. J. 2, 66 (1983).
  3. A. V. Kusikovskii, L. K. Chistyakova, V. I. Kokhanov, “Pulsed Clearing of a Synthetic Aqueous Aerosol by CO2 Laser Radiation,” Sov. J. Quantum Electron. 11, 1277 (1981).
    [CrossRef]
  4. V. E. Zuev, Y. D. Kopytin, “Nonlinear Propagation of Intense Light in a Gaseous Medium with Solid Microfilling,” Sov. Phys. J. 11, 79 (1977).
  5. P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers Within Irradiated Spheres,” J. Opt. Soc. Am. 69, 55 (1979).
    [CrossRef]
  6. J. D. Pendleton, “Water Droplets Irradiated by a Pulsed CO2 Laser: Comparison of Computed Temperature Contours with Explosive Vaporization Patterns,” Appl. Opt. 24, 1631 (1985).
    [CrossRef] [PubMed]
  7. A. P. Prishivalko, “Heating and Destruction of Water Drops on Exposure to Radiation with Inhomogeneous Internal Heat Evolution,” Sov. Phys. J. 26, 142 (1983).
    [CrossRef]
  8. A. P. Prishivalko, S. T. Leiko, “Radiative Heating and Evaporation of Droplets,” J. Appl. Spectrosc. 33, 1137 (1980).
    [CrossRef]
  9. A. Biswas, H. Latifi, P. Shah, L. J. Radziemski, R. L. Armstrong, “Time-Resolved Spectroscopy of Plasmas Initiated on Single, Levitated Aerosol Droplets,” Opt. Lett. 12, 313 (1987).
    [CrossRef] [PubMed]
  10. J. H. Eickmans, S. -F. Hsieh, R. K. Chang, “Laser-Induced Explosion of H2O Droplets: Spatially Resolved Spectra,” Opt. Lett. 12, 22 (1987).
    [CrossRef] [PubMed]
  11. R. G. Pinnick, A. Biswas, P. Chylek, R. L. Armstrong, H. Latifi, E. Creegan, V. Srivastava, M. Jarzembski, G. Fernandez, “Stimulated Raman Scattering from Micron-Sized Droplets: Time Resolved Measurements,” Opt. Lett. 13, 494 (1988).
    [CrossRef] [PubMed]
  12. J. -Z. Zhang, D. H. Leach, R. K. Chang, “Photon Lifetime Within a Droplet: Temporal Determination of Elastic and Stimulated Raman Scattering,” Opt. Lett. 13, 270 (1988).
    [CrossRef] [PubMed]
  13. R. L. Armstrong, “Interactions of Absorbing Aerosols with Intense Light Beams,” J. Appl. Phys. 56, 2142 (1984).
    [CrossRef]
  14. R. Cole, in Advances in Heat Transfer, Vol. 10, J. P. Hartnett, T. F. Irvine, Eds. (Academic, New York, 1974).
    [CrossRef]
  15. V. P. Skirpov, Metastable Liquids (Wiley, New York, 1974).
  16. R. L. Armstrong, “Aerosol Heating and Vaporization by Pulsed Light Beams,” Appl. Opt. 23, 148 (1984).
    [CrossRef] [PubMed]
  17. A. Zardecki, R. L. Armstrong, “Energy Balance in Laser-Irradiated Vaporizing Droplets,” Appl. Opt. 27, 3690 (1988).
    [CrossRef] [PubMed]
  18. S. C. Davies, J. R. Brock, “Laser Evaporation of Droplets,” Appl. Opt. 26, 786 (1987), “Laser Evaporation of Droplets: Errata,” 26, 4036 (1987).
    [CrossRef] [PubMed]
  19. R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Irradiated Droplets,” Phys. Fluids 29, 3573 (1986).
    [CrossRef]
  20. R. L. Armstrong, A. Zardecki, “Diffusive and Convective Vaporization of Irradiated Droplets,” J. Appl. Phys. 62, 4571 (1987).
    [CrossRef]
  21. J. C. Carls, J. R. Brock, “Explosion of a Water Droplet by Pulsed Laser Heating,” Aerosol Sci. Technol. 7, 79 (1987).
    [CrossRef]
  22. N. G. Kondrashov, A. P. Prishivalko, “Approximate Solution for the Evaporation of a Spherical Particle with an Inhomogeneous Distribution of Internal Heat Sources,” Dokl. Akad. Nauk BSSR 19, 984 (1975).
  23. F. A. Williams, “On Vaporization of Mist by Radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
    [CrossRef]
  24. C. J. Knight, “Theoretical Modeling of Rapid Surface Vaporization with Back Pressure,” AIAA J. 17, 519 (1979).
    [CrossRef]
  25. A. V. Kuzikovskii, “Dynamics of a Spherical Particle in an Intense Optical Field,” Sov. Phys. J 13, 615 (1970).
  26. A. P. Prishivalko, M. S. Veremchuk, “Heating and Evaporation of Water Drops with Insoluble Absorbing Core in the Case of Inhomogeneous Internal Heat Liberation,” Dokl. Akad. Nauk BSSR 25, 305 (1981).
  27. L. W. Pinkley, P. P. Sethna, D. Williams, “Optical Constants of Water in the Infrared: Influence of Temperature,” J. Opt. Soc. Am. 67, 494 (1977).
    [CrossRef]

