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Instability of radially spreading extensional flows. Part 1. Experimental analysis

Published online by Cambridge University Press:  25 October 2019

Roiy Sayag Open the ORCID record for Roiy Sayag [Opens in a new window]  and
M. Grae Worster
Roiy Sayag*
Affiliation:
Department of Environmental Physics, Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, 8499000, Israel Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 8410501, Israel
M. Grae Worster
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

We present laboratory experiments that show that fingering patterns can emerge when circular interfaces of strain-rate-softening fluids displace less viscous fluids in extensionally dominated flows. The fingers were separated by regions in which the fluid appeared to be torn apart. Initially, the interface had a large dominant wavenumber, but some of the fingers progressively merged so that the number of fingers gradually declined in time. We find that the transition rate to a lower wavenumber during this cascade is faster the larger is the discharge flux of the displacing fluid. At late times, depending on the discharge flux, the pattern either converged into an integer wavenumber or varied stochastically within a finite range of wavenumbers, implying convergence to a fractional wavenumber. In that stage of the evolution we find that the average wavenumber declines with the discharge flux of the displacing fluid.

JFM classification

Complex Fluids: Complex Fluids Instability: Instability Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 722 - 738
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Copyright
© 2019 Cambridge University Press 

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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
Allouche, M. H., Botton, V., Henry, D., Millet, S., Usha, R. & Hadid, H. B. 2015 Experimental determination of the viscosity at very low shear rate for shear thinning fluids by electrocapillarity. J. Non-Newtonian Fluid Mech. 215, 60–69.Google Scholar
Azaiez, J. & Singh, B. 2002 Stability of miscible displacements of shear thinning fluids in a Hele–Shaw cell. Phys. Fluids 14 (5), 1557–1571.Google Scholar
Bassis, J. N., Fricker, H. A., Coleman, R. & Minster, J.-B. 2008 An investigation into the forces that drive ice-shelf rift propagation on the Amery Ice Shelf, East Antarctica. J. Geol. 54 (184), 17–27.Google Scholar
Bhaskar, K. R, Garik, P., Turner, B. S., Bradley, J. D., Bansil, R., Stanley, H. E. & Lamont, J. T. 1992 Viscous fingering of HCL through gastric mucin. Nature 360 (6403), 458–461.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, 2nd edn., vol. 1. Wiley-Interscience.Google Scholar
Borstad, C., McGrath, D. & Pope, A. 2017 Fracture propagation and stability of ice shelves governed by ice shelf heterogeneity. Geophys. Res. Lett. 44 (9), 4186–4194.Google Scholar
Bzdek, B. R., Power, R. M., Simpson, S. H., Reid, J. P. & Royall, C. P. 2016 Precise, contactless measurements of the surface tension of picolitre aerosol droplets. Chem. Sci. 7 (1), 274–285.Google Scholar
Callan-Jones, A. C., Joanny, J. F. & Prost, J. 2008 Viscous-fingering-like instability of cell fragments. Phys. Rev. Lett. 100 (25), 258106.Google Scholar
Cardoso, S. S. S. & Woods, A. W. 1995 The formation of drops through viscous instability. J. Fluid Mech. 289, 351–378.Google Scholar
Coumou, D., Driesner, T., Geiger, S., Heinrich, C. A. & Matthai, S. 2006 The dynamics of mid-ocean ridge hydrothermal systems: splitting plumes and fluctuating vent temperatures. Earth Planet. Sci. Lett. 245 (1-2), 218–231.Google Scholar
