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Axisymmetric viscous interfacial oscillations – theory and simulations

Published online by Cambridge University Press:  15 August 2017

Palas Kumar Farsoiya ,
Y. S. Mayya  and
Ratul Dasgupta Open the ORCID record for Ratul Dasgupta [Opens in a new window]
Palas Kumar Farsoiya
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India
Y. S. Mayya
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India
Ratul Dasgupta*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology, Bombay Powai, Mumbai 400 076, India
*
†Email address for correspondence: [email protected]
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Abstract

We study axisymmetric, free oscillations driven by gravity and surface tension at the interface of two viscous, immiscible, radially unbounded fluids, analytically and numerically. The interface is perturbed as a zeroth-order Bessel function (in space) and its evolution is obtained as a function of time. In the linearised approximation, we solve the initial value problem (IVP) to obtain an analytic expression for the time evolution of wave amplitude. It is shown that a linearised Bessel mode temporally evolves in exactly the same manner as a Fourier mode in planar geometry. We obtain novel analytical expressions for the time varying vorticity and pressure fields in both fluids. For small initial amplitudes, our analytical results show excellent agreement with those obtained from solving the axisymmetric Navier–Stokes equations numerically. We also compare our results with the normal mode approximation and find the latter to be an accurate representation at very early and late times. The deviation between the normal mode approximation and the IVP solution is found to increase as a function of viscosity ratio. The vorticity field has a jump discontinuity at the interface and we find that this jump depends on the viscosity and the density ratio of the two fluids. Upon increasing the initial perturbation amplitude in the simulations, nonlinearity produces qualitatively new features not present in the analytical IVP solution. Notably, a jet is found to emerge at the axis of symmetry rising to a height greater than the initial perturbation amplitude. Increasing the perturbation amplitude further causes the jet to undergo end pinch off, giving birth to a daughter droplet. This can happen either for an advancing or a receding jet, depending on the viscosity ratio. A relation is found between the maximum jet height and the perturbation amplitude. Hankel transform of the interface demonstrates that at large perturbation amplitudes higher wavenumbers emerge, sharing some of the energy of the lowest mode. When these additional higher modes are present, the interface has pointed crests and rounded troughs.

JFM classification

Waves/Free-surface Flows: Capillary waves Multiphase and Particle-laden Flows: Multiphase flow Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 797 - 818
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Copyright
© 2017 Cambridge University Press 