1988 (3)

1987 (5)

1986 (1)

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Irradiated Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

1985 (2)

1984 (2)

R. L. Armstrong, “Interactions of Absorbing Aerosols with Intense Light Beams,” J. Appl. Phys. 56, 2142 (1984).
[CrossRef]

R. L. Armstrong, “Aerosol Heating and Vaporization by Pulsed Light Beams,” Appl. Opt. 23, 148 (1984).
[CrossRef] [PubMed]

1983 (2)

K. D. Egorov, V. P. Kandidov, S. S. Chesnokov, “Numerical Analysis of the Propagation of High-Intensity Laser Radiation in the Atmosphere,” Sov. Phys. J. 2, 66 (1983).

A. P. Prishivalko, “Heating and Destruction of Water Drops on Exposure to Radiation with Inhomogeneous Internal Heat Evolution,” Sov. Phys. J. 26, 142 (1983).
[CrossRef]

1981 (2)

A. V. Kusikovskii, L. K. Chistyakova, V. I. Kokhanov, “Pulsed Clearing of a Synthetic Aqueous Aerosol by CO2 Laser Radiation,” Sov. J. Quantum Electron. 11, 1277 (1981).
[CrossRef]

A. P. Prishivalko, M. S. Veremchuk, “Heating and Evaporation of Water Drops with Insoluble Absorbing Core in the Case of Inhomogeneous Internal Heat Liberation,” Dokl. Akad. Nauk BSSR 25, 305 (1981).

1980 (1)

A. P. Prishivalko, S. T. Leiko, “Radiative Heating and Evaporation of Droplets,” J. Appl. Spectrosc. 33, 1137 (1980).
[CrossRef]

1979 (2)

P. W. Dusel, M. Kerker, D. D. Cooke, “Distribution of Absorption Centers Within Irradiated Spheres,” J. Opt. Soc. Am. 69, 55 (1979).
[CrossRef]

C. J. Knight, “Theoretical Modeling of Rapid Surface Vaporization with Back Pressure,” AIAA J. 17, 519 (1979).
[CrossRef]

1977 (2)

L. W. Pinkley, P. P. Sethna, D. Williams, “Optical Constants of Water in the Infrared: Influence of Temperature,” J. Opt. Soc. Am. 67, 494 (1977).
[CrossRef]

V. E. Zuev, Y. D. Kopytin, “Nonlinear Propagation of Intense Light in a Gaseous Medium with Solid Microfilling,” Sov. Phys. J. 11, 79 (1977).

1975 (1)

N. G. Kondrashov, A. P. Prishivalko, “Approximate Solution for the Evaporation of a Spherical Particle with an Inhomogeneous Distribution of Internal Heat Sources,” Dokl. Akad. Nauk BSSR 19, 984 (1975).

1970 (1)

A. V. Kuzikovskii, “Dynamics of a Spherical Particle in an Intense Optical Field,” Sov. Phys. J 13, 615 (1970).

1965 (1)

F. A. Williams, “On Vaporization of Mist by Radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
[CrossRef]

Armstrong, R. L.

Biswas, A.

Brock, J. R.

