Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T06:46:48.139Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamic stabilisation of Rayleigh–Plateau modes on a liquid cylinder

Published online by Cambridge University Press:  02 August 2022

Sagar Patankar ,
Saswata Basak  and
Ratul Dasgupta Open the ORCID record for Ratul Dasgupta [Opens in a new window]
Sagar Patankar
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai 400076, India
Saswata Basak
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai 400076, India
Ratul Dasgupta*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay, Powai 400076, India
*
†Email address for correspondence: dasgupta.ratul@iitb.ac.in
Get access Rights & Permissions [Opens in a new window]

Abstract

We demonstrate dynamic stabilisation of axisymmetric Fourier modes susceptible to the classical Rayleigh–Plateau (RP) instability on a liquid cylinder by subjecting it to a radial oscillatory body force. Viscosity is found to play a crucial role in this stabilisation. Linear stability predictions are obtained via Floquet analysis demonstrating that RP unstable modes can be stabilised using radial forcing. We also solve the linearised, viscous initial-value problem for free-surface deformation obtaining an equation governing the amplitude of a three-dimensional Fourier mode. This equation generalizes the Mathieu equation governing Faraday waves on a cylinder derived earlier in Patankar et al. (J. Fluid Mech., vol. 857, 2018, pp. 80–110), is non-local in time and represents the cylindrical analogue of its Cartesian counterpart (Beyer & Friedrich, Phys. Rev. E, vol. 51, issue 2, 1995, p. 1162). The memory term in this equation is physically interpreted and it is shown that, for highly viscous fluids, its contribution can be sizeable. Predictions from the numerical solution to this equation demonstrate the predicted RP mode stabilisation and are in excellent agreement with simulations of the incompressible Navier–Stokes equations (up to the simulation time of several hundred forcing cycles).

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Faraday waves Flow Control: Instability control
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 946 , 10 September 2022 , A2
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