Coussot, P. 1999 Saffman–Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363–376.Google Scholar
Doake, C. S. M. & Vaughan, D. G. 1991 Rapid disintegration of the Wordie Ice Shelf in response to atmospheric warming. Nature 350 (6316), 328–330.Google Scholar
Ferguson, J., Walters, K. & Wolff, C. 1990 Shear and extensional flow of polyacrylamide solutions. Rheol. Acta 29 (6), 571–579.Google Scholar
Glen, J. W. 1955 The creep of polycrystalline ice. Proc. R. Soc. Lond. A 228 (1175), 519–538.Google Scholar
Holdsworth, G. 1983 Dynamics of Erebus Glacier tongue. Ann. Glaciol. 3, 131–137.Google Scholar
Holloway, K. E. & de Bruyn, J. R. 2005 Viscous fingering with a single fluid. Can. J. Phys. 83 (5), 551–564.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous-media. Annu. Rev. Fluid Mech. 19, 271–311.Google Scholar
Hughes, T. 1983 On the disintegration of ice shelves: the role of fracture. J. Glaciol. 29 (101), 98–117.Google Scholar
Jaishankar, A. & McKinley, G. H. 2014 A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids. J. Rheol. 58, 1751–1788.Google Scholar
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2011 Fluid mixing from viscous fingering. Phys. Rev. Lett. 106, 194502.Google Scholar
Jones, D. M., Walters, K. & Williams, P. R. 1987 On the extensional viscosity of mobile polymer solutions. Rheol. Acta 26 (1), 20–30.Google Scholar
Kondic, L., Shelley, M. J. & Palffy-Muhoray, P. 1998 Non-Newtonian Hele–Shaw flow and the Saffman–Taylor instability. Phys. Rev. Lett. 80 (7), 1433–1436.Google Scholar
Kowal, K. N. & Worster, M. G. 2019 Stability of lubricated viscous gravity currents. Part 1. Internal and frontal analyses and stabilisation by horizontal shear. J. Fluid Mech. 871, 970–1006.Google Scholar
Lapasin, R. 1995 Rheology of Industrial Polysaccharides: Theory and Applications, 1st edn. Springer.Google Scholar
Lemaire, E., Levitz, P., Daccord, G. & Vandamme, H. 1991 From viscous fingering to viscoelastic fracturing in colloidal fluids. Phys. Rev. Lett. 67 (15), 2009–2012.Google Scholar
Lindner, A., Bonn, D. & Meunier, J. 2000a Viscous fingering in a shear-thinning fluid. Phys. Fluids 12 (2), 256–261.Google Scholar
Lindner, A., Coussot, P. & Bonn, D. 2000b Viscous fingering in a yield stress fluid. Phys. Rev. Lett. 85 (2), 314–317.Google Scholar
Manickam, O. & Homsy, G. M. 1993 Stability of miscible displacements in porous-media with nonmonotonic viscosity profiles. Phys. Fluids A 5 (6), 1356–1367.Google Scholar
Manickam, O. & Homsy, G. M. 1994 Simulation of viscous fingering in miscible displacements with nonmonotonic viscosity profiles. Phys. Fluids 6 (1), 95–107.Google Scholar
Martin-Alfonso, J. E., Cuadri, A. A., Berta, M. & Stading, M. 2018 Relation between concentration and shear-extensional rheology properties of xanthan and guar gum solutions. Carbohydr. Polym. 181, 63–70.Google Scholar
Mascia, S., Patel, M. J., Rough, S. L., Martin, P. J. & Wilson, D. I. 2006 Liquid phase migration in the extrusion and squeezing of microcrystalline cellulose pastes. Eur. J. Pharmaceut. Sci. 29 (1), 22–34.Google Scholar
Muthamizhi, K., Kalaichelvi, P., Powar, S. T. & Jaishree, R. 2014 Investigation and modelling of surface tension of power-law fluids. RSC Adv. 4, 9771–9776.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Paterson, L. 1985 Fingering with miscible fluids in a Hele Shaw cell. Phys. Fluids 28 (1), 26–30.Google Scholar
Pegler, S. S. & Worster, M. G. 2012 Dynamics of a viscous layer flowing radially over an inviscid ocean. J. Fluid Mech. 696, 152–174.Google Scholar
Pegler, S. S. & Worster, M. G. 2013 An experimental and theoretical study of the dynamics of grounding lines. J. Fluid Mech. 728, 5–28.Google Scholar
