Abstract

The space–time correlation function of the intensity of light scattered off a medium with a constant velocity gradient has been calculated. One property, in particular, of this correlation function, which may be referred to as speckle motion or speckle flow, enables an experimental determination of the tensor components of the velocity gradient. Influence of Brownian motion and finite scattering volume are estimated. Experimental results, obtained at a tube flow, are shown to be in accordance with the theory.

© 1983 Optical Society of America

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References

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  1. G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
    [CrossRef]
  2. F. Böner, W. Staude, Opt. Commun. 40, 407 (1982).
    [CrossRef]
  3. J. Ohtsubo, Opt. Commun. 34, 147 (1980).
    [CrossRef]
  4. B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).
  5. See, for example, E. Jakeman, in Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1973).
  6. R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).
  7. R. T. Foister, T. G. M. van den Ven, J. Fluid Mech. 96, 105 (1980).
    [CrossRef]
  8. M. C. Wang, G. Uhlenbeck, Rev. Mod. Phys. 17, 326 (1945).
    [CrossRef]
  9. K. Weighardt, Theoretische Strömungslehre (Tuebner, Stuttgart, 1974).

1982 (1)

F. Böner, W. Staude, Opt. Commun. 40, 407 (1982).
[CrossRef]

1980 (3)

J. Ohtsubo, Opt. Commun. 34, 147 (1980).
[CrossRef]

R. T. Foister, T. G. M. van den Ven, J. Fluid Mech. 96, 105 (1980).
[CrossRef]

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
[CrossRef]

1945 (1)

M. C. Wang, G. Uhlenbeck, Rev. Mod. Phys. 17, 326 (1945).
[CrossRef]

Bertolotti, M.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Böner, F.

F. Böner, W. Staude, Opt. Commun. 40, 407 (1982).
[CrossRef]

Crosignani, B.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Di Porto, P.

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

Foister, R. T.

R. T. Foister, T. G. M. van den Ven, J. Fluid Mech. 96, 105 (1980).
[CrossRef]

Fuller, G. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Jakeman, E.

See, for example, E. Jakeman, in Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1973).

Leal, L. G.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).

Ohtsubo, J.

J. Ohtsubo, Opt. Commun. 34, 147 (1980).
[CrossRef]

Rallison, J. M.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Schmidt, R. L.

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
[CrossRef]

Staude, W.

F. Böner, W. Staude, Opt. Commun. 40, 407 (1982).
[CrossRef]

Uhlenbeck, G.

M. C. Wang, G. Uhlenbeck, Rev. Mod. Phys. 17, 326 (1945).
[CrossRef]

van den Ven, T. G. M.

R. T. Foister, T. G. M. van den Ven, J. Fluid Mech. 96, 105 (1980).
[CrossRef]

Wang, M. C.

M. C. Wang, G. Uhlenbeck, Rev. Mod. Phys. 17, 326 (1945).
[CrossRef]

Weighardt, K.

K. Weighardt, Theoretische Strömungslehre (Tuebner, Stuttgart, 1974).

J. Fluid Mech. (2)

G. G. Fuller, J. M. Rallison, R. L. Schmidt, L. G. Leal, J. Fluid Mech. 100, 555 (1980).
[CrossRef]

R. T. Foister, T. G. M. van den Ven, J. Fluid Mech. 96, 105 (1980).
[CrossRef]

Opt. Commun. (2)

F. Böner, W. Staude, Opt. Commun. 40, 407 (1982).
[CrossRef]

J. Ohtsubo, Opt. Commun. 34, 147 (1980).
[CrossRef]

Rev. Mod. Phys. (1)

M. C. Wang, G. Uhlenbeck, Rev. Mod. Phys. 17, 326 (1945).
[CrossRef]

Other (4)

K. Weighardt, Theoretische Strömungslehre (Tuebner, Stuttgart, 1974).

B. Crosignani, P. Di Porto, M. Bertolotti, Statistical Properties of Scattered Light (Academic, New York, 1975).

See, for example, E. Jakeman, in Photon Correlation and Light Beating Spectroscopy (Plenum, New York, 1973).

R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1973).

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Figures (6)

Fig. 1
Fig. 1

Scheme of the experimental setup. (The local coordinate system introduced in the theoretical part is not identical with the one shown here but is translated according to the region of interest.)

Fig. 2
Fig. 2

Examples of measured CCF curves and fitted Gaussians. The parameter is the distance from the tube wall. The measured values are normalized according to G plot 2 ( t ) = [ G 2 ( t ) meas G 2 ( ) ] / G 2 ( ).

Fig. 3
Fig. 3

Experimental results of one particular run with geometrical parameters given in the text. The circles refer to shear gradients determined by the CCF, the dots to velocities determined by the heterodyne technique. The solid line is a least-mean-squares fit through the velocity values according to Ref. 8; the dashed line represents the spatial derivative of this curve.

Fig. 4
Fig. 4

Linear dependence of the experimental Γxz values on the flow velocity.

Fig. 5
Fig. 5

Dependence of tM on the angle ϑ. The solid line is a least-mean-square fit according to Eq. (18) through the origin (not drawn).