Adou, A.-higo E. & Tuckerman, L.S 2016 Faraday instability on a sphere: Floquet analysis. J. Fluid Mech. 805, 591–610.CrossRefGoogle Scholar
Arbell, H & Fineberg, J 2000 Temporally harmonic oscillons in newtonian fluids. Phys. Rev. Lett. 85 (4), 756–759.CrossRefGoogle ScholarPubMed
Basak, S., Farsoiya, P.K. & Dasgupta, R. 2021 Jetting in finite-amplitude, free, capillary-gravity waves. J. Fluid Mech. 909, A3.CrossRefGoogle Scholar
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013 The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496–523.Google Scholar
Bechhoefer, J., Ego, V., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288, 325–350.CrossRefGoogle Scholar
Benilov, E.S. 2016 Stability of a liquid bridge under vibration. Phys. Rev. E 93 (6), 063118.CrossRefGoogle ScholarPubMed
Benjamin, T.B. & Ursell, F.J. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225 (1163), 505–515.Google Scholar
Berger, S.A. 1988 Initial-value stability analysis of a liquid jet. SIAM J. Appl. Maths 48 (5), 973–991.CrossRefGoogle Scholar
Beyer, J & Friedrich, R 1995 Faraday instability: linear analysis for viscous fluids. Phys. Rev. E 51 (2), 1162–1168.Google ScholarPubMed
Binz, M., Rohlfs, W. & Kneer, R. 2014 Direct numerical simulations of a thin liquid film coating an axially oscillating cylindrical surface. Fluid Dyn. Res. 46 (4), 041402.Google Scholar
Boffetta, G., Magnani, M. & Musacchio, S. 2019 Suppression of Rayleigh–Taylor turbulence by time-periodic acceleration. Phys. Rev. E 99 (3), 033110.CrossRefGoogle ScholarPubMed
Boronski, P. & Tuckerman, L.S 2007 Poloidal–Toroidal decomposition in a finite cylinder. I: influence matrices for the magnetohydrodynamic equations. J. Comput. Phys. 227 (2), 1523–1543.CrossRefGoogle Scholar
Bostwick, J.B. & Steen, P.H. 2018 Static rivulet instabilities: varicose and sinuous modes. J. Fluid Mech. 837, 819–838.CrossRefGoogle Scholar
Cerda, E. & Tirapegui, E. 1997 Faraday's instability for viscous fluids. Phys. Rev. Lett. 78 (5), 859–862.CrossRefGoogle Scholar
Cerda, E.A. & Tirapegui, E.L. 1998 Faraday's instability in viscous fluid. J. Fluid Mech. 368, 195–228.Google Scholar
Chandrasekhar, S 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chen, T.-Y. & Tsamopoulos, J. 1993 Nonlinear dynamics of capillary bridges: theory. J. Fluid Mech. 255, 373–409.CrossRefGoogle Scholar
Davis, S.H 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98 (2), 225–242.CrossRefGoogle Scholar
Driessen, T. 2013 Drop formation from axi-symmetric fluid jets. Dissertation, University of Twente. doi:10.3990/1.9789036535786.CrossRefGoogle Scholar
Driessen, T., Sleutel, P., Dijksman, F., Jeurissen, R. & Lohse, D. 2014 Control of jet breakup by a superposition of two Rayleigh–Plateau-unstable modes. J. Fluid Mech. 749, 275–296.CrossRefGoogle Scholar
Ebo-Adou, A.-higo, Tuckerman, L.S, Shin, S., Chergui, J. & Juric, D. 2019 Faraday instability on a sphere: numerical simulation. J. Fluid Mech. 870, 433–459.CrossRefGoogle Scholar
Edwards, W.S.tuart & Fauve, S 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.CrossRefGoogle Scholar
Erdelyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F.G 1954 Tables of Integral Transforms: Vol. 2. McGraw-Hill Book Company.Google Scholar
Faraday, M. 1837 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. In Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London, pp. 49–51. The Royal Society London.CrossRefGoogle Scholar
Farsoiya, P.K., Mayya, Y.S. & Dasgupta, R. 2017 Axisymmetric viscous interfacial oscillations–theory and simulations. J. Fluid Mech. 826, 797–818.CrossRefGoogle Scholar
Farsoiya, P.K., Popinet, S. & Deike, L. 2021 Bubble-mediated transfer of dilute gas in turbulence. J. Fluid Mech. 920, A34.CrossRefGoogle Scholar
