Abstract

The temperature and optical path differences are found for the time-dependent conduction equation with axial convection for an arbitrary heat source. The solution satisfies Dirichlet boundary conditions in the radial and axial directions.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Kelly, P. I. Shen, G. C. Valley, American Society of Mechanical Engineers, paper 79-HT-94 (Aug.1979).
  2. G. C. Valley, P. E. Shen, R. E. Kelly, Appl. Opt. 18, 2728 (1979).
    [CrossRef]
  3. P. I. Shen, P. A. Iger, D. R. Regan, J. Phys. (Paris) Colloq. C9, 41, C9-137 (1980), supplement to No. 11.
    [CrossRef]
  4. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 857.
  5. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 864.
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 466.
  7. C. J. Tranter, Integral Transforms in Mathematical Physics (Wiley, New York, 1956), p. 88.

1980 (1)

P. I. Shen, P. A. Iger, D. R. Regan, J. Phys. (Paris) Colloq. C9, 41, C9-137 (1980), supplement to No. 11.
[CrossRef]

1979 (2)

G. C. Valley, P. E. Shen, R. E. Kelly, Appl. Opt. 18, 2728 (1979).
[CrossRef]

R. E. Kelly, P. I. Shen, G. C. Valley, American Society of Mechanical Engineers, paper 79-HT-94 (Aug.1979).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 466.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 864.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 857.

Iger, P. A.

P. I. Shen, P. A. Iger, D. R. Regan, J. Phys. (Paris) Colloq. C9, 41, C9-137 (1980), supplement to No. 11.
[CrossRef]

Kelly, R. E.

G. C. Valley, P. E. Shen, R. E. Kelly, Appl. Opt. 18, 2728 (1979).
[CrossRef]

R. E. Kelly, P. I. Shen, G. C. Valley, American Society of Mechanical Engineers, paper 79-HT-94 (Aug.1979).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 864.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 857.

Regan, D. R.

P. I. Shen, P. A. Iger, D. R. Regan, J. Phys. (Paris) Colloq. C9, 41, C9-137 (1980), supplement to No. 11.
[CrossRef]

Shen, P. E.

Shen, P. I.

P. I. Shen, P. A. Iger, D. R. Regan, J. Phys. (Paris) Colloq. C9, 41, C9-137 (1980), supplement to No. 11.
[CrossRef]

R. E. Kelly, P. I. Shen, G. C. Valley, American Society of Mechanical Engineers, paper 79-HT-94 (Aug.1979).

Tranter, C. J.

C. J. Tranter, Integral Transforms in Mathematical Physics (Wiley, New York, 1956), p. 88.

Valley, G. C.

R. E. Kelly, P. I. Shen, G. C. Valley, American Society of Mechanical Engineers, paper 79-HT-94 (Aug.1979).

G. C. Valley, P. E. Shen, R. E. Kelly, Appl. Opt. 18, 2728 (1979).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 466.

American Society of Mechanical Engineers (1)

R. E. Kelly, P. I. Shen, G. C. Valley, American Society of Mechanical Engineers, paper 79-HT-94 (Aug.1979).

Appl. Opt. (1)

J. Phys. (Paris) Colloq. C9 (1)

P. I. Shen, P. A. Iger, D. R. Regan, J. Phys. (Paris) Colloq. C9, 41, C9-137 (1980), supplement to No. 11.
[CrossRef]

Other (4)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 857.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 864.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 466.

C. J. Tranter, Integral Transforms in Mathematical Physics (Wiley, New York, 1956), p. 88.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Transient radial temperature difference (°C) as a function of axial distance z (cm) for various times.

Fig. 2
Fig. 2

Transient OPD (cm) as a function of accumulated phase-length L (cm) for various times.

Equations (66)

Equations on this page are rendered with MathJax. Learn more.

