Hostname: page-component-848d4c4894-2xdlg Total loading time: 0 Render date: 2024-07-02T21:11:29.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Lubricated viscous gravity currents of power-law fluids. Part 2. Stability analysis

Published online by Cambridge University Press:  11 April 2022

Lucas Tsun-yin Leung  and
Katarzyna N. Kowal Open the ORCID record for Katarzyna N. Kowal [Opens in a new window]
Lucas Tsun-yin Leung
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Trinity College, University of Cambridge, Cambridge CB2 1TQ, UK
Katarzyna N. Kowal*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the stability of radially spreading, gravity-driven thin films of power-law fluids, lubricated from below by another power-law viscous fluid. Such flows are susceptible to a viscous fingering instability, also known as a non-porous viscous fingering instability, when a less viscous fluid intrudes beneath a more viscous fluid. In contrast to the Saffman–Taylor instability, such instabilities originate from a jump in hydrostatic pressure gradient across the intrusion front, associated with gradients in the upper surface. These are stabilised by buoyancy forces associated with the lower layer near its nose, and all instabilities are suppressed above a critical density difference. We find that shear-thinning flows are more prone to instability than Newtonian and shear-thickening flows. Lower consistency ratios are sufficient for the onset of instability of shear-thinning flows, and the stabilising influences of buoyancy forces are suppressed. As such, higher density differences are required to suppress the instability completely.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Lubrication theory Non-Newtonian Flows: Non-Newtonian Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 940 , 10 June 2022 , A27
Check for updates
Copyright
© The Author(s), 2022. 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

Al-Housseiny, T.T., Tsai, P.A. & Stone, H.A. 2012 Control of interfacial instabilities using flow geometry. Nat. Phys. 8, 747–750.CrossRefGoogle Scholar
Balmforth, N.J. & Craster, R.V. 2000 Dynamics of cooling domes of viscoplastic fluid. J. Fluid Mech. 422, 225–248.CrossRefGoogle Scholar
Balmforth, N.J., Craster, R.V. & Toniolo, C. 2003 Interfacial instability in non-Newtonian fluid layers. Phys. Fluids 15 (11), 3370–3384.CrossRefGoogle Scholar
Ben-Jacob, E., Godbey, R., Goldenfeld, N.D., Koplik, J., Levine, H., Mueller, T. & Sander, L.M. 1985 Experimental demonstration of the role of anisotropy in interfacial pattern formation. Phys. Rev. Lett. 55, 1892.CrossRefGoogle ScholarPubMed
Buka, A., Kertész, J. & Vicsek, T. 1986 Transitions of viscous fingering patterns in nematic liquid crystals. Nature 323, 424–425.CrossRefGoogle Scholar
Cazabat, A.M., Heslot, F., Troian, S.M. & Carles, P. 1990 Fingering instability of thin spreading films driven by temperature gradients. Nature 346 (6287), 824–826.CrossRefGoogle Scholar
Charru, F. & Hinch, E.J. 2000 Phase diagram of interfacial instabilities in a two-layer Couette flow and mechanism of the long-wave instability. J. Fluid Mech. 414, 195–223.CrossRefGoogle Scholar
Chen, K.P. 1993 Wave formation in the gravity driven low Reynolds number flow of two liquid films down an inclined plane. Phys. Fluids A 5 (12), 3038–3048.CrossRefGoogle Scholar