Farsoiya, P.K., Roy, A. & Dasgupta, R. 2020 Azimuthal capillary waves on a hollow filament – the discrete and the continuous spectrum. J. Fluid Mech. 883, A21.CrossRefGoogle Scholar
Fauve, S 1998 Waves on interfaces. In Free Surface Flows (ed. H.C. Kuhlmann & H.-J. Rath), pp. 1–44. Springer.Google Scholar
García, F.J. & González, H 2008 Normal-mode linear analysis and initial conditions of capillary jets. J. Fluid Mech. 602, 81–117.CrossRefGoogle Scholar
Goren, S.L 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12 (2), 309–319.CrossRefGoogle Scholar
Haefner, S., Benzaquen, M., Bäumchen, O., Salez, T., Peters, R., McGraw, J.D., Jacobs, K., Raphaël, E. & Dalnoki-Veress, K. 2015 Influence of slip on the Plateau–Rayleigh instability on a fibre. Nat. Commun. 6 (1), 7409.CrossRefGoogle ScholarPubMed
Haynes, M, Vega, E.J., Herrada, M.A., Benilov, E.S. & Montanero, J.M. 2018 Stabilization of axisymmetric liquid bridges through vibration-induced pressure fields. J. Colloid Interface Sci. 513, 409–417.CrossRefGoogle ScholarPubMed
Holt, R.G.lynn & Trinh, E.H 1996 Faraday wave turbulence on a spherical liquid shell. Phys. Rev. Lett. 77 (7), 1274–1277.CrossRefGoogle ScholarPubMed
Jacqmin, D. & Duval, W.M.B. 1988 Instabilities caused by oscillating accelerations normal to a viscous fluid–fluid interface. J. Fluid Mech. 196, 495–511.CrossRefGoogle Scholar
Kudrolli, A & Gollub, J.P 1996 Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97 (1–3), 133–154.CrossRefGoogle Scholar
Kumar, S. 2000 Mechanism for the Faraday instability in viscous liquids. Phys. Rev. E 62, 1416–1419.CrossRefGoogle ScholarPubMed
Kumar, K. & Tuckerman, L.S 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Liu, F., Kang, N., Li, Y. & Wu, Q. 2019 Experimental investigation on the atomization of a spherical droplet induced by Faraday instability. Expl Therm. Fluid Sci. 100, 311–318.CrossRefGoogle Scholar
Liu, Z. & Liu, Z. 2006 Linear analysis of three-dimensional instability of non-newtonian liquid jets. J. Fluid Mech. 559, 451–459.CrossRefGoogle Scholar
Lowry, B.J. & Steen, P.H. 1994 Stabilization of an axisymmetric liquid bridge by viscous flow. Intl J. Multiphase Flow 20 (2), 439–443.CrossRefGoogle Scholar
Lowry, B.J & Steen, P.H 1995 Flow-influenced stabilization of liquid columns. J. Colloid Interface Sci. 170 (1), 38–43.CrossRefGoogle Scholar
Lowry, B.J & Steen, P.H 1997 Stability of slender liquid bridges subjected to axial flows. J. Fluid Mech. 330, 189–213.CrossRefGoogle Scholar
Maity, D.K. 2021 Floquet analysis on a viscous cylindrical fluid surface subject to a time-periodic radial acceleration. Theor. Comput. Fluid Dyn. 35 (1), 93–107.CrossRefGoogle Scholar
Maity, D.K., Kumar, K. & Khastgir, S.P. 2020 Instability of a horizontal water half-cylinder under vertical vibration. Exp. Fluids 61 (2), 25.CrossRefGoogle Scholar
Marqués, F. 1990 On boundary conditions for velocity potentials in confined flows: application to Couette flow. Phys. Fluids A 2 (5), 729–737.CrossRefGoogle Scholar
Marr-Lyon, M.J, Thiessen, D.B & Marston, P.L 1997 Stabilization of a cylindrical capillary bridge far beyond the Rayleigh–Plateau limit using acoustic radiation pressure and active feedback. J. Fluid Mech. 351, 345–357.CrossRefGoogle Scholar
Marr-Lyon, M.J, Thiessen, D.B & Marston, P.L 2001 Passive stabilization of capillary bridges in air with acoustic radiation pressure. Phys. Rev. Lett. 86 (11), 2293–2296.CrossRefGoogle ScholarPubMed
Matthiessen, L. 1868 Akustische versuche, die kleinsten transversalwellen der flüssigkeiten betreffend. Ann. Phys. 210 (5), 107–117.CrossRefGoogle Scholar
Melde, F. 1860 Ueber die erregung stehender wellen eines fadenförmigen körpers. Ann. Phys. 187 (12), 513–537.CrossRefGoogle Scholar