ρ c p T t + ρ c p v T ρ c p ɛ 2 T = Q ˙
OPD ( L , t ) = ( n 1 ) T s s 0 L [ T ( r , z , t ) T ( r = 0 , z , t ) ] d z ,
( t 2 + z 2 ) T ( r , θ , z ) υ ɛ T z = Q ˙ ( r , θ , z ) ρ c p ɛ ,
t 2 G ( r , z , r , z ) + 2 G z 2 υ ɛ G z = δ ( r r ) δ ( z z ) ,
T ( r , θ , z ) = z l z + d z d s Q ˙ ( r , z ) ρ c p ɛ G ( r , z | r , z ) + υ ɛ d s [ T ( r , z ) G ( r , z | r , z ) ] z = z l + d s [ T ( r , z ) G z G T z ] z = z l + [ G t T ( r , z ) T t G ] d A t ,
G ( r , θ , z | r , θ , z ) = U n ( r , θ ) U n * ( r , θ ) C n ( z , z )
G ( r , θ + 2 π , z ) = G ( r , θ , z ) , G ( 0 , θ , z ) = finite , G ( r , θ , ) = finite , G ( r = R , θ , z ) = 0 . }
( t 2 + k n 2 ) U n ( r , θ ) = 0 ,
U n U m * d υ = δ n m ,
U n ( r , θ + 2 π ) = U n ( r , θ ) ,
U n ( 0 , θ ) = finite ,
U n ( R , θ ) = 0 ,
2 C n ( z , z ) 2 z 2 υ ɛ C n z k n 2 C n = δ ( z z ) .
U n m = [ 1 π R 2 J m + 1 2 ( κ m n R ) ] 1 / 2 exp ( i m θ ) J m ( κ m n r ) ,
J m ( κ m n R ) = 0 ,
C n ( λ ) = exp ( i λ z ) ( λ + λ 0 ) 2 + c 2 ,
C n ( λ ) = C n ( z ) exp ( i λ z ) d z ,
λ 0 = i υ 2 ɛ , c 2 = κ m n 2 + υ 2 2 ɛ .
C m n ( z , z ) = exp [ υ 2 ɛ ( z , z ) ] exp ( c | z z | ) 2 c .
ω ± = υ 2 ɛ { 1 ± [ 1 + ( 2 ɛ κ m n υ ) 2 ] 1 / 2 } ,
C m n ( z , z ) = 1 ω + ω { exp [ ( z , z ) ω , z > z ] exp [ ( z , z ) ω + , z < z ] .
T ( r , θ , z ) = z + d z d s Q ˙ ( r , θ , z ) ρ c p ɛ m , n 1 π R 2 J m + 1 2 ( k m n R ) e [ i m ( θ θ ) ] × J m ( κ m n r ) J m ( κ m n r ) [ exp [ υ 2 ɛ ( z z ) ] exp ( c | z z | ) 2 c ]
g 1 / 2 ( z | z ) = exp [ υ 2 ɛ ( z z ) ] 2 c [ exp ( c | z z | ) exp ( c | z + z | ) ] = 1 ω + ω { exp ( ω z ) [ exp ( ω z ) exp ( ω + z ) ] , z > z exp ( ω + z ) [ exp ( ω + z ) exp ( ω z ) ] , z < z ,
T ( r , θ , z ) = 1 π R 2 d s Q ˙ ( r , θ ) ρ c p ɛ m , n × exp [ i m ( θ θ ) ] J m ( κ m n r ) J m ( κ m n r ) κ m n 2 J m + 1 2 ( κ m n R ) [ 1 exp ( ω z ) ] .
Q ˙ m ( κ m n ) = 0 2 π d θ 0 R r d r Q ˙ ( r , θ ) exp ( i m θ ) J m ( κ m n r ) ,
Q ˙ ( r , θ ) = 2 2 π R 2 m , n Q ˙ m ( κ m n ) J m + 1 2 ( κ m n R ) exp ( i m θ ) J m ( κ m n r )
T ( r , θ , z ) = 1 π ρ c p ɛ m , n exp ( i m θ ) J m ( κ m n r ) Q ˙ m ( κ m n ) R 2 J m + 1 2 ( κ m n R ) κ m n 2 [ 1 exp ( ω z ) ] ,
Q ( r , θ ) = μ , l Q μ l exp ( i l θ ) R μ l ( ρ ) ; ρ = r / R .
Q m ( κ m n ) = 2 π R 2 μ Q μ m ( 1 ) ( μ m ) / 2 J μ + 1 ( κ m n R ) κ m n R .
T λ l ( z ) = 0 2 π d θ 0 1 ρ d ρ T ( R ρ , θ , z ) exp ( i l θ ) R λ l ( ρ )
T λ l ( z ) = 2 ρ c ρ ɛ R 2 n Q ˙ l ( κ l n ) [ 1 exp ( ω z ) ] J l + 1 2 ( κ l n R ) κ l n 2 ( 1 ) ( λ l ) / 2 J λ + 1 ( κ l n R ) κ l n R
T ( ρ , z ) = 2 R 2 ρ c ρ ɛ s J 0 ( s ρ ) Q ˙ 0 ( s ) s 2 J 1 2 ( s ) [ 1 exp ( ω z ) ] ,
Q ˙ 0 ( κ o n ) = 2 π R 2 Q ˙ 0 ( s ) , J 0 ( κ o n R ) = J 0 ( s ) = 0 , Q ˙ 0 ( s ) = 0 1 ρ d ρ Q ˙ ( ρ ) J 0 ( s ρ ) , ρ = r / R .
ɛ [ 1 r r ( r T r ) + 1 r 2 2 T θ 2 ] ɛ 2 T z 2 + υ T z + T t = Q ˙ ( r , z , t ) ρ c p ,
T t + υ T z ɛ 2 T z 2 + ɛ κ m n 2 T = Q ˙ ρ c p
g t ( z , t | z , t ) + υ g z ɛ 2 g z 2 + ɛ κ m n 2 g = δ ( z z ) δ ( t t ) ;
T ( z , t ) = 1 ρ c p 0 t + d t z l z d z g ( z , t | z , t ) Q ˙ ( z , t ) + υ 0 t + d t [ T ( z , t ) g ( z , t | z , t ) ] z = z l + ɛ 0 t + d t [ T ( z , t ) z g ( z , t | z , t ) g ( z , t | z , t ) T z ( z , t ) ] z = z l + 0 d z [ T ( z , t ) g ( z , t | z , t ) ] t = 0 .
g ( z , t | z , t ) = 1 2 π + d λ e [ i λ ( z z ) ] g λ ( t | t ) .
g t λ + ( ɛ λ 2 + i λ υ + β ) g = δ ( t t ) ,
g λ ( t | t ) = exp { [ ɛ λ 2 + i υ λ + β ] ( t t ) } S ( t t ) ,
g ( z , t | z , t ) = 1 2 π ɛ ( t t ) exp [ β ( t t ) ] exp { [ 1 4 ɛ ( t t ) ] [ ( z z ) υ ( t t ) ] 2 } S ( t t )
T ( r , θ , z , t ) = 1 π ρ c p R 2 m n z + d z 0 t d t 1 2 π ɛ ( t t ) exp [ ɛ κ m n 2 ( t t ) ] × exp [ 1 4 ɛ ( t t ) [ ( z z ) υ ( t t ) ] 2 ] Q ˙ m ( κ m n ; z , t ) J m + 1 2 J m ( κ m n r ) J m + 1 2 ( κ m n R ) exp ( i m θ ) ,
T ( z , t ) = 1 2 π ρ c p z + Q ˙ ( z ) d z + d λ × exp [ i λ ( z z ) ] ( 1 exp { ɛ [ ( λ + λ 0 ) 2 + c 2 ] t } ( ɛ [ ( λ + λ 0 ) 2 + c 2 ] ) ) ,
λ 0 = i υ 2 ɛ , c 2 = β ɛ + ( υ 2 ɛ ) 2 , β = ɛ κ m n 2 .
T m ( κ m n ; z ) = 1 2 c ρ c p ɛ z + d z Q ˙ m ( κ m n ; z ) × exp [ υ 2 ɛ ( z z ) ] exp ( c | z z | )