J. C. Carls, J. R. Brock, “Explosion of a Water Droplet by Pulsed Laser Heating,” Aerosol Sci. Technol. 7, 79 (1987).
[CrossRef]

S. C. Davies, J. R. Brock, “Laser Evaporation of Droplets,” Appl. Opt. 26, 786 (1987), “Laser Evaporation of Droplets: Errata,” 26, 4036 (1987).
[CrossRef] [PubMed]

Carls, J. C.

J. C. Carls, J. R. Brock, “Explosion of a Water Droplet by Pulsed Laser Heating,” Aerosol Sci. Technol. 7, 79 (1987).
[CrossRef]

Chang, R. K.

Chesnokov, S. S.

K. D. Egorov, V. P. Kandidov, S. S. Chesnokov, “Numerical Analysis of the Propagation of High-Intensity Laser Radiation in the Atmosphere,” Sov. Phys. J. 2, 66 (1983).

Chistyakova, L. K.

A. V. Kusikovskii, L. K. Chistyakova, V. I. Kokhanov, “Pulsed Clearing of a Synthetic Aqueous Aerosol by CO2 Laser Radiation,” Sov. J. Quantum Electron. 11, 1277 (1981).
[CrossRef]

Chylek, P.

Cole, R.

R. Cole, in Advances in Heat Transfer, Vol. 10, J. P. Hartnett, T. F. Irvine, Eds. (Academic, New York, 1974).
[CrossRef]

Cooke, D. D.

Creegan, E.

Davies, S. C.

Dusel, P. W.

Egorov, K. D.

K. D. Egorov, V. P. Kandidov, S. S. Chesnokov, “Numerical Analysis of the Propagation of High-Intensity Laser Radiation in the Atmosphere,” Sov. Phys. J. 2, 66 (1983).

Eickmans, J. H.

Fernandez, G.

Gerstl, S. A. W.

Hsieh, S. -F.

Jarzembski, M.

Kandidov, V. P.

K. D. Egorov, V. P. Kandidov, S. S. Chesnokov, “Numerical Analysis of the Propagation of High-Intensity Laser Radiation in the Atmosphere,” Sov. Phys. J. 2, 66 (1983).

Kerker, M.

Knight, C. J.

C. J. Knight, “Theoretical Modeling of Rapid Surface Vaporization with Back Pressure,” AIAA J. 17, 519 (1979).
[CrossRef]

Kokhanov, V. I.

A. V. Kusikovskii, L. K. Chistyakova, V. I. Kokhanov, “Pulsed Clearing of a Synthetic Aqueous Aerosol by CO2 Laser Radiation,” Sov. J. Quantum Electron. 11, 1277 (1981).
[CrossRef]

Kondrashov, N. G.

N. G. Kondrashov, A. P. Prishivalko, “Approximate Solution for the Evaporation of a Spherical Particle with an Inhomogeneous Distribution of Internal Heat Sources,” Dokl. Akad. Nauk BSSR 19, 984 (1975).

Kopytin, Y. D.

V. E. Zuev, Y. D. Kopytin, “Nonlinear Propagation of Intense Light in a Gaseous Medium with Solid Microfilling,” Sov. Phys. J. 11, 79 (1977).

Kusikovskii, A. V.

A. V. Kusikovskii, L. K. Chistyakova, V. I. Kokhanov, “Pulsed Clearing of a Synthetic Aqueous Aerosol by CO2 Laser Radiation,” Sov. J. Quantum Electron. 11, 1277 (1981).
[CrossRef]

Kuzikovskii, A. V.

A. V. Kuzikovskii, “Dynamics of a Spherical Particle in an Intense Optical Field,” Sov. Phys. J 13, 615 (1970).

Latifi, H.

Leach, D. H.

Leiko, S. T.

A. P. Prishivalko, S. T. Leiko, “Radiative Heating and Evaporation of Droplets,” J. Appl. Spectrosc. 33, 1137 (1980).
[CrossRef]

O’Rourke, P. J.

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Irradiated Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

Pendleton, J. D.

Pinkley, L. W.

Pinnick, R. G.

Prishivalko, A. P.

A. P. Prishivalko, “Heating and Destruction of Water Drops on Exposure to Radiation with Inhomogeneous Internal Heat Evolution,” Sov. Phys. J. 26, 142 (1983).
[CrossRef]

A. P. Prishivalko, M. S. Veremchuk, “Heating and Evaporation of Water Drops with Insoluble Absorbing Core in the Case of Inhomogeneous Internal Heat Liberation,” Dokl. Akad. Nauk BSSR 25, 305 (1981).

