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Emergent order in rheoscopic swirls

Published online by Cambridge University Press:  29 November 2010

MICHAEL WILKINSON ,
VLAD BEZUGLYY  and
BERNHARD MEHLIG
MICHAEL WILKINSON*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
VLAD BEZUGLYY
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK
BERNHARD MEHLIG
Affiliation:
Department of Physics, Göteborg University, 41296 Gothenburg, Sweden
*
‡Email address for correspondence: [email protected]
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Abstract

We consider the ordering of particles in a rheoscopic fluid (a suspension of microscopic rod-like particles) in a steady two-dimensional flow, and discuss its consequences for the reflection of light. The ordering is described by an order parameter which is a non-oriented vector, obtained by averaging solutions of a nonlinear equation containing the strain rate of the fluid flow. Exact solutions of this equation are obtained from solutions of a linear equation which are analogous to Bloch bands for a one-dimensional Schrödinger equation with a periodic potential. On some contours of the stream function, the order parameter approaches a limit, and on others it depends increasingly sensitively upon position. However, in the long-time limit a local average of the order parameter is a smooth function of position in both cases. We analyse the topology of the order parameter and the structure of the generic zeros of the order parameter field.

JFM classification

Mathematical Foundations: General fluid mechanics Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 667 , 25 January 2011 , pp. 158 - 187
Copyright
Copyright © Cambridge University Press 2010

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Footnotes

†

Present address: Laboratoire d'Energétique et de Mécanique Théorique et Appliquée, Nancy- Université, CNRS, 2 avenue de la forêt de Haye, BP 160, 54504 Vandoeuvre CEDEX, France.

References

REFERENCES

Ballal, B. Y. & Rivlin, R. S. 1976 Flow of a Newtonian fluid between eccentric rotating cylinders: inertial effects. Arch. Ration. Mech. Anal. 62, 237–294.CrossRefGoogle Scholar
Berry, M. V. & Hannay, J. H. 1977 Umbilic points on Gaussian random surfaces. J. Phys. A: Math. Gen. 10, 1809–1821.CrossRefGoogle Scholar
Bezuglyy, V., Wilkinson, M. & Mehlig, B. 2010 Poincaré indices of rheoscopic visualisations. Europhys. Lett. 89, 34003.CrossRefGoogle Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low-Reynolds number. J. Fluid Mech. 14, 284–304.CrossRefGoogle Scholar
Dennis, M. R. 2008 Polarization singularity anisotropy: determining monstardom. Optics Lett. 33, 2572–2574.CrossRefGoogle ScholarPubMed
Gauthier, G., Gondoret, P. & Rabaud, M. 1998 Motion of anisotropic particles: application to visualisation of three-dimensional flows. Phys. Fluids 10, 2147–2154.CrossRefGoogle Scholar
Henry, E. R. 1900 Classification and Uses of Finger Prints. Routledge.Google Scholar
Jeffery, G. B. 1922 a The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond., Ser. A 102, 161.Google Scholar
Jeffery, G. B. 1922 b The rotation of two circular cylinders in a viscous fluid. Proc. R. Soc. Lond. A101, 169–174.Google Scholar
Lipscomb, G. G., Denn, M. M., Hur, D. U. & Boger, D. V. 1988 The flow of fiber suspensions in complex geometries. J. Non-Newton. Fluid Mech. 26, 297–325.CrossRefGoogle Scholar
Mallier, R. & Maxey, M. R. 1991 The settling of non-spherical particles in a cellular flow field, Phys. Fluids A3, 1481.CrossRefGoogle Scholar
Matisse, P. & Gorman, M. 1984 Neutrally buoyant anisotropic particles for flow visualization. Phys. Fluids 27, 759.CrossRefGoogle Scholar
Mott, N. F. & Twose, W. D. 1961 The theory of impurity conduction. Adv. Phys. 10, 107.CrossRefGoogle Scholar
Müller, W. 1942 An article on the theory of the low current of two excentric circular cylinders in viscous fluid. Z. Angew. Math. Mech. 22, 177–189.CrossRefGoogle Scholar
Preston, J. C. 1984 patent: 4438096. Available at: http://www.freepatentsonline.com/4438096.htmlGoogle Scholar
Reed, C. (2002) Pure Fabrications. Harvard Magazine. Available at: http://harvardmagazine.com/2002/05/pure-fabrications.htmlGoogle Scholar
Savaş, Ö. 1985 On flow visualization using reflective flakes. J. Fluid Mech. 152, 235–248.CrossRefGoogle Scholar
Shin, H. & Maxey, M. R. 1991 Chaotic sedimentation of spheroidal particles in a cellular flow field. Phys. Fluids A3, 1434.CrossRefGoogle Scholar
Shin, H. & Maxey, M. R. 1997 Chaotic motion of nonspherical particles settling in a cellular flow field. Phys. Rev. E 56, 5431.Google Scholar
Szeri, A. J. 1993 Pattern formation in recirculating flows of suspensions of orientable particles. Phil. Trans. R. Soc. Lond. A345, 477–508.Google Scholar
Szeri, A. J. & Leal, L. G. 1994 Orientation dynamics and stretching of particles in unsteady, three-dimensional fluid flows: unsteady attractors. Chaos Solitons Fractals 4, 913–927.CrossRefGoogle Scholar
Szeri, A. J., Wiggins, S. & Leal, L. G. 1991 On the dynamics of suspended microstructure in unsteady, spatially inhomogeneous, two-dimensional fluid flows. J. Fluid Mech. 228, 207.Google Scholar
Thoroddsen, S. T. & Bauer, J. M. 1999 Qualitative flow visualization using colored lights and reflective flakes. Phys. Fluids 11, 1702.CrossRefGoogle Scholar
Wannier, G. H. 1950 A contribution to the hydrodynamics of lubrication. Q. Appl. Math. 8, 1–32.CrossRefGoogle Scholar
Wilkinson, M., Bezuglyy, V. & Mehlig, B. 2009 Fingerprints of random flows. Phys. Fluids 21, 043304.CrossRefGoogle Scholar
Ziman, J. M. 1979 Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems. Cambridge University Press.Google Scholar