- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Short-term oscillation and falling dynamics for a water drop dripping in quiescent air

B. Zhang and Y. Ling*

P.-H. Tsai and A.-B. Wang†

S. Popinet and S. Zaleski

  • Department of Mechanical Engineering, Baylor University, Texas 76798, USA

  • Institute of Applied Mechanics, National Taiwan University, Taipei, 10617, Taiwan, Republic of China

  • Sorbonne Universités and CNRS, Institut Jean le Rond d'Alembert, UMR 7190, F-75005, Paris, France

  • *stanley_ling@baylor.edu
  • †abwang@iam.ntu.edu.tw
PDF

Phys. Rev. Fluids 4, 123604 – Published 18 December, 2019

DOI: https://doi.org/10.1103/PhysRevFluids.4.123604

Abstract

The short-term transient falling dynamics of a dripping water drop in quiescent air has been investigated through both simulation and experiment. A representative case with a low inflow rate in the dripping regime is considered. The focus is on the short-term behavior and the time range considered covers about eight dominant second-mode oscillations of the drop after it is formed. Due to the small fluid inertia at the inlet, the growth of the drop is quasistatic and is well captured by the static pendant drop theory. Nevertheless, it is demonstrated that the pinching dynamics and the resulting postformation state of the drop trigger a nonlinear oscillation when the drop falls. The initial shape of the drop when it is just formed is decomposed into spherical harmonic modes. The initial mode amplitudes, characterized by the Fourier-Legendre coefficients, are found to be finite for up to the tenth mode. The pinching dynamics such as interface overturning introduces small-scale variation on the drop contour, which in turn contributes to the finite amplitudes of the higher-order modes. Furthermore, the initial kinetic energy when the droplet is just formed is as important as the initial surface energy contained in the drop shape, and is found to amplify the initial oscillation amplitude and to induce a phase shift in the oscillation of all the modes. By incorporating both the initial surface and kinetic energy, the linear model for a free drop oscillation yields very good predictions for the second and third modes. The mode amplitude spectra show both the primary frequencies that are consistent with the Lamb's theory and the secondary frequencies arising from different modes due to nonlinear intermode coupling. Moreover, it is worth to noting that the nonlinear effect is most profound for the fourth mode owing to its resonant coupling with the dominant second mode. The complex transient flow inside and outside the drop is induced by the interaction between the falling motion and the nonlinear oscillation. The streamlines indicate that the internal flow is substantially different from the Hill vortex for a falling drop without oscillation. The temporal evolutions of both the internal flow and the wake morphology follow the dominant second oscillation mode. In the oblate-to-prolate deformation, the internal flow goes against the external flow. As a result, a saddle point arises in the drop, which gives rise to two counterrotating vortices. The vortex dynamics are visualized by the swirling-strength vortex identification criterion and the vorticity. Whereas the potential flow changes direction during a second-mode oscillation cycle, the rotating directions of the vortices remain the same.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (58)