A. P. Prishivalko, S. T. Leiko, “Radiative Heating and Evaporation of Droplets,” J. Appl. Spectrosc. 33, 1137 (1980).
[CrossRef]

N. G. Kondrashov, A. P. Prishivalko, “Approximate Solution for the Evaporation of a Spherical Particle with an Inhomogeneous Distribution of Internal Heat Sources,” Dokl. Akad. Nauk BSSR 19, 984 (1975).

Radziemski, L. J.

Sethna, P. P.

Shah, P.

Skirpov, V. P.

V. P. Skirpov, Metastable Liquids (Wiley, New York, 1974).

Srivastava, V.

Veremchuk, M. S.

A. P. Prishivalko, M. S. Veremchuk, “Heating and Evaporation of Water Drops with Insoluble Absorbing Core in the Case of Inhomogeneous Internal Heat Liberation,” Dokl. Akad. Nauk BSSR 25, 305 (1981).

Williams, D.

Williams, F. A.

F. A. Williams, “On Vaporization of Mist by Radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
[CrossRef]

Zardecki, A.

A. Zardecki, R. L. Armstrong, “Energy Balance in Laser-Irradiated Vaporizing Droplets,” Appl. Opt. 27, 3690 (1988).
[CrossRef] [PubMed]

R. L. Armstrong, A. Zardecki, “Diffusive and Convective Vaporization of Irradiated Droplets,” J. Appl. Phys. 62, 4571 (1987).
[CrossRef]

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Irradiated Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

R. L. Armstrong, S. A. W. Gerstl, A. Zardecki, “Nonlinear Pulse Propagation in the Presence of Evaporating Aerosols,” J. Opt. Soc. Am. A 2, 1739 (1985).
[CrossRef]

Zhang, J. -Z.

Zuev, V. E.

V. E. Zuev, Y. D. Kopytin, “Nonlinear Propagation of Intense Light in a Gaseous Medium with Solid Microfilling,” Sov. Phys. J. 11, 79 (1977).

Aerosol Sci. Technol. (1)

J. C. Carls, J. R. Brock, “Explosion of a Water Droplet by Pulsed Laser Heating,” Aerosol Sci. Technol. 7, 79 (1987).
[CrossRef]

AIAA J. (1)

C. J. Knight, “Theoretical Modeling of Rapid Surface Vaporization with Back Pressure,” AIAA J. 17, 519 (1979).
[CrossRef]

Appl. Opt. (4)

Dokl. Akad. Nauk BSSR (2)

A. P. Prishivalko, M. S. Veremchuk, “Heating and Evaporation of Water Drops with Insoluble Absorbing Core in the Case of Inhomogeneous Internal Heat Liberation,” Dokl. Akad. Nauk BSSR 25, 305 (1981).

N. G. Kondrashov, A. P. Prishivalko, “Approximate Solution for the Evaporation of a Spherical Particle with an Inhomogeneous Distribution of Internal Heat Sources,” Dokl. Akad. Nauk BSSR 19, 984 (1975).

Int. J. Heat Mass Transfer (1)

F. A. Williams, “On Vaporization of Mist by Radiation,” Int. J. Heat Mass Transfer 8, 575 (1965).
[CrossRef]

J. Appl. Phys. (2)

R. L. Armstrong, A. Zardecki, “Diffusive and Convective Vaporization of Irradiated Droplets,” J. Appl. Phys. 62, 4571 (1987).
[CrossRef]

R. L. Armstrong, “Interactions of Absorbing Aerosols with Intense Light Beams,” J. Appl. Phys. 56, 2142 (1984).
[CrossRef]

J. Appl. Spectrosc. (1)

A. P. Prishivalko, S. T. Leiko, “Radiative Heating and Evaporation of Droplets,” J. Appl. Spectrosc. 33, 1137 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (4)

Phys. Fluids (1)

R. L. Armstrong, P. J. O’Rourke, A. Zardecki, “Vaporization of Irradiated Droplets,” Phys. Fluids 29, 3573 (1986).
[CrossRef]

Sov. J. Quantum Electron. (1)

A. V. Kusikovskii, L. K. Chistyakova, V. I. Kokhanov, “Pulsed Clearing of a Synthetic Aqueous Aerosol by CO2 Laser Radiation,” Sov. J. Quantum Electron. 11, 1277 (1981).
[CrossRef]

Sov. Phys. J (1)

A. V. Kuzikovskii, “Dynamics of a Spherical Particle in an Intense Optical Field,” Sov. Phys. J 13, 615 (1970).

Sov. Phys. J. (3)

V. E. Zuev, Y. D. Kopytin, “Nonlinear Propagation of Intense Light in a Gaseous Medium with Solid Microfilling,” Sov. Phys. J. 11, 79 (1977).

K. D. Egorov, V. P. Kandidov, S. S. Chesnokov, “Numerical Analysis of the Propagation of High-Intensity Laser Radiation in the Atmosphere,” Sov. Phys. J. 2, 66 (1983).

A. P. Prishivalko, “Heating and Destruction of Water Drops on Exposure to Radiation with Inhomogeneous Internal Heat Evolution,” Sov. Phys. J. 26, 142 (1983).
[CrossRef]

Other (2)

R. Cole, in Advances in Heat Transfer, Vol. 10, J. P. Hartnett, T. F. Irvine, Eds. (Academic, New York, 1974).
[CrossRef]

V. P. Skirpov, Metastable Liquids (Wiley, New York, 1974).

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Figures (13)

Fig. 1
Fig. 1

Source function distribution for λ0 = 10.6 μm along the droplet diameter parallel to the propagation direction of the incident laser beam (from left to right). The numbers next to the curve are the values of R0 in μm. In the figure, the radius is normalized to 1, i.e., r ¯ = r / R 0.

Fig. 2
Fig. 2

Temperature distribution along the droplet diameter parallel to the incident laser at times, t = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, and 1.41 μs λ0 = 10.6 μm. R0 = 10.0 μm. α = 1.0. I0 = 106 W/cm2. Hypothetical source function S = 1.0 inside of the droplet.

Fig. 3
Fig. 3

Surface temperature variation with time under the same conditions of Fig. 2.

Fig. 4
Fig. 4

Energy loss rate through each surface term in Eq. (4) with time under the same conditions of Fig. 2. Vaporization (a), shrinkage (b) and convection (c).

Fig. 5
Fig. 5

Temperature distribution along the droplet diameter parallel to the incident laser at times, t = .1, .2, .3, .4, .5, .6 and .7 ms. (b) For t = 0.7, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8 and 2.0 ms. λ = 10.6 μm. R0 = 10.0 μm. I0 = 103W/cm2.

Fig. 6
Fig. 6

Surface temperature variation with time under the same conditions of Fig. 5.

Fig. 7
Fig. 7

Energy loss rate through each surface term in Eq. (4) with time under the same conditions of Fig. 5. Vaporization (a), shrinkage (b) and conduction (d).

Fig. 8
Fig. 8

Temperature distribution along the droplet diameter parallel to the incident laser at times, t = 0.2, 0.4, 0.6 and 0.8 μs. λ0 = 10.6 μm. R0 = 10.0 μm. I0 = 106 W/cm2. α = 1.0.

Fig. 9
Fig. 9

Surface temperature variation at ( r ¯ , θ ) = ( 1 . 0 , 0 . 0 ) with time under the same conditions of Fig. 8.

Fig. 10
Fig. 10

Energy loss rate at ( r ¯ , θ ) = ( 1 . 0 , 0 . 0 ) through each term in Eq. (4) with time under the same conditions of Fig. 8. Vaporization (a), shrinkage (b) and convection (c).

Fig. 11
Fig. 11

Temperature distribution along the droplet diameter parallel to the incident laser at times, t = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 ms. λ0 = 10.6 μm. R0 = 10.0 μm. I0 = 103 W/cm2.

Fig. 12
Fig. 12

Surface temperature variation at ( r ¯ , θ ) = ( 1 . 0 , 0 . 0 ) with time under the same conditions of Fig. 11.

Fig. 13
Fig. 13

Energy loss rate at ( r ¯ , θ ) = ( 1 . 0 , 0 . 0 ) through each surface term in Eq. (4) with time under the same conditions of Fig. 11. Vaporization (a), shrinkage (b) and conduction (d).