Moldavsky, L., Fichman, M. & Oron, A. 2007 Dynamics of thin liquid films falling on vertical cylindrical surfaces subjected to ultrasound forcing. Phys. Rev. E 76 (4), 045301.CrossRefGoogle ScholarPubMed
Mollot, D.J., Tsamopoulos, J, Chen, T-Y & Ashgriz, N 1993 Nonlinear dynamics of capillary bridges: experiments. J. Fluid Mech. 255, 411–435.CrossRefGoogle Scholar
Monin, A.S. & Yaglom, A.M. 2007 Statistical Fluid Mechanics, Volume I: Mechanics of Turbulence. Dover.Google Scholar
Mostert, W. & Deike, L. 2020 Inertial energy dissipation in shallow-water breaking waves. J. Fluid Mech. 890, A12.CrossRefGoogle Scholar
Nicolás, J.A. 1992 Magnetohydrodynamic stability of cylindrical liquid bridges under a uniform axial magnetic field. Phys. Fluids A 4 (11), 2573–2577.CrossRefGoogle Scholar
Olver, F.W.J. (Eds), et al. 2021 Nist digital library of mathematical functions. http://dlmf.nist.gov/, Release 1.1.3 of 2021-09-15.Google Scholar
Patankar, S., Basak, S. & Dasgupta, R. 2019 Fragmenting a viscous cylindrical fluid filament using the faraday instability. In APS Division of Fluid Dynamics Meeting Abstracts, pp. S34–001.Google Scholar
Patankar, S., Basak, S. & Dasgupta, R. 2022 Dynamic stabilisation of Rayleigh–Plateau modes on a liquid cylinder. arxiv.org.2202.03102.Google Scholar
Patankar, S., Basak, S., Farsoiya, P.K. & Dasgupta, R. 2020 Viscous stabilisation of Rayleigh–Plateau modes on a cylindrical filament through radial oscillatory forcing. https://gfm.aps.org/meetings/dfd-2020/5f5f0e8d199e4c091e67bdbd. In 73th Annual Meeting of the APS Division of Fluid Dynamics. APS Division of Fluid Dynamics.Google Scholar
Patankar, S., Farsoiya, P.K. & Dasgupta, R. 2018 Faraday waves on a cylindrical fluid filament–generalised equation and simulations. J. Fluid Mech. 857, 80–110.CrossRefGoogle Scholar
Piriz, A.R., Prieto, G.R.odriguez, Diaz, I.M.uñoz, Cela, J.J.L. & Tahir, N.A. 2010 Dynamic stabilization of Rayleigh–Taylor instability in newtonian fluids. Phys. Rev. E 82 (2), 026317.CrossRefGoogle ScholarPubMed
Plateau, J. 1873 a Experimental and theoretical statics of liquids subject to molecular forces only.Google Scholar
Plateau, J.A.F. 1873 b Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires, vol. 2. Gauthier-Villars.Google Scholar
Popinet, S. 2014 Basilisk. http://basilisk.fr.Google Scholar
Prosperetti, A. 1976 Viscous effects on small-amplitude surface waves. Phys. Fluids 19 (2), 195–203.CrossRefGoogle Scholar
Prosperetti, A. 1981 Motion of two superposed viscous fluids. Phys. Fluids 24 (7), 1217–1223.CrossRefGoogle Scholar
Prosperetti, A. 2011 Advanced Mathematics for Applications. Cambridge University Press.CrossRefGoogle Scholar
Raco, R.J 1968 Electrically supported column of liquid. Science 160 (3825), 311–312.CrossRefGoogle Scholar
Ramadugu, R., Perlekar, P. & Govindarajan, R. 2022 Surface tension as the destabiliser of a vortical interface. J. Fluid Mech. 936, A45.CrossRefGoogle Scholar
Raman, C.V. 1909 The maintenance of forced oscillations of a new type. Nature 82 (2093), 156–157.Google Scholar
Raman, C.V. 1912 Experimental investigations on the maintenance of vibrations. Proc. Indian Assoc. Cultiv. Sci. Bull. 6, 1–40.Google Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 1 (1), 4–13.CrossRefGoogle Scholar
Rayleigh, Lord 1883 XXXIII. On maintained vibrations. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 15 (94), 229–235.Google Scholar
Rayleigh, Lord 1887 XVII. On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 24 (147), 145–159.CrossRefGoogle Scholar
Rayleigh, Lord 1892 a On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. 34 (207), 145–154.CrossRefGoogle Scholar
