Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T11:26:07.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous fingering and dendritic growth under an elastic membrane

Published online by Cambridge University Press:  02 August 2017

Lucie Ducloué Open the ORCID record for Lucie Ducloué [Opens in a new window] ,
Andrew L. Hazel Open the ORCID record for Andrew L. Hazel [Opens in a new window] ,
Draga Pihler-Puzović  and
Anne Juel Open the ORCID record for Anne Juel [Opens in a new window]
Lucie Ducloué
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Andrew L. Hazel
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Draga Pihler-Puzović
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Anne Juel
Affiliation:
Manchester Centre for Nonlinear Dynamics and School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We present an experimental investigation of interfacial fingering instabilities in a compliant channel, where the interface can adopt a planar front orthogonal to the direction of propagation over most of the channel width. Finite-length fingers develop on that front, similarly to the previously studied radial configuration with injection of air at constant flow rate (Pihler-Puzović et al., Phys. Rev. Lett., vol. 108 (7), 2012, 074502), but, unlike the radial case, the interface propagates steadily. This allows us to present the first quantification of the nonlinearly saturated fingering pattern and to demonstrate that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction. Furthermore, we find that changes to the geometry of the channel cross-section lead to a rich variety of possible interfacial patterns.

JFM classification

Interfacial Flows (free surface): Fingering instability Nonlinear Dynamical Systems: Pattern formation
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , R2
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8 (10), 747–750.Google Scholar
Arneodo, A., Couder, Y., Grasseau, G., Hakim, V. & Rabaud, M. 1989 Uncovering the analytical Saffman–Taylor finger in unstable viscous fingering and diffusion-limited aggregation. Phys. Rev. Lett. 63 (9), 984–987.CrossRefGoogle Scholar
Bachmann, J. & van der Ploeg, R. R. 2002 A review on recent developments in soil water retention theory: interfacial tension and temperature effects. J. Plant Nutr. Soil Sci. 165 (4), 468–478.Google Scholar
Ben-Jacob, E. & Garik, P. 1990 The formation of patterns in non-equilibrium growth. Nature 343, 523–530.Google Scholar
Ben-Jacob, E., Shmueli, H., Shochet, O. & Tenenbaum, A. 1992 Adaptive self-organization during growth of bacterial colonies. Phys. A (Amsterdam, Neth.) 187 (3), 378–424.Google Scholar
Bischofberger, I., Ramachandran, R. & Nagel, S. R. 2014 Fingering versus stability in the limit of zero interfacial tension. Nat. Commun. 5, 5265.Google Scholar
Chaudhury, M. K., Chakrabarti, A. & Ghatak, A. 2015 Adhesion-induced instabilities and pattern formation in thin films of elastomers and gels. Eur. Phys. J. E 38 (7), 1–26.Google Scholar
Couder, Y. 2000 Viscous fingering as an archetype for growth patterns. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 53–104. Cambridge University Press.Google Scholar
Couder, Y., Cardoso, O., Dupuy, D., Tavernier, P. & Thom, W. 1986 Dendritic growth in the Saffman–Taylor experiment. Europhys. Lett. 2 (6), 437–443.Google Scholar
Coyle, D. J., Macosko, C. W. & Scriven, L. E. 1986 Film-splitting flows in forward roll coating. J. Fluid Mech. 171, 183–207.Google Scholar
Ducloué, L., Hazel, A. L., Thompson, A. B. & Juel, A. 2017 Reopening modes of a collapsed elasto-rigid channel. J. Fluid Mech. 819, 121–146.Google Scholar
Franco-Gómez, A., Thompson, A. B., Hazel, A. L. & Juel, A. 2016 Sensitivity of Saffman–Taylor fingers to channel-depth perturbations. J. Fluid Mech. 794, 343–368.CrossRefGoogle Scholar
Hakim, V., Rabaud, M., Thomé, H. & Couder, Y. 1990 Directional growth in viscous fingering. In New Trends in Nonlinear Dynamics and Pattern Forming Phenomena (ed. Coullet, P. & Huerre, P.), pp. 327–337. Plenum.CrossRefGoogle Scholar
Heil, M. & Hazel, A. L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.CrossRefGoogle Scholar
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255–272.CrossRefGoogle Scholar
Jensen, O. E., Horsburgh, M. K., Halpern, D. & Gaver, D. P. 2002 The steady propagation of a bubble in a flexible-walled channel: asymptotic and computational models. Phys. Fluids 14, 443–457.Google Scholar
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26 (01), 1–15.Google Scholar
Pailha, M., Hazel, A. L., Glendinning, P. A. & Juel, A. 2012 Oscillatory bubbles induced by geometrical constraint. Phys. Fluids 24 (2), 021702.CrossRefGoogle Scholar
Peng, G. G., Pihler-Puzović, D., Juel, A., Heil, M. & Lister, J. R. 2015 Displacement flows under elastic membranes. Part 2. Analysis of interfacial effects. J. Fluid Mech. 784, 512–547.Google Scholar
Pihler-Puzović, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108 (7), 074502.Google Scholar
Pihler-Puzović, D., Juel, A., Peng, G. G., Lister, J. R. & Heil, M. 2015 Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations. J. Fluid Mech. 784, 487–511.Google Scholar
Pihler-Puzović, D., Périllat, R., Russell, M., Juel, A. & Heil, M. 2013 Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells. J. Fluid Mech. 731, 162–183.Google Scholar
Rabaud, M., Couder, Y. & Michalland, S. 1991 Wavelength selection and transients in the one-dimensional array of cells of the printers instability. Eur. J. Mech. (B/Fluids) 10, 253–260.Google Scholar
Rabaud, M., Michalland, S. & Couder, Y. 1990 Dynamical regimes of directional viscous fingering: spatiotemporal chaos and wave propagation. Phys. Rev. Lett. 64 (2), 184–187.Google Scholar
Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Sheng, J. 2010 Modern Chemical Enhanced Oil Recovery: Theory and Practice. Gulf Professional Publishing.Google Scholar
Thompson, A. B., Juel, A. & Hazel, A. L. 2014 Multiple finger propagation modes in Hele-Shaw channels of variable depth. J. Fluid Mech. 746, 123–164.CrossRefGoogle Scholar

Ducloué et al. supplementary movie 1

Aerial view of an air finger steadily propagating under a latex membrane (b=210 μm) topping a partially occluded channel. The fingers regularly split, maintaining a constant average width and length. Block dimensions: 30 mm wide and 900 μm high; h = 150 μm. The video has been slowed down four times.

Download Ducloué et al. supplementary movie 1(Video)
Video 906.1 KB

Ducloué et al. supplementary movie 2

Side view of an air finger propagating under a thin (b = 100 μm) silicone membrane. The transparent membrane has been dusted with fine powder to make the laser line visible (bright line). The straight section of interface propagating on the block does not undergo fingering. Block dimensions: 30 mm wide and 900 μm high; h = 150 μm. The video has been slowed down twice.

Download Ducloué et al. supplementary movie 2(Video)
Video 3.6 MB