Rabbani, H. S., Or, D., Liu, Y., Lai, C.-Y., Lu, N. B., Datta, S. S., Stone, H. A. & Shokri, N. 2018 Suppressing viscous fingering in structured porous media. Proc. Natl Acad. Sci. USA 115 (19), 4833–4838.Google Scholar
Robison, R. A. V., Huppert, H. E. & Worster, M. G. 2010 Dynamics of viscous grounding lines. J. Fluid Mech. 648, 363–380.Google Scholar
Roussel, N., Lanos, C. & Toutou, Z. 2006 Identification of Bingham fluid flow parameters using a simple squeeze test. J. Non-Newtonian Fluid Mech. 135 (1), 1–7.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 312–329.Google Scholar
Sayag, R. & Worster, M. G. 2013 Axisymmetric gravity currents of power-law fluids over a rigid horizontal surface. J. Fluid Mech. 716, R5.Google Scholar
Sayag, R. & Worster, M. G. 2019 Instability of radially spreading extensional flows. Part 2. Theoretical analysis. J. Fluid Mech. 881, 739–771.Google Scholar
Song, K. W., Kim, Y. S. & Chang, G. S. 2006a Rheology of concentrated xanthan gum solutions: Steady shear flow behavior. Fibers Polym. 7 (2), 129–138.Google Scholar
Song, K. W., Kuk, H. Y. & Chang, G. S. 2006b Rheology of concentrated xanthan gum solutions: oscillatory shear flow behavior. Korea–Austral. Rheol. J. 18 (2), 67–81.Google Scholar
Spiegelman, M., Kelemen, P. B. & Aharonov, E. 2001 Causes and consequences of flow organization during melt transport: the reaction infiltration instability in compactible media. J. Geophys. Res. 106 (B2), 2061–2077.Google Scholar
Stelter, M. & Brenn, G. 2002 Elongational rheometry for the characterization of viscoelastic liquids. Chem. Engng Technol. 25, 30–35.Google Scholar
Stelter, M., Brenn, G., Yarin, A. L., Singh, R. P. & Durst, F. 2002 Investigation of the elongational behavior of polymer solutions by means of an elongational rheometer. J. Rheol. 46, 507–527.Google Scholar
Stokes, J. R., Macakova, L., Chojnicka-Paszun, A., de Kruif, C. G. & de Jongh, H. H. J. 2011 Lubrication, adsorption, and rheology of aqueous polysaccharide solutions. Langmuir 27, 3474–3484.Google Scholar
Vandenberghe, N., Vermorel, R. & Villermaux, E. 2013 Star-shaped crack pattern of broken windows. Phys. Rev. Lett. 110 (17).Google Scholar
Vermorel, R., Vandenberghe, N. & Villermaux, E. 2010 Radial cracks in perforated thin sheets. Phys. Rev. Lett. 104 (17).Google Scholar
Walker, C. C., Bassis, J. N., Fricker, H. A. & Czerwinski, R. J. 2013 Structural and environmental controls on Antarctic ice shelf rift propagation inferred from satellite monitoring. J. Geophys. Res. 118 (4), 2354–2364.Google Scholar
Wooding, R. A. & Morelseytoux, H. J. 1976 Multiphase fluid flow through porous media. Annu. Rev. Fluid Mech. 8, 233–274.Google Scholar
Wyatt, N. B. & Liberatore, M. W. 2009 Rheology and viscosity scaling of the polyelectrolyte xanthan gum. J. Appl. Polym. Sci. 114 (6), 4076–4084.Google Scholar
Zhao, B., MacMinn, C. W. & Juanes, R. 2016 Wettability control on multiphase flow in patterned microfluidics. Proc. Natl Acad. Sci. USA 113 (37), 10251–10256.Google Scholar
Zhao, H. & Maher, J. V. 1993 Associating-polymer effects in a Hele-Shaw experiment. Phys. Rev. E 47 (6), 4278–4283.Google Scholar

Sayag et al. supplementary movie 1

Movie of experiment with source flux Q = 3.9 g/cm^3

Download Sayag et al. supplementary movie 1(Video)
Video 3.3 MB

Sayag et al. supplementary movie 2

Movie of experiment with source flux Q = 2.64 g/cm^3

Download Sayag et al. supplementary movie 2(Video)
Video 4.8 MB

Sayag et al. supplementary movie 3

Movie of experiment with source flux Q = 1.36 g/cm^3

Download Sayag et al. supplementary movie 3(Video)
Video 2.8 MB