  1. J. Q. Feng and K. V. Beard, A perturbation model of raindrop oscillation characteristics with aerodynamic effects, J. Atmos. Sci. 48, 1856 (1991).
  2. O. A. Basaran, H. Gao, and P. P. Bhat, Nonstandard inkjets, Annu. Rev. Fluid Mech. 45, 85 (2013).
  3. C. Clanet and J. C. Lasheras, Transition from dripping to jetting, J. Fluid Mech. 383, 307 (1999).
  4. B. Lalanne, S. Tanguy, and F. Risso, Effect of rising motion on the damped shape oscillations of drops and bubbles, Phys. Fluids 25, 112107 (2013).
  5. M. Agrawal, A. R. Premlata, M. K. Tripathi, B. Karri, and K. C. Sahu, Nonspherical liquid droplet falling in air, Phys. Rev. E 95, 033111 (2017).
  6. K. Bergeles, Y. Hardalupas, and A. M. K. P. Taylor, On the transient flow inside and around a deforming millimetre class oil droplet falling under the action of gravity in stagnant air, Phys. Fluids 30, 013305 (2018).
  7. J. F. Padday and A. R. Pitt, The stability of axisymmetric menisci, Philos. Trans. R. Soc. London A 275, 489 (1973).
  8. P. T. Sumesh and R. Govindarajan, The possible equilibrium shapes of static pendant drops, J. Chem. Phys. 133, 144707 (2010).
  9. R. M. S. M. Schulkes, The evolution and bifurcation of a pendant drop, J. Fluid Mech. 278, 83 (1994).
  10. P. Coullet, L. Mahadevan, and C. S. Riera, Hydrodynamical models for the chaotic dripping faucet, J. Fluid Mech. 526, 1 (2005).
  11. J. R. Castrejón-Pita, A. A. Castrejón-Pita, S. S. Thete, K. Sambath, I. M. Hutchings, J. Hinch, J. R. Lister, and O. A. Basaran, Plethora of transitions during breakup of liquid filaments, Proc. Natl. Acad. Sci. USA 112, 4582 (2015).
  12. J. Eggers, Universal Pinching of 3D Axisymmetric Free-Surface Flow, Phys. Rev. Lett. 71, 3458 (1993).
  13. J. Eggers and T. F. Dupont, Drop formation in a one-dimensional approximation of the Navier–Stokes equation, J. Fluid Mech. 262, 205 (1994).
  14. D. T. Papageorgiou, On the breakup of viscous liquid threads, Phys. Fluids 7, 1529 (1995).
  15. R. F. Day, E. J. Hinch, and J. R. Lister, Self-Similar Capillary Pinchoff of an Inviscid Fluid, Phys. Rev. Lett. 80, 704 (1998).
  16. B. W. Zeff, B. Kleber, J. Fineberg, and D. P. Lathrop, Singularity dynamics in curvature collapse and jet eruption on a fluid surface, Nature (London) 403, 401 (2000).
  17. A. U. Chen, P. K. Notz, and O. A. Basaran, Computational and Experimental Analysis of Pinch-Off and Scaling, Phys. Rev. Lett. 88, 174501 (2002).
  18. P. Doshi, I. Cohen, W. W. Zhang, M. Siegel, P. Howell, O. A. Basaran, and S. R. Nagel, Persistence of memory in drop breakup: The breakdown of universality, Science 302, 1185 (2003).
  19. J. R. Castrejon-Pita, A. A. Castrejon-Pita, E. J. Hinch, J. R. Lister, and I. M. Hutchings, Self-similar breakup of near-inviscid liquids, Phys. Rev. E 86, 015301(R) (2012).
  20. E. D. Wilkes, S. D. Phillips, and O. A. Basaran, Computational and experimental analysis of dynamics of drop formation, Phys. Fluids 11, 3577 (1999).
  21. A. van der Bos, M.-J. van der Meulen, T. Driessen, M. van den Berg, H. Reinten, H. Wijshoff, M. Versluis, and D. Lohse, Velocity Profile Inside Piezoacoustic Inkjet Droplets in Flight: Comparison Between Experiment and Numerical Simulation, Phys. Rev. Applied 1, 014004 (2014).
  22. M. P. Borthakur, G. Biswas, and D. Bandyopadhyay, Formation of liquid drops at an orifice and dynamics of pinch-off in liquid jets, Phys. Rev. E 96, 013115 (2017).
  23. X. Zhang, Dynamics of drop formation in viscous flows, Chem. Eng. Sci. 54, 1759 (1999).
  24. L. Rayleigh, On the capillary phenomena of jets, Proc. R. Soc. London 29, 71 (1879).
  25. H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932).
  26. W. H. Reid, The oscillations of a viscous liquid drop, Q. Appl. Math. 18, 86 (1960).
  27. C. A. Miller and L. E. Scriven, The oscillations of a fluid droplet immersed in another fluid, J. Fluid Mech. 32, 417 (1968).
  28. A. Prosperetti, Normal-mode analysis for the oscillations of a viscous-liquid drop in an immiscible liquid, J. Méc. 19, 149 (1980).
  29. J. A. Tsamopoulos and R. A. Brown, Nonlinear oscillations of inviscid drops and bubbles, J. Fluid Mech. 127, 519 (1983).
  30. R. Natarajan and R. A. Brown, Third-order resonance effects and the nonlinear stability of drop oscillations, J. Fluid Mech. 183, 95 (1987).