Cinar, Y., Riaz, A. & Tchelepi, H.A. 2009 Experimental study of CO2 injection into saline formations. Soc. Petrol. Engng J. 14, 589–594.Google Scholar
Dias, E.O., Alvarez-Lacalle, E., Carvalho, M.S. & Miranda, J.A. 2012 Minimization of viscous fluid fingering: a variational scheme for optimal flow rates. Phys. Rev. Lett. 109, 144502.CrossRefGoogle ScholarPubMed
Dias, E.O. & Miranda, J.A. 2013 Taper-induced control of viscous fingering in variable-gap Hele-Shaw flows. Phys. Rev. E 87, 053015.CrossRefGoogle ScholarPubMed
Engelhardt, H., Humphrey, N., Kamb, B. & Fahnestock, M. 1990 Physical conditions at the base of a fast moving Antarctic ice stream. Science 248, 57–59.CrossRefGoogle ScholarPubMed
Fast, P., Kondic, L., Shelley, M.J. & Palffy-Muhoray, P. 2001 Pattern formation in non-Newtonian Hele-Shaw flow. Phys. Fluids 13, 1191–1212.CrossRefGoogle Scholar
Fink, J.H. & Griffiths, R.W. 1990 Radial spreading of viscous-gravity currents with solidifying crust. J. Fluid Mech. 221, 485–509.CrossRefGoogle Scholar
Fink, J.H. & Griffiths, R.W. 1998 Morphology, eruption rates and rheology of lava domes: insights from laboratory models. J. Geophys. Res. 103, 527–545.CrossRefGoogle Scholar
Hewitt, I.J. & Schoof, C. 2017 Models for polythermal glaciers and ice sheets. Cryosphere 11, 541–551.CrossRefGoogle Scholar
Hindmarsh, R.C.A. 2004 Thermoviscous stability of ice-sheet flows. J. Fluid Mech. 502, 17–40.CrossRefGoogle Scholar
Hindmarsh, R.C.A. 2009 Consistent generation of ice-streams via thermo-viscous instabilities modulated by membrane stresses. Geophys. Res. Lett. 36, L06502.CrossRefGoogle Scholar
Hooper, A.P. & Boyd, W.G.C. 1983 Shear-flow instability at the interface between two viscous fluids. J. Fluid Mech. 128, 507–528.CrossRefGoogle Scholar
Hull, D. 1999 Fractology. Cambridge University Press.Google Scholar
Huppert, H.E. 1982 Flow and instability of a viscous current down a slope. Nature 300, 427–429.CrossRefGoogle Scholar
Juel, A. 2012 Flattened fingers. Nat. Phys. 8, 706–707.CrossRefGoogle Scholar
Kagei, N., Kanie, D. & Kawaguchi, M. 2005 Viscous fingering in shear thickening silica suspensions. Phys. Fluids 17, 054103.CrossRefGoogle Scholar
Kamb, B. 1970 Sliding motion of glaciers – theory and observation. Rev. Geophys. Space Phys. 8 (4), 673–728.CrossRefGoogle Scholar
Kataoka, D.E. & Troian, S.M. 1999 Patterning liquid flow on the microscopic scale. Nature 402, 794–797.CrossRefGoogle Scholar
Kondic, L., Shelley, M.J. & Palffy-Muhoray, P. 1998 Non-Newtonian Hele-Shaw flow and the Saffman–Taylor instability. Phys. Rev. Lett. 80, 1433–1436.CrossRefGoogle Scholar
Kowal, K.N. 2021 Viscous banding instabilities: non-porous viscous fingering. J. Fluid Mech. 926, A4.CrossRefGoogle Scholar
Kowal, K.N. & Worster, M.G. 2015 Lubricated viscous gravity currents. J. Fluid Mech. 766, 626–655.CrossRefGoogle Scholar
Kowal, K.N. & Worster, M.G. 2019 a Stability of lubricated viscous gravity currents. Part 1. Internal and frontal analyses and stabilisation by horizontal shear. J. Fluid Mech. 871, 970–1006.CrossRefGoogle Scholar
Kowal, K.N. & Worster, M.G. 2019 b Stability of lubricated viscous gravity currents. Part 2. Global analysis and stabilisation by buoyancy forces. J. Fluid Mech. 871, 1007–1027.CrossRefGoogle Scholar