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References

Abate, J. & Whitt, W. 2006 A unified framework for numerically inverting laplace transforms. INFORMS J. Comput. 18 (4), 408–421.Google Scholar
Abramowitz, M. & Segun, I.-A. 1970 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table. Dover.Google Scholar
Balmforth, N. J. & Morrison, P. J.1995 Singular eigenfunctions for shearing fluids i. https://web2.ph.utexas.edu/m̃orrison/95IFSR692_morrison.pdf.Google Scholar
Bartolo, D., Josserand, C. & Bonn, D. 2006 Singular jets and bubbles in drop impact. Phys. Rev. Lett. 96 (12), 124501.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
Becker, E., Hiller, W. J. & Kowalewski, T. A. 1991 Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets. J. Fluid Mech. 231, 189–210.Google Scholar
Beetham, E., Kench, P. S., O’Callaghan, J. & Popinet, S. 2016 Wave transformation and shoreline water level on funafuti atoll, tuvalu. J. Geophys. Res. 121 (1), 311–326.Google Scholar
Bell, J. B, Colella, P. & Glaz, H. M. 1989 A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85 (2), 257–283.CrossRefGoogle Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.Google Scholar
Brøns, M., Thompson, M. C., Leweke, T. & Hourigan, K. 2014 Vorticity generation and conservation for two-dimensional interfaces and boundaries. J. Fluid Mech. 758, 63–93.Google Scholar
Bush, J.2013 Lecture notes – 18.357 interfacial phenomena, 5.4 appendix b, pp. 21. https://ocw.mit.edu/courses/mathematics/18-357-interfacial-phenomena-fall-2010/lecture-notes/MIT18_357F10_lec_all.pdf.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Courier Dover Publications.Google Scholar
Dasgupta, R., Tomar, G. & Govindarajan, R. 2015 Numerical study of laminar, standing hydraulic jumps in a planar geometry. Eur. Phys. J. E 38 (5), 1–14.Google Scholar
Debnath, L. 1994 Nonlinear Water Waves. Academic.Google Scholar
Denner, F. 2016 Frequency dispersion of small-amplitude capillary waves in viscous fluids. Phys. Rev. E 94 (2), 023110.Google ScholarPubMed
Denner, F., Paré, G. & Zaleski, S. 2017a Dispersion and viscous attenuation of capillary waves with finite amplitude. Eur. Phys. J. 226, 1229–1238.Google Scholar
Denner, F., Paré, G. & Zaleski, S.2017b Dispersion and viscous attenuation of capillary waves with finite amplitude (supporting data). https://zenodo.org/record/259434#.WRmAkhJ97vt.Google Scholar
Dixit, H. N. & Govindarajan, R. 2010 Vortex-induced instabilities and accelerated collapse due to inertial effects of density stratification. J. Fluid Mech. 646, 415–439.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–340.Google Scholar
Farsoiya, P. K., Mayya, Y. S. & Dasgupta, R.2017 Supplementary material. doi:10.1017/jfm.2017.443.Google Scholar
Gekle, S. & Gordillo, J. M. 2010 Generation and breakup of worthington jets after cavity collapse. Part 1. Jet formation. J. Fluid Mech. 663, 293–330.Google Scholar
Goodridge, C. L., Shi, W. T., Hentschel, H. G. E. & Lathrop, D. P. 1997 Viscous effects in droplet-ejecting capillary waves. Phys. Rev. E 56 (1), 472–475.Google Scholar
Harrison, W. J. 1908 The influence of viscosity on the oscillations of superposed fluids. Proc. Lond. Math. Soc. 2 (1), 396–405.Google Scholar
Henderson, D. M. & Miles, J. W. 1994 Surface-wave damping in a circular cylinder with a fixed line. J. Fluid Mech. 275, 285–299.Google Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (vof) method for the dynamics of free boundaries. J. Comput. Phys. 39 (1), 201–225.Google Scholar
James, A. J., Vukasinovic, B., Smith, M. K. & Glezer, A. 2003 Vibration-induced drop atomization and bursting. J. Fluid Mech. 476, 1–28.Google Scholar
Joseph, D. D. 2003 Viscous potential flow. J. Fluid Mech. 479, 191–197.Google Scholar
Kundu, P. K. & Cohen, I. M. 2002 Fluid Mechanics, 2nd edn. chap. Appendix B, pp. 710–714. Academic.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Li, Y. & Umemura, A. 2014 Threshold condition for spray formation by faraday instability. J. Fluid Mech. 759, 73–103.Google Scholar
Longuet-Higgins, M. S. 2001 Vertical jets from standing waves. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 457, pp. 495–510. The Royal Society.Google Scholar
Mack, L. R. 1962 Periodic, finite-amplitude, axisymmetric gravity waves. J. Geophys. Res. 67 (2), 829–843.CrossRefGoogle Scholar
Mathematica2017 Perform numerical hankel transform. https://www.wolfram.com/mathematica/new-in-8/new-and-improved-core-algorithms/perform-numerical-hankel-transform.html.Google Scholar
MATLAB 2015 MATLAB and Statistics Toolbox Release 2015b. The MathWorks Inc.Google Scholar
McClure, T.2016 Numerical inverse laplace transform. https://in.mathworks.com/matlabcentral/fileexchange/39035-numerical-inverse-laplace-transform.Google Scholar
Menikoff, R., Mjolsness, R. C., Sharp, D. H., Zemach, C. & Doyle, B. J. 1978 Initial value problem for Rayleigh–Taylor instability of viscous fluids. Phys. Fluids 21 (10), 1674–1687.CrossRefGoogle Scholar
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22 (1), 143–165.CrossRefGoogle Scholar
Miles, J. W. 1968 The cauchy-poisson problem for a viscous liquid. J. Fluid Mech. 34 (2), 359–370.Google Scholar
Perinet, N., Juric, D. & Tuckerman, L. S. 2009 Numerical simulation of faraday waves. J. Fluid Mech. 635, 1–26.Google Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries. J. Comput. Phys. 190 (2), 572–600.Google Scholar
Popinet, S.2013, Air-water capillary wave. http://gfs.sourceforge.net/tests/tests/capwave.html#toc15.Google Scholar
Popinet, S.2014 Basilisk. http://basilisk.fr.Google Scholar
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the serre–green–naghdi equations. J. Comput. Phys. 302, 336–358.Google Scholar
Prosperetti, A. 1976 Viscous effects on small-amplitude surface waves. Phys. Fluids 19 (2), 195–203.CrossRefGoogle Scholar
Prosperetti, A. 1980 Normal-mode analysis for the oscillations of a viscous-liquid drop in an immiscible liquid. J. de Méc. 19 (1), 149–182.Google Scholar
Prosperetti, A. 1981 Motion of two superposed viscous fluids. Phys. Fluids 24 (7), 1217–1223.Google Scholar
Puthenveettil, B. A. & Hopfinger, E. J. 2009 Evolution and breaking of parametrically forced capillary waves in a circular cylinder. J. Fluid Mech. 633, 355–379.Google Scholar
Rajchenbach, J. & Clamond, D. 2015 Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited. J. Fluid Mech. 777, R2.CrossRefGoogle Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8.Google Scholar
Sweeney, H., Kerswell, R. R. & Mullin, T. 2013 Rayleigh-Taylor instability in a finite cylinder: linear stability analysis and long-time fingering solutions. J. Fluid Mech. 734, 338.Google Scholar
Taylor, G. 1953 An experimental study of standing waves. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 218, pp. 44–59. The Royal Society.Google Scholar
Thorpe, S. A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32 (03), 489–528.Google Scholar
Trinh, E. & Wang, T. G. 1982 Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315–338.Google Scholar
Tripathi, M. K., Sahu, K. C. & Govindarajan, R. 2014 Why a falling drop does not in general behave like a rising bubble? Sci. Rep. 4, 4771.Google Scholar
Tsai, C. S., Mao, R. W., Lin, S. K., Zhu, Y. & Tsai, S. C. 2014 Faraday instability-based micro droplet ejection for inhalation drug delivery. Technology 2 (01), 75–81.Google Scholar
Wolfram Research Inc. 2010 Mathematica 10.0.Google Scholar
Zeff, B. W., Kleber, B., Fineberg, J. & Lathrop, D. P. 2000 Singularity dynamics in curvature collapse and jet eruption on a fluid surface. Nature 403 (6768), 401–404.Google Scholar
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