- Editors' Suggestion
- Access by Xinjiang University
Revisiting the Taylor-Culick approximation. II. Retraction of a viscous sheet
Phys. Rev. Fluids 5, 093603 – Published 18 September, 2020
DOI: https://doi.org/10.1103/PhysRevFluids.5.093603
Abstract
We study the retraction of a viscous liquid sheet of finite length with negligible effect of the ambient medium. Using the long-wavelength model we derive the scaling laws and similarity solution for the interface profile of the retracting sheet. Far from the tip, the similarity solution for the interface profiles converges to an asymptotic value of 1/4. Direct numerical simulations are performed to compare the theoretical results with the simulations. When the inertia is negligible, the interface profiles remain flat during the retraction process which is in agreement with the self-similar solution. Using this similarity solution we derive the expression for the temporal variation of the tip speed for finite liquid sheets. We demonstrate that unlike an infinite sheet where the sheet retracts with a steady speed (known as Taylor-Culick speed), the tip speed decreases as a function of time for a finite liquid sheet. This is true when the viscous effects are larger than or of the same order with the inertia effects. Otherwise, the sheet retracts with the formation of a bulbous tip whose speed reaches a value closer to the Taylor-Culick speed.
Physics Subject Headings (PhySH)
Article Text
References (32)
- E. Villermaux and C. Clanet, Life of a flapping liquid sheet, J. Fluid Mech. 462, 341 (2002).
- N. Bremond, C. Clanet, and E. Villermaux, Atomization of undulating liquid sheets, J. Fluid Mech. 585, 421 (2007).
- J. Eggers and E. Villermaux, Physics of liquid jets, Rep. Prog. Phys. 71, 036601 (2008).
- X. Hu and A. M. Jacobi, The intertube falling film: Part 1—Flow characteristics, mode transitions, and hysteresis, J. Heat Trans. 118, 616 (1996).
- D. S. Finnicum, S. J. Weinstein, and K. J. Ruschak, The effect of applied pressure on the shape of a two-dimensional liquid curtain falling under the influence of gravity, J. Fluid Mech. 255, 647 (1993).
- G. Sünderhauf, H. Raszillier, and F. Durst, The retraction of the edge of a planar liquid sheet, Phys. Fluids 14, 198 (2002).
- A. Dupré, Théorie mécanique de la chaleur, Ann. Chim. Phys. 11, 194 (1867).
- W. E. Ranz, Some experiments on the dynamics of liquid films, J. Appl. Phys. 30, 1950 (1959).
- G. I. Taylor, The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets, Proc. R. Soc. Lond. A 253, 313 (1959).
- F. E. C. Culick, Comments on a ruptured soap film, J. Appl. Phys. 31, 1128 (1960).
- W. R. McEntee and K. J. Mysels, Bursting of soap films. I. An experimental study, J. Phys. Chem. 73, 3018 (1969).
- J. B. Keller, Breaking of liquid films and threads, Phys. Fluids 26, 3451 (1983).
- A. B. Pandit and J. F. Davidson, Hydrodynamics of the rupture of thin liquid films, J. Fluid Mech. 212, 11 (1990).
- N. Savva and J. W. M. Bush, Viscous sheet retraction, J. Fluid Mech. 626, 211 (2009).
- L. Gordillo, G. Agbaglah, L. Duchemin, and C. Josserand, Asymptotic behavior of a retracting two-dimensional fluid sheet, Phys. Fluids 23, 122101 (2011).
- M. Murano and K. Okumura, Bursting dynamics of viscous film without circular symmetry: The effect of confinement, Phys. Rev. Fluids 3, 031601 (2018).
- M. P. Brenner and D. Gueyffier, On the bursting of viscous films, Phys. Fluids 11, 737 (1999).
- J.-L. Pierson, J. Magnaudet, E. J. Soares, and S. Popinet, Revisiting the Taylor-Culick approximation: Retraction of an axisymmetric filament, Phys. Rev. Fluids 5, 073602 (2020).
- T. Erneux and S. H. Davis, Nonlinear rupture of free films, Phys. Fluids A: Fluid Dyn. 5, 1117 (1993).
- A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997).
- G. I. Barenblatt, Scaling, Self-similarity and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics, Vol. 14 (Cambridge University Press, Cambridge, UK, 1996).
- J. Eggers, Post-breakup solutions of Navier-Stokes and Stokes threads, Phys. Fluids 26, 072104 (2014).
- J. P. Munro and J. R. Lister, Capillary retraction of the edge of a stretched viscous sheet, J. Fluid Mech. 844, R1 (2018).
- J. Eggers and M. A. Fontelos, Singularities: Formation, Structure, and Propagation (Cambridge University Press, Cambridge, UK, 2015), pp. 1–453.
- S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys. 228, 5838 (2009).
- S. Popinet, A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations, J. Comput. Phys. 302, 336 (2015).
- Basilisk, http://basilisk.fr.
- H. Deka, J.-L. Pierson, and E. J. Soares, Retraction of a viscoplastic liquid sheet, J. Non-Newt. Fluid Mech. 272, 104172 (2019).
- H. Lhuissier and E. Villermaux, Bursting bubble aerosols, J. Fluid Mech. 696, 5 (2012).
- G. Debrégeas, P.-G. de Gennes, and F. Brochard-Wyart, The life and death of “bare” viscous bubbles, Science 279, 1704 (1998).
- C. R. Constante-Amores, L. Kahouadji, A. Batchvarov, S. Shin, J. Chergui, D. Juric, and O. K. Matar, Dynamics of retracting surfactant-laden ligaments at intermediate Ohnesorge number, Phys. Rev. Fluids 5, 084007 (2020).
- E. De Malmazet, F. Risso, O. Masbernat, and V. Pauchard, Coalescence of contaminated water drops at an oil/water interface: Influence of micro-particles, Colloids Surf., A 482, 514 (2015).