Kumar, P., Zuri, S., Kogan, D., Gottlieb, M. & Sayag, R. 2021 Lubricated gravity currents of power-law fluids. J. Fluid Mech. 916, A33.CrossRefGoogle Scholar
Kyrke-Smith, T.M., Katz, R.F. & Fowler, A.C. 2014 Subglacial hydrology and the formation of ice streams. Proc. R. Soc. A 470, 20130494.CrossRefGoogle ScholarPubMed
Kyrke-Smith, T.M., Katz, R.F. & Fowler, A.C. 2015 Subglacial hydrology as a control on emergence, scale, and spacing of ice streams. J. Geophys. Res. 120, 1501–1514.CrossRefGoogle Scholar
Leung, L.T. & Kowal, K.N. 2022 Lubricated viscous gravity currents of power-law fluids. Part 1. Self-similar flow regimes. J. Fluid Mech. 940, A26.Google Scholar
Li, S., Lowengrub, J.S., Fontana, J. & Palffy-Muhoray, P. 2009 Control of viscous fingering patterns in a radial Hele-Shaw cell. Phys. Rev. Lett. 102, 174501.CrossRefGoogle Scholar
Loewenherz, D.S. & Lawrence, C.J. 1989 The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number. Phys. Fluids A 1 (10), 1686–1693.CrossRefGoogle Scholar
Nase, J., Derks, D. & Lindner, A. 2011 Dynamic evolution of fingering patterns in a lifted Hele-Shaw cell. Phys. Fluids 23, 123101.CrossRefGoogle Scholar
Nye, J.F. 1969 A calculation on sliding of ice over a wavy surface using a Newtonian viscous approximation. Proc. R. Soc. Lond. A 311 (1506), 445–467.Google Scholar
Orr, F.M. & Taber, J.J. 1984 Use of carbon dioxide in enhanced oil recovery. Science 224, 563–569.CrossRefGoogle ScholarPubMed
Pihler-Puzovic, 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, 074502.CrossRefGoogle Scholar
Pihler-Puzovic, D., Juel, A. & Heil, M. 2014 The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Phys. Fluids 26, 022102.CrossRefGoogle Scholar
Pihler-Puzovic, D., Perillat, 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.CrossRefGoogle Scholar
Pouliquen, O., Delour, J. & Savage, S.B. 1997 Fingering in granular flows. Nature 386 (6627), 816–817.CrossRefGoogle Scholar
Reinelt, D.A. 1995 The primary and inverse instabilities of directional fingering. J. Fluid Mech. 285, 303–327.CrossRefGoogle Scholar
Sayag, R. & Tziperman, E. 2008 Spontaneous generation of pure ice streams via flow instability: role of longitudinal shear stresses and subglacial till. J. Geophys. Res. 113, B05411.Google Scholar
Schoof, C. & Mantelli, E. 2021 The role of sliding in ice stream formation. Proc. R. Soc. A 477, 20200870.CrossRefGoogle Scholar
Snyder, D. & Tait, S. 1998 A flow-front instability in viscous gravity currents. J. Fluid Mech. 369, 1–21.CrossRefGoogle Scholar
Stasiuk, M.V., Jaupart, C. & Sparks, R.S.J. 1993 Influence of cooling on lava-flow dynamics. Geology 21, 335–338.2.3.CO;2>CrossRefGoogle Scholar
Taylor, G.I. 1963 Cavitation of a viscous fluid in narrow passages. J. Fluid Mech. 16, 595–619.CrossRefGoogle Scholar
Troian, S.M., Herbolzheimer, E., Safran, S.A. & Joann, J.F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10 (1), 25–30.CrossRefGoogle Scholar
Weertman, J. 1957 On the sliding of glaciers. J. Glaciol. 3, 33–38.CrossRefGoogle Scholar
Whitehead, J.A. & Helfrich, K.R. 1991 Instability of flow with temperature-dependent viscosity – a model of magma dynamics. J. Geophys. Res. 96, 4145–4155.CrossRefGoogle Scholar
Yih, C.S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.CrossRefGoogle Scholar