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Advection and diffusion in a chemically induced compressible flow

Published online by Cambridge University Press:  21 May 2018

Florence Raynal Open the ORCID record for Florence Raynal [Opens in a new window] ,
Mickael Bourgoin ,
Cécile Cottin-Bizonne ,
Christophe Ybert  and
Romain Volk
Florence Raynal*
Affiliation:
LMFA, Univ Lyon, École Centrale Lyon, INSA Lyon, Université Lyon 1, CNRS, F-69134 Écully, France
Mickael Bourgoin
Affiliation:
Laboratoire de Physique, ENS de Lyon, Univ Lyon, CNRS, 69364 Lyon CEDEX 07, France
Cécile Cottin-Bizonne
Affiliation:
Institut Lumière Matière, Univ Lyon, Université Lyon 1, CNRS, F-69622 Villeurbanne CEDEX, France
Christophe Ybert
Affiliation:
Institut Lumière Matière, Univ Lyon, Université Lyon 1, CNRS, F-69622 Villeurbanne CEDEX, France
Romain Volk*
Affiliation:
Laboratoire de Physique, ENS de Lyon, Univ Lyon, CNRS, 69364 Lyon CEDEX 07, France
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

We study analytically the joint dispersion of Gaussian patches of salt and colloids in linear flows, and how salt gradients accelerate or delay colloid spreading by diffusiophoretic effects. Because these flows have constant gradients in space, the problem can be solved almost entirely for any set of parameters, leading to predictions of how the mixing time and the Batchelor scale are modified by diffusiophoresis. We observe that the evolution of global concentrations, defined as the inverse of the patches’ areas, are very similar to those obtained experimentally in chaotic advection. They are quantitatively explained by examining the area dilatation factor, in which diffusive and diffusiophoretic effects are shown to be additive and appear as the divergence of a diffusive contribution or of a drift velocity. An analysis based on compressibility is developed in the salt-attracting case, for which colloids are first compressed before dispersion, to predict the maximal colloid concentration as a function of the parameters. This maximum is found not to depend on the flow stretching rate nor on its topology (strain or shear flow), but only on the characteristics of salt and colloids (diffusion coefficients and diffusiophoretic constant) and the initial size of the patches.

JFM classification

Complex Fluids: Colloids Mathematical Foundations: Mathematical Foundations Mixing: Mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 228 - 243
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Copyright
© 2018 Cambridge University Press 

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References

Abécassis, B., Cottin-Bizonne, C., Ybert, C., Ajdari, A. & Bocquet, L. 2009 Osmotic manipulation of particles for microfluidic applications. New J. Phys. 11 (7), 075022.Google Scholar
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid. Mech. 21, 61–99.Google Scholar
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. Ser. A 235 (1200), 67–77.Google Scholar
Bakunin, O. G.(Ed.) 2011 Chaotic Flows. Springer.CrossRefGoogle Scholar
Banerjee, A., Williams, I., Azevedo, R. N., Helgeson, M. E. & Squires, T. M. 2016 Soluto-inertial phenomena: designing long-range, long-lasting, surface-specific interactions in suspensions. Proc. Natl Acad. Sci. USA 113 (31), 8612–8617.Google Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5 (1), 113–133.Google Scholar
Birch, D. A., Young, W. R. & Franks, P. J. S. 2008 Thin layers of plankton: formation by shear and death by diffusion. Deep-Sea Res. I 55 (3), 277–295.Google Scholar
Deseigne, J., Cottin-Bizonne, C., Stroock, A. D., Bocquet, L. & Ybert, C. 2014 How a “pinch of salt” can tune chaotic mixing of colloidal suspensions. Soft Matt. 10, 4795–4799.CrossRefGoogle Scholar
Mauger, C., Volk, R., Machicoane, N., Bourgoin, M., Cottin-Bizonne, C., Ybert, C. & Raynal, F. 2016 Diffusiophoresis at the macroscale. Phys. Rev. Fluids 1, 034001.Google Scholar
Metcalfe, G., Speetjens, M. F. M., Lester, D. R. & Clercx, H. J. H. 2012 Beyond passive: chaotic transport in stirred fluids. In Advances in Applied Mechanics, vol. 45, pp. 109–188. Elsevier.Google Scholar
Okubo, A. 1967 The effect of shear in an oscillatory current on horizontal diffusion from an instantaneous source. Intl J. Oceanol. Limnol. 1, 194–204.Google Scholar
Rhines, P. B. & Young, W. R. 1983 How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133, 133145.Google Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219 (1137), 186–203.Google Scholar
Townsend, A. A. 1951 The diffusion of heat spots in isotropic turbulence. Proc. R. Soc. Lond. A 209 (1098), 418–430.Google Scholar
Volk, R., Mauger, C., Bourgoin, M., Cottin-Bizonne, C., Ybert, C. & Raynal, F. 2014 Chaotic mixing in effective compressible flows. Phys. Rev. E 90, 013027.Google Scholar
Young, W. R., Rhines, P. B. & Garrett, C. J. R. 1982 Shear-flow dispersion, internal waves and horizontal mixing in the ocean. J. Phys. Oceanogr. 12 (6), 515–527.Google Scholar