Skip to main content
×
×
Home

Viscous fingering and dendritic growth under an elastic membrane

  • Lucie Ducloué (a1), Andrew L. Hazel (a2), Draga Pihler-Puzović (a1) and Anne Juel (a1)
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.

Copyright
References
Hide All
Al-Housseiny, T. T., Tsai, P. A. & Stone, H. A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8 (10), 747750.
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), 984987.
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), 468478.
Ben-Jacob, E. & Garik, P. 1990 The formation of patterns in non-equilibrium growth. Nature 343, 523530.
Ben-Jacob, E., Shmueli, H., Shochet, O. & Tenenbaum, A. 1992 Adaptive self-organization during growth of bacterial colonies. Phys. A (Amsterdam, Neth.) 187 (3), 378424.
Bischofberger, I., Ramachandran, R. & Nagel, S. R. 2014 Fingering versus stability in the limit of zero interfacial tension. Nat. Commun. 5, 5265.
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), 126.
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. 53104. Cambridge University Press.
Couder, Y., Cardoso, O., Dupuy, D., Tavernier, P. & Thom, W. 1986 Dendritic growth in the Saffman–Taylor experiment. Europhys. Lett. 2 (6), 437443.
Coyle, D. J., Macosko, C. W. & Scriven, L. E. 1986 Film-splitting flows in forward roll coating. J. Fluid Mech. 171, 183207.
Ducloué, L., Hazel, A. L., Thompson, A. B. & Juel, A. 2017 Reopening modes of a collapsed elasto-rigid channel. J. Fluid Mech. 819, 121146.
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, 343368.
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. 327337. Plenum.
Heil, M. & Hazel, A. L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141162.
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255272.
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, 443457.
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26 (01), 115.
Pailha, M., Hazel, A. L., Glendinning, P. A. & Juel, A. 2012 Oscillatory bubbles induced by geometrical constraint. Phys. Fluids 24 (2), 021702.
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, 512547.
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.
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, 487511.
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, 162183.
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, 253260.
Rabaud, M., Michalland, S. & Couder, Y. 1990 Dynamical regimes of directional viscous fingering: spatiotemporal chaos and wave propagation. Phys. Rev. Lett. 64 (2), 184187.
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), 312329.
Sheng, J. 2010 Modern Chemical Enhanced Oil Recovery: Theory and Practice. Gulf Professional Publishing.
Thompson, A. B., Juel, A. & Hazel, A. L. 2014 Multiple finger propagation modes in Hele-Shaw channels of variable depth. J. Fluid Mech. 746, 123164.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed