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Spreading or contraction of viscous drops between plates: single, multiple or annular drops

Published online by Cambridge University Press:  26 August 2021

H.K. Moffatt Open the ORCID record for H.K. Moffatt [Opens in a new window] ,
Howard Guest  and
Herbert E. Huppert Open the ORCID record for Herbert E. Huppert [Opens in a new window]
H.K. Moffatt*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, CambridgeCB3 0WA, UK
Howard Guest
Affiliation:
St John's Innovation Centre, Cowley Rd, Milton, CambridgeCB4 0WS, UK
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, King's College, CambridgeCB2 1ST, UK
*
†Email address for correspondence: [email protected]
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Abstract

The behaviour of a viscous drop squeezed between two horizontal planes (a squeezed Hele-Shaw cell) is treated by both theory and experiment. When the squeezing force $F$ is constant and surface tension is neglected, the theory predicts ultimate growth of the radius $a\sim t^{1/8}$ with time $t$. This theory is first reviewed and found to be in excellent agreement with experiment. Surface tension at the drop boundary reduces the interior pressure, and this effect is included in the analysis, although it is negligibly small in the squeezing experiments. An initially elliptic drop tends to become circular as $t$ increases. More generally, the circular evolution is found to be stable under small perturbations. If, on the other hand, the force is reversed ($F<0$), so that the plates are drawn apart (the ‘contraction’, or ‘lifting plate’, problem), the boundary of the drop is subject to a fingering instability on a scale determined by surface tension. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a ‘one-eighth’ power law, but this is unstable, the instability originating at the boundary of the air bubble, i.e. the inner boundary of the annulus. The air bubble is realised experimentally in two ways: first by simply starting with the drop in the form of an annulus, as nearly circular as possible; and second by forcing four initially separate drops to expand and merge, a process that involves the resolution of ‘contact singularities’ by surface tension. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, before ultimate rupture of the film and complete separation of the plates, fingering spreads also from the boundary of any interior trapped air bubble, and small cavitation bubbles appear in the very low-pressure region, far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis.

JFM classification

Interfacial Flows (free surface): Liquid bridges Interfacial Flows (free surface): Fingering instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 925 , 25 October 2021 , A26
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Anjos, P.H.A. & Li, S. 2020 Weakly nonlinear analysis of the Saffman–Taylor problem in a radially spreading fluid annulus. Phys. Rev. Fluids 5, 054002.CrossRefGoogle Scholar
Ball, T.V. & Huppert, H.E. 2019 Similarity solutions and viscous gravity current adjustment times. J. Fluid Mech. 874, 285–298.CrossRefGoogle Scholar
Beeson-Jones, T.H. & Woods, A.W. 2015 On the selection of viscosity to suppress the Saffman–Taylor instability in a radially spreading annulus. J. Fluid Mech. 782, 127–143.CrossRefGoogle Scholar
Ben-Jacob, E., Godbey, R., Goldenfeld, N.D., Levine, H., Mueller, T. & Sander, L.M. 1985 Experimental demonstration of the role of anisotropy in interfacial pattern formation. Phys. Rev. Lett. 55, 1315–1318.CrossRefGoogle ScholarPubMed
Bikerman, J.J. 1947 The fundamentals of tackiness and adhesion. J. Colloid Sci. 2, 163–175.CrossRefGoogle Scholar
Bretherton, F.P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Brulin, S., Roisman, I.V. & Tropea, C. 2020 Fingering instability of a viscous liquid bridge stretched by an accelerating substrate. J. Fluid Mech. 899, A1.CrossRefGoogle Scholar
Cardoso, S.S.S. & Woods, A.W. 1995 The formation of drops through viscous instability. J. Fluid Mech. 289, 351–378.CrossRefGoogle Scholar
Daripa, P. 2008 a Studies on stability in three-layer Hele–Shaw flows. Phys. Fluids 20, 112101.CrossRefGoogle Scholar
Daripa, P. 2008 b Hydrodynamic stability of multi-layer Hele–Shaw flows. J. Stat. Mech. 2008, P12005.CrossRefGoogle Scholar
Daripa, P. & Ding, X. 2012 Universal stability properties for multi-layer Hele–Shaw flows and application to instability control. SIAM J. Appl. Maths 72, 1667–1685.CrossRefGoogle Scholar
Davis, P.J. 1974 The Schwarz Function and its Applications. The Carus Mathematical Monographs, vol. 17, pp. 228. Math. Assoc. of America.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M.A. 2015 Singulariities: formation, structure, and propagation. Cambridge Texts in Applied Mathematics, pp. 453. Cambridge University Press.CrossRefGoogle Scholar
Gaudet, S., McKinley, G.H. & Stone, H.A. 1996 Extensional deformation of Newtonian liquid bridges. Phys. Fluids 8, 2568–2579.CrossRefGoogle Scholar
Gay, C. 2002 Stickiness — some fundamentals of adhesion. Integr. Compar. Biol. 42, 1123–1126.CrossRefGoogle ScholarPubMed
Gin, C. & Daripa, P. 2015 Stability results for multi-layer radial Hele–Shaw and porous media flows. Phys. Fluids 27, 012101.CrossRefGoogle Scholar
Gustafsson, B. & Vasiliev, A. 2006 Conformal and Potential Analysis in Hele–Shaw Cells, p. 183. Birkhäuser.Google Scholar
Howison, S.D. 2000 A note on the two-phase Hele–Shaw problem. J. Fluid Mech. 409, 243–249.CrossRefGoogle Scholar
Huppert, H.E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H.E., Shepherd, J.B., Sigurdsson, H. & Sparks, S.J. 1982 On lava dome growth, with application to the 1979 lava extrusion of the Soufrière of St Vincent. J. Volcanol. Geotherm. Res. 14, 199–222.CrossRefGoogle Scholar
Kanhurkar, S.D., Patankar, V., ul Islam, T. & Gandhi, P.S. 2019 Stability of viscous fingering in lifted Hele–Shaw cells with a hole. Phys. Rev. Fluids 4, 094003.CrossRefGoogle Scholar
Kelly, E.D. & Hinch, E.J. 1997 Numerical simulations of sink flow in the Hele–Shaw cell with small surface tension. Eur. J. Appl. Maths 8, 533–550.CrossRefGoogle Scholar
La Roche, H., Fernández, J.F., Octavio, M., Loeser, A.G. & Lobb, C.J. 1991 Diffusion-limited- aggregation model for Poisson growth. Phys. Rev. A 44, R6185(R).CrossRefGoogle ScholarPubMed
Lakrout, H., Sergot, P. & Creton, C. 1999 Direct observation of cavitation and fibrillation in a probe tack experiment on model acrylic pressure-sensitive adhesives. J. Adhes. 69, 307–359.CrossRefGoogle Scholar
Moffatt, H.K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 1–18.CrossRefGoogle Scholar
Moffatt, H.K. 1977 Six lectures on general fluid dynamics and two on hydromagnetic dynamo theory. In Fluid Dynamics (eds. R. Balian & J.-L. Peube), pp. 149–233. Gordon and Breach.Google Scholar
Paterson, L. 1981 Radial fingering in a Hele Shaw cell. J. Fluid Mech. 113, 513–529.CrossRefGoogle Scholar
Paterson, L. 1985 Fingering with miscible fluids in a Hele Shaw cell. Phys. Fluids 28, 26–30.CrossRefGoogle Scholar
Pihler-Puzović, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous liquid by elastic membranes. Phys. Rev. Lett. 108, 074502.CrossRefGoogle Scholar
Poivet, S., Nallet, F., Gay, C., Teissiere, J. & Fabre, P. 2004 Force response of a viscous liquid in a probe-tack geometry: fingering versus cavitation. Eur. Phys. J. E 15, 97–116.CrossRefGoogle Scholar
Rabaut, M., Couder, Y. & Gerard, N. 1988 Dynamics and stability of anomalous Saffman–Taylor fingers. Phys. Rev. A 37, 935–947.CrossRefGoogle Scholar
Saffman, P.G. & Taylor, G.I. 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, 312–329.Google Scholar
Savina, T.V. 2021 On a two-fluid Hele–Shaw problem with an elliptical interface. J. Differ. Equ. 270, 787–808.CrossRefGoogle Scholar
Savina, T.V. & Nepomnyashchy, A.A. 2015 On a Hele–Shaw flow with a time-dependent gap in the presence of surface tension. J. Phys. A: Math. Theor. 48, 125501.CrossRefGoogle Scholar
Shelley, M.J., Tiany, F.-R. & Wlodarski, K. 1997 Hele–Shaw flow and pattern formation in a time-dependent gap. Nonlinearity 10, 1471–1495.CrossRefGoogle Scholar
Stefan, J. 1874 Veruche über die scheinbare Adhäsion. Sitzber. Akad. WisE. Wien, Math. naturw. Kl. A 69, 713.Google Scholar
Tanveer, S. 2000 Surprises in viscous fingering. J. Fluid Mech. 409, 273–308.CrossRefGoogle Scholar
Ward, T. 2006 Radial spreading of a viscous drop between parallel-plane surfaces. Phys. Fluids 354, 816–824.Google Scholar
Ward, T. 2011 Capillary-pressure driven adhesion of rigid-planar surfaces. J. Colloid Interface Sci. 18, 093101.Google Scholar
Webber, J.J. & Huppert, H.E. 2020 Time to approach similarity. Q. J. Mech. Appl. Maths 73, 1–23.Google Scholar
Zhang, S.-Z., Louis, E., Pla, O. & Guinea, F. 1998 Linear stability analysis of the Hele–Shaw cell with lifting plates. Eur. Phys. J. B 1, 123–127.CrossRefGoogle Scholar
Zhao, M., Anjos, P.H.A., Lowengrub, J. & Li, S. 2020 Pattern formation of the three-layer Saffman–Taylor problem in a radial Hele–Shaw cell. Phys. Rev. Fluids 5, 124005.CrossRefGoogle Scholar

