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Dynamics of a viscous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: breakup and touchdown singularities

Published online by Cambridge University Press:  02 August 2011

Q. Wang  and
D. T. Papageorgiou
Q. Wang
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, NJ 07102, USA
D. T. Papageorgiou*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: d.papageorgiou@imperial.ac.uk
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Abstract

The nonlinear dynamics of a viscous filament surrounded by a second viscous fluid arranged in a core-annular configuration when a radial electric field acts in the annular region, are studied analytically and computationally using boundary element methods. The flow is characterized by the viscosity ratio, an electric Weber number measuring the strength of the electric field, a geometrical parameter measuring the thickness of the undisturbed annular region, as well as a computational parameter that fixes the wavenumber of the undulations. Axisymmetric solutions are computed by direct numerical simulations in the Stokes limit for general values of the parameters when the two fluids have equal viscosities, and an asymptotic theory is carried out to produce a novel evolution equation for thin film dynamics valid when the undisturbed annular thickness is small and the viscosity ratio is of order one. It is established (in agreement with previous computations in the absence of electric fields) that a sufficiently thick annulus enables thread breakup while a sufficiently thin one (approximately one fifth of the undisturbed thread radius for the case of equal viscosities, for instance) suppresses pinching and drives the interface to approach the tube wall asymptotically without actually touching it. The present simulations show that the electric field affects the dynamics drastically in several ways. First, it promotes interfacial wall touchdown in finite time and a comparison between direct simulations and the asymptotic solutions are in fair agreement. Second, the electric field acts to suppress pinching in the sense that solutions that lead to jet breakup due to a thick enough viscous annulus are driven to wall touchdown. When pinching takes place we find that the ultimate pinching solutions are self-similar and recover the non-electrified ones to leading order for the range of parameters studied.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 683 , 25 September 2011 , pp. 27 - 56
Copyright
Copyright © Cambridge University Press 2011