Tables (2)

Tables Icon

Table I Values of I s f for 10 μm Water Droplet

Tables Icon

Table II Physical Constants for Water Droplet–Air System

Equations (41)

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t ρ ( U + 1 2 υ 2 ) + · ρ υ ( H + 1 2 υ 2 ) + · ( K T ) = W ,
4 π 3 R 0 3 ρ c d T d t + 4 π R 0 2 m [ L + c ( T T 0 ) ] 4 π R 0 2 K ( T r ) | r = R 0 + 4 π R 0 2 m 3 2 p 2 = π R 0 2 Q a I .
c ( T ) ρ ( T ) T t = · ( K T ) + f f = 4 π Re ( n ) Im ( n ) λ 0 I S S = | E | 2 | E inc | 2 ,
K ( T ) ( T r 1 r 2 R θ T θ ) [ 1 + 1 r 2 ( R θ ) 2 ] 1 / 2 | r ¯ R = K ( T r 1 r 2 R θ T θ ) [ 1 + 1 r 2 ( R θ ) ] 1 / 2 | r ¯ R + + m L + m c ( T T 0 ) + m 3 2 ρ 2 ,
R t = m ρ
K ( T ) ( T r 1 r 2 R θ T θ ) [ 1 + 1 r 2 ( R θ ) 2 ] 1 / 2 = m L ( m L ) | t ¯ 0 .
R t = C exp [ ρ ( T ) L ( T ) n k k T ] ,
c ρ T t = · ( K T ) + f 1 r 2 r ( K r 2 T r ) + f .
K Δ T f ( Δ R ) 2 = K Δ T λ 0 4 π I S ( Δ R ) 2 Re ( n ) Im ( n ) ,
I s f = K Δ T λ 0 4 π S ( Δ R ) 2 Re ( n ) Im ( n ) .
Y t + υ Y r = D ρ r 2 r ( r 2 Y r ) ,
T t + υ T r = K ρ c r 2 r ( r 2 T r ) ,
m = D R ln [ 1 Y 0 1 Y 0 exp ( L M R g T 0 L M R g T ) ] ,
K T r | r = R + = m c ( T T 0 ) exp ( m R g c K ) 1 ,
m = ν α p υ c υ ,
ν = α P e 2 π ( R g M ) T 0 ,
ρ υ = P υ ( R g M ) T 0 c υ = T 0 2 π ( R g M ) P υ = P e
P e = P r exp ( L M R g T r ) exp ( L M R g T e ) ,
m = α P r exp ( L M R g T r ) 2 π R g M [ exp ( L M R g T ) T exp ( L M R g T 0 ) T 0 ] ,
c ( T ) ρ ( T ) T t = 1 r 2 r [ K ( T ) r 2 T r ] + 1 r 2 sin 2 θ θ × [ K ( T ) sin θ T θ ] + f ( r , θ , T , n ; t ) ,
f = 4 π Re ( n ) Im ( n ) λ 0 I S S ( r , θ , n ; t ) = | E ( r , θ , n ; t ) | 2 | E inc ( r , θ ; t ) | 2 , K ( T ) ( T r 1 r 2 R θ T θ ) [ 1 + 1 r 2 ( R θ ) 2 ] 1 / 2 | r ¯ R = K ( T ) ( T r 1 r 2 R θ T θ ) [ 1 + 1 r 2 ( R θ ) 2 ] 1 / 2 | r ¯ R + + m L + m c ( T T 0 ) + m 3 2 ρ 2 ,
R t = m ρ .
T ( r , θ ; 0 ) = T 0 ,
R ( θ ; 0 ) = R 0 ,
T ( r , θ ; t ) < ,
T θ | θ = 0 = T θ | θ = π = 0 ,
r i = i h r , h r = 1 / N , r N = 1 , i = 1 , 2 , , N θ j = ( j 1 2 ) h θ , h θ = 1 / L , j = 1 , 2 , , L Δ t = t k t k 1 , k = 1 , 2 , 3 , ,
θ j 1 / 2 = 1 2 ( θ j θ j 1 ) , K i , j + 1 / 2 = 1 2 ( K i , j + K i , j + 2 1 ) , R ˙ j k = 1 2 ( R j + 1 k R j 1 k h θ ) , R ¨ j k = R j + 1 k 2 R j k + R j 1 k h θ 2 , r ¯ i = r i r i 1 , T i , j , R j 0 = 1 , R j 0 = 0 , R ˙ j 0 = 0 , R ¨ j 0 = 0 .
