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Instability of radially spreading extensional flows. Part 2. Theoretical 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: roiy@bgu.ac.il
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Abstract

The interface of a strain-rate-softening fluid that displaces a low-viscosity fluid in a circular geometry with negligible drag can develop finger-like patterns separated by regions in which the fluid appears to be torn apart. Such patterns were observed and explored experimentally in Part 1 using polymeric solutions. They do not occur when the viscosity of the displacing fluid is constant, or when the displacing fluid has no-slip conditions along its boundaries. We investigate theoretically the formation of tongues at the interface of an axisymmetric initial state. We show that finger-like patterns can emerge when circular interfaces of strain-rate-softening fluids displace low-viscosity fluids between stress-free boundaries. The instability, which is fundamentally different from the classical Saffman–Taylor viscous fingering, is driven by the tension that builds up along the circular front of the propagating fluid. That destabilising tension is a geometrical consequence and is present independently of the nonlinear properties of the fluid. Shear stresses stabilise the growth either along extended circumferential streamlines or through a street of vortices. However, such stabilising processes become weaker, thereby allowing the instability to develop, the more strain-rate-softening the fluid is. The theoretical model that we present predicts the main experimental observations made in Part 1. In particular, the patterns we predict using linear-stability theory are consistent with the strongly nonlinear experimental patterns. Our model depends on a single dimensionless number representing the power-law exponent, which implies that the instability we describe could arise in any extensional flow of strain-rate-softening material, ranging from suspensions that rupture in squeeze experiments to rifts formed in ice shelves.

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. 739 - 771
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Copyright
© 2019 Cambridge University Press 

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References

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
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
Cardoso, S. S. S. & Woods, A. W. 1995 The formation of drops through viscous instability. J. Fluid Mech. 289, 351–378.Google Scholar
Coussot, P. 1999 Saffman-Taylor instability in yield-stress fluids. J. Fluid Mech. 380, 363–376.10.1017/S002211209800370XGoogle Scholar
Dallaston, M. C. & Hewitt, I. J. 2014 Free-boundary models of a meltwater conduit. Phys. Fluids 26 (8), 083101.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
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
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2011 Fluid mixing from viscous fingering. Phys. Rev. Lett. 106, 194502.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
Lindner, A., Bonn, D. & Meunier, J. 2000a Viscous fingering in a shear-thinning fluid. Phys. Fluids 12 (2), 256–261.10.1063/1.870303Google 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
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. Pharm. Sci. 29 (1), 22–34.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
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, R5.10.1017/jfm.2012.545Google Scholar
Sayag, R. & Worster, M. G. 2019 Instability of radially spreading extensional flows. Part 1. Experimental analysis. J. Fluid Mech. 881, 722–738.Google Scholar
Vandenberghe, N., Vermorel, R. & Villermaux, E. 2013 Star-shaped crack pattern of broken windows. Phys. Rev. Lett. 110, 174302.Google Scholar
Vermorel, R., Vandenberghe, N. & Villermaux, E. 2010 Radial cracks in perforated thin sheets. Phys. Rev. Lett. 104, 175502.Google Scholar
Wooding, R. A. & Morelseytoux, H. J. 1976 Multiphase fluid-flow through porous-media. Annu. Rev. Fluid Mech. 8, 233–274.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

Emergence of vortices in the secondary flow as the wavenumber k grows (n=100, δ=0.75).

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