T m ( κ m n ; z , t ) = 1 4 c ρ c p ɛ z + d z exp [ υ 2 ɛ ( z z ) ] Q ˙ m ( κ m n ; z ) × ( exp [ c ( z z ) ] erfc { ɛ t [ c + ( z z ) 2 ɛ t ] } + exp [ c ( z z ) ] erfc { ɛ t [ c ( z z ) 2 ɛ t ] } ) ,
c 2 = κ m n 2 + ( υ 2 ɛ ) 2 .
g 1 / 2 ( z , t | z , t ) = exp [ υ 2 ɛ ( z z ) ] 2 π ɛ ( t t ) { exp [ ( z z ) 2 4 ɛ ( t t ) ] exp [ ( z z ) 2 4 ɛ ( t t ) ] } exp { [ ( β + υ 2 4 ɛ ) ( t t ) ] } S ( t t )
g 1 / 2 ( 0 , t | z , t ) = g 1 / 2 ( z , t | 0 , t ) = 0 , g 1 / 2 ( z , t | z , t ) = g 1 / 2 ( z , t | z , t ) , limit t + 0 t + g 1 / 2 ( z , t | z , t ) d t = steady state , Eq . ( 18 ) .
T m , 1 / 2 ( κ m n ; z , ) = 1 2 c ρ c p ɛ z + d z exp [ υ 2 ɛ ( z z ) ] × Q m ( κ m n ; z , ) [ exp ( c | z z | ) exp ( c | z + z ) ]
T m , 1 / 2 ( κ m n , z , t ) = 1 4 c ρ c p ɛ z + d z exp [ υ 2 ɛ ( z z ) ] Q ˙ m ( κ m n ; z ) { exp [ c ( z z ) ] erfc [ ɛ t ( c + ( z z ) 2 ɛ t ) ] + exp [ c ( z z ) erfc { ɛ t [ c ( z z ) 2 ɛ t ] } ( z z z + z ) } ,
Δ T ( Δ r , z , t ) = T ( r , z , t ) T ( 0 , z , t ) .
Q 0 ( κ 0 n ) = 2 π Q κ 0 n [ r 0 J 1 ( κ 0 n r 0 ) r i J 1 ( κ 0 n r i ) ] ,
Q = α P π ( R 0 2 R i 2 ) .
Δ T ( Δ r , z , t ) = 1 π R 2 n T ( κ 0 n , z , t ) J 1 2 ( κ 0 n R ) [ J 0 ( κ 0 n r ) J 0 ( 0 ) ] ,
Δ T ( z = 50 . , t = 0.5 ) Δ T ( z = 50 ) = 4 × 10 3 , steady state
Δ T ( z = 100 , t = 1 ) Δ T ( z = 50 ) = 2.5 × 10 3 . steady state
2 G x 2 + 2 G y 2 + 2 G z 2 υ ɛ G x 1 ɛ G t = δ ( r r ) δ ( t t ) .
G ( r , t | r , t ) = 1 ( 2 π ) 3 d 2 p × exp ( p ρ ) d λ exp [ i λ ( x x ) ] g λ ( t | t ) ,
g λ ( t | t ) = ɛ exp { [ λ 2 ɛ + i λ υ + p 2 ɛ ] ( t t ) } S ( t t ) .
G ( r , t | r , t ) = ɛ S ( t t ) [ 4 π ( t t ) ɛ ] 3 × exp [ υ 2 2 ɛ ( t t ) + υ 2 ɛ ( x x ) R 2 4 ( t t ) ɛ ]
G ( r | r ) = limit t 0 t G ( r , t | r , t ) d t .
G ( r | r ) = [ υ 2 ɛ ( x x ) ] exp ( υ 2 ɛ R ) 4 π R
G ( r , t | r , t ) = δ [ ( x x ) υ ( t t ) ] S ( t t ) 4 π ( t t ) exp [ ρ 2 4 ( t t ) ɛ ] ,
G ( r | r ) = 1 4 π ( x x ) exp [ υ 4 ɛ ( x x ) ρ 2 ] ,
T ( x , y , z ) = 1 C p υ ρ d x Q ˙ ( x , y , z ) .

Metrics