## Diffusing-wave spectroscopy in a shear flow

JOSA B, Vol. 7, Issue 1, pp. 15-20 (1990)

http://dx.doi.org/10.1364/JOSAB.7.000015

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### Abstract

We present a new technique for measuring velocity gradients for laminar shear flow, using dynamic light scattering in the strongly multiple-scattering regime. We derive temporal autocorrelation functions for multiply scattered light, taking into account particle displacements arising from deterministic shear flow and random Brownian motion. The laminar shear flow and Brownian motion are characterized by the relaxation rates
*τ _{B}*

^{−1}=

*Dk*

_{0}

^{2}, respectively, where

*k*

_{0}= 2

*πn/λ*is the wave number in the scattering medium,

*l** is the transport mean free path of the photons, and

*D*is the diffusion coefficient of the scatterers. We obtain excellent agreement between theory and experiment over a wide range of shear rates,

*τ*is much less than

_{S}*τ*. Thus the mean shear rate can be simply determined by a single measurement.

_{B}© 1990 Optical Society of America

**History**

Original Manuscript: June 16, 1989

Manuscript Accepted: September 28, 1989

Published: January 1, 1990

**Citation**

X-L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, and D. A. Weitz, "Diffusing-wave spectroscopy in a shear flow," J. Opt. Soc. Am. B **7**, 15-20 (1990)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-7-1-15

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### References

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- The correction to Brownian motion that is due to the convective flow is called Taylor dispersion. Taylor dispersion modifies particle diffusion in the direction of velocity gradient. In the entire range of shear rate in this experiment the correction (Γτ)2/3 is much smaller than 1. This justifies our approximation that the particle diffusion and convective shear are decoupled.
- For some scattering geometries there will be an additional term in the sum corresponding to the difference between the input and output wave vectors. This term is proportional to the velocity (rather than to the velocity gradient) and does not contribute to the homodyne correlation function. We also note that for the common case that the flow direction is perpendicular to the input and output wave vectors, this term is identically zero.
- G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge U. Press, Cambridge, 1977), p. 83.
- More generally, to include the effects of particle interactions, one must replace F(q) by the full scattering function S(q)F(q), where S(q) is the structure factor. We note that in these experiments, however, the volume fraction of PSS’s is low (ϕ= 0.02) and the Coulomb interaction between spheres is highly screened. Under these conditions, S(q) ≃ 1.
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- We note that the average intensity of transmitted light, 〈I〉, through the sample did not vary with Γ¯. Since 〈I〉 ∼ l*/L, we conclude that l*, and hence P(s), does not vary with Γ¯ for our samples (see Refs. 6 and 8). This also suggests that S(q) is essentially independent of Γ for the weakly interacting samples used in this study.
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- S. H. Chen, T. L. Lin, J. S. Huang, in Physics of Complex and Supermolecular Fluids (Wiley, New York, 1987), p. 285.

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