c i , j ρ i , j ( T i , j k T i , j k 1 / 2 Δ t ) = + 1 a ( R j k ) 2 r i 2 h r 2 [ K i + 1 / 2 j r ¯ i + 1 2 ( T i + 1 , j k T i , j k ) K i 1 / 2 , j r ¯ i 2 ( T i , j k T i 1 , j k ) ] + ( R ˙ i k ) 2 a π 2 ( R j k ) 4 r i h r 2 [ K i + 1 / 2 , j r ¯ i + 1 ( T i + 1 , j k T i , j k ) K i 1 / 2 , j r ¯ i ( T i , j k T i 1 , j k ) ] + 1 2 h r ( T i + 1 , j k T i 1 , j k ) [ c i , j ρ i , j r i R j k R j k K i , j cos ( π θ j ) R ˙ j k a π ( R j k ) 3 r i sin ( π θ j ) K i , j R ¨ j k a π 2 ( R j k ) 3 r i + K i , j ( R ˙ j k ) 2 a π 2 ( R j k ) 4 r i ] R ˙ j k a π 2 ( R j k ) 3 r i h r h θ [ K i + 1 , j 1 / 2 ( T i + 1 , j k 1 / 2 T i + 1 , j 1 k 1 / 2 ) K i , j 1 / 2 ( T i , j k 1 / 2 T i , j 1 k 1 / 2 ) ] + R 0 2 a T 0 f i , j k .
c i , j ρ i , j ( T i , j k 1 / 2 T i , j k 1 Δ t ) = + 1 a π 2 ( R j k ) 2 r i 2 sin ( π θ j ) h θ 2 [ K i , j + 1 / 2 sin ( π θ j + 1 / 2 ) ( T i , j + 1 k 1 / 2 T i , j k 1 / 2 ) K i , j 1 / 2 sin ( π θ j 1 / 2 ) ( T i , j k 1 / 2 T i , j 1 k 1 / 2 ) ] R ˙ j k a π 2 ( R j k ) 3 r i h r h θ [ K i 1 / 2 , j ( T i , j + 1 k 1 T i 1 , j + 1 k 1 ) K i 1 / 2 , j ( T i , j k 1 T i 1 , j k 1 ) ] .
m a K N , j T 0 [ 1 + ( R ˙ j k ) 2 π 2 ( R j k ) 2 r n 2 ] 1 / 2 [ c N , j ( T N , j k 1 ) T 0 exp ( m R g c N , j K N , j ) 1 + L N , j k + c N , j ( T N , j k 1 ) T 0 + m 2 2 ρ 2 ] = 1 R j k h r ( T N , j k T N 1 , j k ) R ˙ j k π 2 ( R j k ) 2 r N 2 h θ ( T N , j + 1 k 1 / 2 T N , j k 1 / 2 ) + ( R ˙ j k ) 2 π 2 ( R j k ) 3 r N h r ( T N , j h , T N 1 , j 4 ) .
R j k = R 0 a m ρ .
R j k = R j k 1 + Δ t R j k 1 .
L ( T ) ( cal / g ) = 30 . 46 × [ exp ( 0 . 000803 T ) ] × 374 T ,
ρ ( T ) ( g / cm 3 ) = 1 0 . 7792 × 10 5 × T 1 . 8478 ,
K ( T ) ( cal / scm ° C ) = ( 1 . 373 + 0 . 385 × 10 7 × T 0 . 138 x 10 4 × T 2 ) × 10 3 ,
c ( J / g ° C ) = A ( 0 . 996185 + 0 . 0002874 × ( T + 100 100 ) 5 . 26 + 0 . 11160 × 10 0 . 036 T ) ,
[ Re ( n ) ] 2 1 [ Re ( n ) ] 2 + 2 ( 1 ρ ) = C ,
Re [ n ( T ) ] = 1 + 2 C ρ ( T ) 1 C ρ ( T ) .
Im [ n ( T ) ] = Im [ n ( T 0 ) ] ( T T 0 ) b ,
Im [ n ( T ) ] = 0 . 07158 ( 273 T ) 0 . 7298 + 0 . 004 .

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