Moffatt et al. supplementary movie 1

This movie was created in three stages, using the same camera for each stage: initial stage 1 is a video of duration 174s; squeezing stage 2 is a succession of stills at 10s intervals, lasting for 7m 50s; levering (fingering) stage 3 is a video of duration 77s; the whole is compressed for aesthetic reasons.This movie was created in three stages, using the same camera for each stage: initial stage 1 is a video of duration 174s; squeezing stage 2 is a succession of stills at 10s intervals, lasting for 7m 50s; levering (fingering) stage 3 is a video of duration 77s; the whole is compressed for aesthetic reasons.

Download Moffatt et al. supplementary movie 1(Video)
Video 59.2 MB

Moffatt et al. supplementary movie 2

This movie was created in three stages, using the same camera for each stage: initial stage 1 is a video of duration 215s; squeezing stage 2 is a succession of stills at 10s intervals, lasting for 7h 42m 24s; levering (fingering) stage 3 is a video of duration 109s; the whole is compressed for aesthetic reasons.

Download Moffatt et al. supplementary movie 2(Video)
Video 110.9 MB

Moffatt et al. supplementary movie 3

This movie was created in three stages, using the same camera for each stage: initial stage 1 is a video of duration 179s; squeezing stage 2 is a succession of stills at 10s intervals, lasting for 7m 58s; levering (fingering) stage 3 is a video of duration 76s; the whole is compressed for aesthetic reasons.

Download Moffatt et al. supplementary movie 3(Video)
Video 48.5 MB