  31. E. Becker, W. J. Hiller, and T. A. Kowalewski, Experimental and theoretical investigation of large-amplitude oscillations of liquid droplets, J. Fluid Mech. 231, 189 (1991).
  32. O. A. Basaran, Nonlinear oscillations of viscous liquid drops, J. Fluid Mech. 241, 169 (1992).
  33. E. Becker, W. J. Hiller, and T. A. Kowalewski, Nonlinear dynamics of viscous droplets, J. Fluid Mech. 258, 191 (1994).
  34. D. J. Gunn, Transfer of heat or mass to particles in fixed and fluidised beds, Int. J. Heat Mass Transfer 21, 467 (1978).
  35. B. T. Helenbrook and C. F. Edwards, Quasi-steady deformation and drag of uncontaminated liquid drops, Int. J. Multiphase Flow 28, 1631 (2002).
  36. J. Q. Feng, A deformable liquid drop falling through a quiescent gas at terminal velocity, J. Fluid Mech. 658, 438 (2010).
  37. H. J. J. Staat, A. van der Bos, M. van den Berg, H. Reinten, H. Wijshoff, M. Versluis, and D. Lohse, Ultrafast imaging method to measure surface tension and viscosity of inkjet-printed droplets in flight, Exp. Fluids 58, 2 (2017).
  38. M. K. Tripathi, K. C. Sahu, and R. Govindarajan, Why a falling drop does not in general behave like a rising bubble, Sci. Rep. 4, 4771 (2014).
  39. P. Ern, F. Risso, D. Fabre, and J. Magnaudet, Wake-induced oscillatory paths of bodies freely rising or falling in fluids, Annu. Rev. Fluid Mech. 44, 97 (2012).
  40. S. K. Chung and E. H. Trinh, Internal flow of an electrostatically levitated droplet undergoing resonant shape oscillation, Phys. Fluids 12, 249 (2000).
  41. G. Tryggvason, R. Scardovelli, and S. Zaleski, Direct Numerical Simulations of Gas-Liquid Multiphase Flows (Cambridge University Press, Cambridge, 2011).
  42. X. Zhang, Dynamics of growth and breakup of viscous pendant drops into air, J. Colloid Interface Sci. 212, 107 (1999).
  43. D. Gueyffier, J. Li, A. Nadim, R. Scardovelli, and S. Zaleski, Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. Comput. Phys. 152, 423 (1999).
  44. H. Deka, B. Ray, G. Biswas, A. Dalal, P.-H. Tsai, and A.-B. Wang, The regime of large bubble entrapment during a single drop impact on a liquid pool, Phys. Fluids 29, 092101 (2017).
  45. R. Gunn and G. D. Kinzer, The terminal velocity of fall for water droplets in stagnant air, J. Meteorol. 6, 243 (1949).
  46. S. Popinet, Gerris: A tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comput. Phys. 190, 572 (2003).
  47. S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys. 228, 5838 (2009).
  48. R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow, Annu. Rev. Fluid Mech. 31, 567 (1999).
  49. M. M. Francois, S. J. Cummins, E. D. Dendy, D. B. Kothe, J. M. Sicilian, and M. W. Williams, A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework, J. Comput. Phys. 213, 141 (2006).
  50. B. Ambravaneswaran, E. D. Wilkes, and O. A. Basaran, Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops, Phys. Fluids 14, 2606 (2002).
  51. X. Zhang and O. A. Basaran, An experimental study of dynamics of drop formation, Phys. Fluids 7, 1184 (1995).
  52. Y. Ling, J.-M. Fullana, S. Popinet, and C. Josserand, Droplet migration in a Hele–Shaw cell: Effect of the lubrication film on the droplet dynamics, Phys. Fluids 28, 062001 (2016).
  53. M. J. M. Hill, On a spherical vortex, Philos. Trans. R. Soc. London A 185, 213 (1894).
  54. J. Zhou, R. J. Adrian, S. Balachandar, and T. M. Kendall, Mechanisms for generating coherent packets of hairpin vortices in channel flow, J. Fluid Mech. 387, 353 (1999).
  55. P. Chakraborty, S. Balachandar, and R. J. Adrian, On the relationships between local vortex identification schemes, J. Fluid Mech. 535, 189 (2005).
  56. J. Hoepffner and G. Paré, Recoil of a liquid filament: Escape from pinch-off through creation of a vortex ring, J. Fluid Mech. 734, 183 (2013).
  57. H. Aref and S. Balachandar, Chaotic advection in a Stokes flow, Phys. Fluids 29, 3515 (1986).
  58. J. R. Angilella and J. P. Brancher, Note on chaotic advection in an oscillating drop, Phys. Fluids 15, 261 (2003).

Outline

Information

Phys. Rev. Fluids 4, 123604– Published 18 December, 2019
  • Received 13 August 2019
Article part of CHORUS
Crossmark - Check for updates button

©2019 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation