Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-12T06:58:39.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Instability of sliding viscoplastic films

Published online by Cambridge University Press:  11 February 2021

Thomasina V. Ball Open the ORCID record for Thomasina V. Ball [Opens in a new window]  and
Neil J. Balmforth Open the ORCID record for Neil J. Balmforth [Opens in a new window]
Thomasina V. Ball*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada
Neil J. Balmforth
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BCV6T 1Z2, Canada
*
†Email address for correspondence: tvball@math.ubc.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of sliding spreading films of Herschel–Bulkley fluid is investigated theoretically, motivated by a dramatic fingering pattern observed experimentally and proposed theoretically to originate from an extensional flow instability of shear-thinning fluids. Considering the thin-film limit, we construct axisymmetric base states and then test their stability towards non-axisymmetric perturbations by numerically solving the initial-value problem. We complement the numerics with analytical solutions for early and late times. The stability analysis demonstrates that spreading thinning films are unstable. At late times, where the spreading of the base state becomes self-similar, non-axisymmetric patterns can develop strongly if the fluid has a yield stress or is sufficiently shear thinning.

JFM classification

Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 912 , 10 April 2021 , A23
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Ball, T.V., Balmforth, N.J. & Dufresne, A.P. 2021 a Viscoplastic fingers and fractures in a Hele-Shaw cell. J.Non-Newtonian Fluid Mech. 289, 104492.CrossRefGoogle Scholar
Ball, T.V., Balmforth, N.J., Morris, S.W. & Dufresne, A.P. 2021 b Fracture patterns in viscoplastic gravity currents? In preparation.Google Scholar
Balmforth, N.J. 2018 Viscoplastic asymptotics and other techniques. In Viscoplastic Fluids: From Theory to Application (ed. G. Ovarlez & S. Hormozi), CISM. Springer.CrossRefGoogle Scholar
Balmforth, N.J., Frigaard, I. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121–146.CrossRefGoogle Scholar
Craster, R.V. & Matar, O.K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.CrossRefGoogle Scholar
England, P. & McKenzie, D. 1982 A thin viscous sheet model for continental deformation. Geophys. J. Intl 70 (2), 295–321.CrossRefGoogle Scholar
England, P. & McKenzie, D. 1983 Correction to: a thin viscous sheet model for continental deformation. Geophys. J. Intl 73 (2), 523–532.CrossRefGoogle Scholar
Koch, D.M. & Koch, D.L. 1995 Numerical and theoretical solutions for a drop spreading below a free fluid surface. J.Fluid Mech. 287, 251–278.CrossRefGoogle Scholar
Liu, Y., Balmforth, N.J. & Hormozi, S. 2018 Axisymmetric viscoplastic dambreaks and the slump test. J.Non-Newtonian Fluid Mech. 258, 45–57.CrossRefGoogle Scholar
Luu, L.-H. & Forterre, Y. 2009 Drop impact of yield-stress fluids. J.Fluid Mech. 632, 301–327.CrossRefGoogle Scholar
Luu, L.-H. & Forterre, Y. 2013 Giant drag reduction in complex fluid drops on rough hydrophobic surfaces. Phys. Rev. Lett. 110 (18), 184501.CrossRefGoogle ScholarPubMed
MacAyeal, D.R. 1989 Large-scale ice flow over a viscous basal sediment: theory and application to ice stream B, Antarctica. J.Geophys. Res. 94 (B4), 4071–4087.CrossRefGoogle Scholar
MacAyeal, D.R. & Barcilon, V. 1988 Ice-shelf response to ice-stream discharge fluctuations: I. Unconfined ice tongues. J.Glaciol. 34 (116), 121–127.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.CrossRefGoogle Scholar
Pegler, S.S., Lister, J.R. & Worster, M.G. 2012 Release of a viscous power-law fluid over an inviscid ocean. J.Fluid Mech. 700, 63–76.CrossRefGoogle 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.CrossRefGoogle Scholar
di Pietro, N.D. & Cox, R.G. 1979 The spreading of a very viscous liquid on a quiescent water surface. Q. J. Mech. Appl. Maths 32 (4), 355–381.CrossRefGoogle Scholar
Prager, W. & Hodge, P.G. 1951 Theory of Perfectly Plastic Solids. Wiley.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.CrossRefGoogle Scholar
Sayag, R. 2019 Rifting of extensional flows on a sphere. Phys. Rev. Lett. 123 (21), 214502.CrossRefGoogle Scholar
Sayag, R., Pegler, S.S. & Worster, M.G. 2012 Floating extensional flows. Phys. Fluids 24 (9), 091111.CrossRefGoogle Scholar
Sayag, R. & Worster, M.G. 2019 a Instability of radially spreading extensional flows. Part 1. Experimental analysis. J.Fluid Mech. 881, 722–738.CrossRefGoogle Scholar
Sayag, R. & Worster, M.G. 2019 b Instability of radially spreading extensional flows. Part 2. Theoretical analysis. J.Fluid Mech. 881, 739–771.CrossRefGoogle Scholar
Schoof, C. & Hewitt, I. 2013 Ice-sheet dynamics. Annu. Rev. Fluid Mech. 45, 217–239.CrossRefGoogle Scholar