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References

1. Acrivos, A. & Rallison, J. M. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.Google Scholar
2. Ambravaneswaran, B., Wilkes, E. D. & Basaran, O. A. 2002 Drop formation from a capillary tube: comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. Phys. Fluids 14, 2606–2621.CrossRefGoogle Scholar
3. Basset, A. B. 1894 Waves and jets in a viscous liquid. Am. J. Math. 16, 93–110.CrossRefGoogle Scholar
4. Brenner, M. P., Lister, J. R. & Stone, H. A. 1996 Pinching threads, singularities and the number 0.0304. Phys. Fluids 8, 2827–2836.CrossRefGoogle Scholar
5. Chen, A. U., Notz, P. K. & Basaran, O. A. 2002 Computational and experimental analysis of pinch-off and scaling. Phys. Rev. Lett. 88, 4501.CrossRefGoogle ScholarPubMed
6. Cloupeau, M. & Prunet-Foch, B. 1989 Electrostatic spraying in cone-jet mode. J. Electrostat. 22, 135–159.CrossRefGoogle Scholar
7. Cohen, I., Brenner, M. P., Eggers, J. & Nagel, S. R. 1999 Two fluid drop snap-off problem: experiments and theory. Phys. Rev. Lett. 83, 1147–1150.CrossRefGoogle Scholar
8. Cohen, I. & Nagel, S. R. 2001 Testing for scaling behaviour dependence on geometrical and fluid parameters in the two fluid drop snap-off problem. Phys. Fluids 13, 3533–3541.CrossRefGoogle Scholar
9. Collins, R. T., Harris, M. T. & Basaran, O. A. 2007 Breakup of electrified jets. J. Fluid Mech. 588, 75–129.CrossRefGoogle Scholar
10. Conroy, D. T., Craster, R. V., Matar, O. & Papageorgiou, D. T. 2010 Dynamics and stability of an annular electrolyte film. J. Fluid Mech. 656, 481–506.CrossRefGoogle Scholar
11. Craster, R. V. & Matar, O. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
12. Craster, R. V., Matar, O. & Papageorgiou, D. T. 2002 Pinchoff and satellite formation in surfactant covered viscous threads. Phys. Fluids 14, 1364–1376.CrossRefGoogle Scholar
13. Craster, R. V., Matar, O. & Papageorgiou, D. T. 2003 Pinchoff and satellite formation in compound viscous threads. Phys. Fluids 15, 3409–3428.CrossRefGoogle Scholar
14. Craster, R. V., Matar, O. & Papageorgiou, D. T. 2005 On compound threads with large viscosity contrast. J. Fluid Mech. 533, 95–124.CrossRefGoogle Scholar
15. Dubash, N. & Mestel, A. J. 2007 Behaviour of a conducting drop in a highly viscous fluid subject to an electric field. J. Fluid Mech. 581, 469–493.CrossRefGoogle Scholar
16. Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.CrossRefGoogle ScholarPubMed
17. Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865.CrossRefGoogle Scholar
18. Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equations. J. Fluid Mech. 262, 205–221.CrossRefGoogle Scholar
19. Eggers, J. & Villermeaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
20. Georgiou, E., Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1991 The double layer–capillary stability of an annular electrolyte film surrounding a dielectric-fluid core in a tube. J. Fluid Mech. 226, 149–174.CrossRefGoogle Scholar
21. Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12, 309–319.CrossRefGoogle Scholar
22. Hameed, M., Young, Y. N., Booty, M., Li, J., Siegel, M. & Papageorgiou, D. T. 2008 Influence of surfactant on the deformation and breakup of a bubble in a viscous surrounding. J. Fluid Mech. 594, 307–340.CrossRefGoogle Scholar
23. Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384.CrossRefGoogle Scholar
24. Huebner, A. L. & Chu, H. N. 1971 Instability and breakup of charged liquid jets. J. Fluid Mech. 49, 361–372.CrossRefGoogle Scholar
25. Kelly, A. J. 1994 On the statistical quantum and practical mechanics of electrostatic atomization. J. Aerosol Sci. 25, 1159–1177.CrossRefGoogle Scholar
26. Kwak, S. & Pozrikidis, C. 2001 Effect of surfactants on the instability of a liquid thread or annular layer. Part I. Quiescent fluids. Intl J. Multiphase Flow 27, 1–37.CrossRefGoogle Scholar
27. Lac, E. & Homsy, G. M. 2007 Axisymmetric deformation and stability of a viscous drop in a steady electric field. J. Fluid Mech. 590, 239–264.CrossRefGoogle Scholar
28. Lister, J. R., Rallison, J. M., King, A. A., Cummings, L. J. & Jensen, O. E. 2006 Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.CrossRefGoogle Scholar
29. Lister, J. R. & Stone, H. A. 1998 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 10, 2758–2764.CrossRefGoogle Scholar
30. López-Herrera, J. M. & Gañán-Calvo, A. M. 2004 A note on charged capillary jet breakup of conducting liquids: experimental validation of a viscous one-dimensional model. J. Fluid Mech. 501, 303–326.CrossRefGoogle Scholar
31. López-Herrera, J. M., Gañán-Calvo, A. M. & Perez-Saborid, M. 1999 One-dimensional simulation of the breakup of capillary jets of conducting liquids: application to E.H.D. spraying. J. Aerosol Sci. 30, 895–912.CrossRefGoogle Scholar
32. Magarvey, R. H. & Outhouse, L. E. 1962 Note on the break-up of a charged liquid jet. J. Fluid Mech. 13, 151–157.CrossRefGoogle Scholar
33. Mestel, A. J. 1996 Electrohydrodynamic stability of a highly viscous jet. J. Fluid Mech. 312, 311–326.CrossRefGoogle Scholar
34. Newhouse, L. A. & Pozrikidis, C. 1992 The capillary instability of annular layers and liquid threads. J. Fluid Mech. 242, 193–209.CrossRefGoogle Scholar
35. Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223–256.CrossRefGoogle Scholar
36. Notz, P. K., Chen, A. U. & Basaran, O. A. 2001 Satellite drops: unexpected dynamics and change of scaling during pinch-off. Phys. Fluids 13, 549–551.CrossRefGoogle Scholar
37. Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7, 1529–1544.CrossRefGoogle Scholar
38. Plateau, J. 1863 Experimental and theoretical researches on the figures of equilibrium of a liquid mass drawn from the action of gravity. In Annual Report of the Board of Regents of the Smithsonian Institution, pp. 270–283.Google Scholar
39. Pozrikidis, C. 1992 Boundary Integral and Singularity Method for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
40. Pozrikidis, C. 1999 Capillary instability and breakup of a viscous thread. J. Engng Math. 36, 255–275.CrossRefGoogle Scholar
41. Rayleigh, L. 1879 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.Google Scholar
42. Rayleigh, L. 1882 On the equilibrium of liquid conducting masses charged with electricity. Phil. Mag. 14, 184–186.CrossRefGoogle Scholar
43. Saville, D. A. 1971 Stability of electrically charged viscous cylinders. Phys. Fluids 14, 1095–1099.CrossRefGoogle Scholar
44. Schneider, J. M., Lindbald, N. R., Hendricks, C. E. & Crowley, J. M. 1967 Stability of an electrified jet. J. Appl. Phys. 38, 2599–2605.CrossRefGoogle Scholar
45. Setiawan, E. R. & Heister, S. D. 1997 Nonlinear modeling of an infinite electrified jet. J. Electrostat. 42, 243–257.CrossRefGoogle Scholar
46. Sierou, A. & Lister, J. R. 2003 Self-similar solutions for viscous capillary pinch-off. J. Fluid Mech. 497, 381–403.CrossRefGoogle Scholar
47. Song, H., Chen, D. L. & Ismagilov, R. F. 2006 Reactions in droplets in microfluidic channels. Angew. Chem. Int. Ed. 45, 7336–7356.CrossRefGoogle ScholarPubMed
48. 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
49. Taylor, G. I. 1969 Electrically driven jets. Proc. R. Soc. Lond. A 313, 453–475.Google Scholar
50. Timmermans, M.-L. & Lister, J. R. 2002 The effect of surfactant on the stability of a liquid thread. J. Fluid Mech. 459, 289–306.CrossRefGoogle Scholar
51. Tjahjadi, M., Stone, H. A. & Otino, J. M. 1992 Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297–317.CrossRefGoogle Scholar
52. Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. A 150, 322–337.Google Scholar
53. Tseluiko, D. & Papageorgiou, D. T. 2007 Nonlinear dynamics of electrified thin liquid films. SIAM J. Appl. Math. 67, 1310–1329.CrossRefGoogle Scholar
54. Wang, Q. 2010 Nonlinear evolution of annular layers and threads in electric fields. PhD thesis, NJIT.Google Scholar
55. Wang, Q., Mählmann, S. & Papageorgiou, D. T. 2009 Dynamics of liquid jets and threads under the action of radial electric fields: microthread formation and touchdown singularities. Phys. Fluids 21, 032109.CrossRefGoogle Scholar