Fig. 6
Fig. 6

Example of an experimentally determined autocorrelation function using two laser beams. Geometry similar to that of Fig. 1; the two laser beams have small, opposite kx values (angle between the beams is ~0.7°).

Equations (32)

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G 2 ( r 1 , r 2 , t ) = I ( r 1 , t ) · I ( r 2 , O ) ,
v ( r ) = v 0 + Γ · r ,
G 2 ( q , q 2 , t ) = I ( q 1 , t ) I ( q 2 , O ) .
G 2 ( q 1 , q 2 , t ) = I ( q 1 , t ) I ( q 2 , O ) + | G 1 ( q 1 , q 2 , t ) | 2 .
G 1 ( q 1 , q 2 , t ) = 1 2 0 c E ( q 1 , t ) · E * ( q 2 , O ) .
| G 1 ( q 1 , q 2 , t ) | = I | d 3 r d 3 r L q 1 ( r ) f ( q 1 ) L q 2 ( r ) f ( q 2 ) × n ( r , t ) n ( r , O ) exp [ i ( q 1 · r q 2 · r ) ] | .
n ( r , t ) n ( r , O ) = ρ P ( r , t | r , O ) + ρ 2 ,
P ( r , t | r , O ) = 1 ( 2 π ) 3 exp ( D q · A q + i v 0 · S T q + i r · R T q i r · q ) d 3 q ,
A = 0 t exp Γ t exp ( Γ T t ) d t , S = 0 t exp ( Γ t ) d t , R = exp ( Γ t ) , and
P ( r , t | r , O ) = 1 ( 4 π D ) 3 / 2 1 | A | · exp [ 1 4 D ( r R r S v 0 ) · A 1 ( r R r S v 0 ) ] ,
r = P 1 ( ξ η ) + 1 2 S v 0 , r = P ( ξ + η ) + 1 2 S v 0 ,
| G 1 ( q 1 , q 2 , t ) | = ρ I · 1 π 3 | A q 1 [ P ( ξ + η ) + 1 2 S v 0 ] × A q 2 [ P 1 ( ξ η ) + 1 2 S v 0 ] × exp [ i ( P T q 1 P T q 2 ) · ξ ] × exp [ i ( P T q 1 + P T q 2 ) · η ] × exp ( D q · A q i 2 η · P T q ) d 3 ξ d 3 η d 3 q | .
| G 1 ( q 1 , q 2 , t ) | = ρ I ( π D ) 3 / 2 1 | A | | A q 1 [ P ( ξ + η ) + 1 2 S v 0 ] × A q 2 [ P 1 ( ξ η ) + 1 2 S v 0 ] exp ( i d · ξ ) × exp ( i s · η ) exp ( 1 D η · P T A 1 P η ) d 3 ξ d 3 η ,
| G 1 ( q 1 , q 2 , t ) | = ρ I L 2 ( 2 π ) 3 δ ( d ) exp ( D 4 s · P 1 A P T s ) f ( q 1 ) f ( q 2 ) .
d = P T q 1 P T q 2 = 0.
q 1 = R T q 2 .
q 1 ( q 2 , t ) = R T ( t ) q 2 .
| G 1 ( q 1 , q 2 , t ) | = ρ I | A q 1 ( P ξ + 1 2 S v 0 ) A q 2 ( P 1 ξ + 1 2 S v 0 ) × exp ( i d · ξ ) d 3 ξ | exp ( D 4 s · P 1 A P T s ) .
| D 4 t ( s · ( P 1 A P T ) s ) | + | O ˙ | | d ˙ | · W d ˙ .
O ˙ = t ln [ A q 1 ( R ξ + S v 0 ) A q 2 ( ξ ) d 3 ξ ] .
W d ˙ = 1 | d ˙ | [ d ˙ ( ξ ξ ¯ ) ] 2 ¯ ,
O ˙ = A q 1 ( R ξ + S v 0 ) · ( Γ ξ + v 0 ) · ξ A q 2 ( ξ ) d 3 ξ .
t M < t 0 ,
t 0 Γ 1 , which implies exp ( Γ t 0 ) I + Γ t 0 ,
D t 0 min ( W ) ,
υ 0 W v 0 | Γ T q 2 | W d ˙ ,
D q 2 | Γ T q 2 | W d ˙ .
v ( r ) = ( υ 0 + Γ x z · x ) z ˆ ,
| G 1 ( q 1 , q 2 , t ) | = ρ I f 2 β 2 π | l ( y , z + t / 2 υ 0 ) · l ( y , z t / 2 υ 0 ) × exp [ i ( κ y y + κ z z ) ] d y d z × exp ( D t 4 | q 1 + q 2 | 2 ) × exp [ 1 8 β ( κ x + t 2 Γ x z K z ) 2 ] ,
exp [ ( 2 κ x Γ x z K z + t ) 2 / 2 β Γ x z 2 K z 2 ] .
k = 2 π λ ( 0 , cos ϑ , sin ϑ ) , k 1,2 = 2 π λ ( ± sin φ , cos φ , 0 ) ,
Γ x z = 2 sin φ sin ϑ · 1 t M ,

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