Rayleigh, Lord 1892 b XVI. On the instability of a cylinder of viscous liquid under capillary force. Lond. Edinb. Dubl. Phil. Mag. J. Sci. 34 (207), 145–154.CrossRefGoogle Scholar
Rohlfs, W., Binz, M. & Kneer, R. 2014 On the stabilizing effect of a liquid film on a cylindrical core by oscillatory motions. Phys. Fluids 26 (2), 022101.CrossRefGoogle Scholar
Rutland, D.F. & Jameson, G.J. 1971 A non-linear effect in the capillary instability of liquid jets. J. Fluid Mech. 46 (2), 267–271.CrossRefGoogle Scholar
Sankaran, S. & Saville, D.A. 1993 Experiments on the stability of a liquid bridge in an axial electric field. Phys. Fluids A 5 (4), 1081–1083.CrossRefGoogle Scholar
Sanz, A. 1985 The influence of the outer bath in the dynamics of axisymmetric liquid bridges. J. Fluid Mech. 156, 101–140.CrossRefGoogle Scholar
Shats, M., Francois, N., Xia, H. & Punzmann, H. 2014 Turbulence driven by faraday surface waves. In International Journal of Modern Physics: Conference Series, vol. 34, p. 1460379. World Scientific.Google Scholar
Singh, M., Farsoiya, P.K. & Dasgupta, R. 2019 Test cases for comparison of two interfacial solvers. Intl J. Multiphase Flow 115, 75–92.CrossRefGoogle Scholar
Song, M., Kartawira, K., Hillaire, K.D, Li, C., Eaker, C.B, Kiani, A., Daniels, K.E & Dickey, M.D 2020 Overcoming Rayleigh–Plateau instabilities: stabilizing and destabilizing liquid–metal streams via electrochemical oxidation. Proc. Natl Acad. Sci. 117 (32), 19026–19032.CrossRefGoogle ScholarPubMed
Sterman-Cohen, E., Bestehorn, M. & Oron, A. 2017 Rayleigh–Taylor instability in thin liquid films subjected to harmonic vibration. Phys. Fluids 29 (5), 052105.CrossRefGoogle Scholar
Stone, H.A, Stroock, A.D & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.CrossRefGoogle Scholar
Taylor, G.I. 1969 Electrically driven jets. Proc. R. Soc. Lond. A 313 (1515), 453–475.Google Scholar
Thiele, U., Vega, J.M & Knobloch, E. 2006 Long-wave Marangoni instability with vibration. J. Fluid Mech. 546, 61–87.CrossRefGoogle Scholar
Thiessen, D.B, Marr-Lyon, M.J & Marston, P.L 2002 Active electrostatic stabilization of liquid bridges in low gravity. J. Fluid Mech. 457, 285–294.CrossRefGoogle Scholar
Troyon, F. & Gruber, R. 1971 Theory of the dynamic stabilization of the Rayleigh–Taylor instability. Phys. Fluids 14 (10), 2069–2073.Google Scholar
Tyndall, J. 1901 Sound, vol. 7. Collier.Google Scholar
Vega, E.J. & Montanero, J.M. 2009 Damping of linear oscillations in axisymmetric liquid bridges. Phys. Fluids 21 (9), 092101.CrossRefGoogle Scholar
Vukasinovic, B., Smith, M.K & Glezer, A.R.I 2007 Dynamics of a sessile drop in forced vibration. J. Fluid Mech. 587, 395–423.CrossRefGoogle Scholar
Wang, J., Joseph, D.D & Funada, T. 2005 Pressure corrections for potential flow analysis of capillary instability of viscous fluids. J. Fluid Mech. 522, 383–394.CrossRefGoogle Scholar
Weber, C. 1931 Zum zerfall eines flüssigkeitsstrahles. Z. Angew. Math. Mech. 11 (2), 136–154.Google Scholar
Wolf, G.H. 1969 The dynamic stabilization of the Rayleigh–Taylor instability and the corresponding dynamic equilibrium. Z. Phys. 227 (3), 291–300.CrossRefGoogle Scholar
Wolf, G.H. 1970 Dynamic stabilization of the interchange instability of a liquid-gas interface. Phys. Rev. Lett. 24 (9), 444–446.CrossRefGoogle Scholar
Wolfram Research, Inc. 2017 Mathematica 11.Google Scholar
Woods, D.R & Lin, S.P. 1995 Instability of a liquid film flow over a vibrating inclined plane. J. Fluid Mech. 294, 391–407.CrossRefGoogle Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (2), 337–352.CrossRefGoogle Scholar
Supplementary material: File

Patankar et al. supplementary material

Patankar et al. supplementary material

Download Patankar et al. supplementary material(File)
File 6.5 MB