Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-23T05:14:59.453Z Has data issue: false hasContentIssue false
  • English
  • Français

Time-evolving bubbles in two-dimensional Stokes flow

Published online by Cambridge University Press:  26 April 2006

Saleh Tanveer  and
Giovani L. Vasconcelos
Saleh Tanveer
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA
Giovani L. Vasconcelos
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA Present address: Departamento de Física, Universidade Federal de Pernambuco, 50732-910 Recife, Brazil.
Get access Rights & Permissions [Opens in a new window]

Abstract

A general class of exact solutions is presented for a time-evolving bubble in a two-dimensional slow viscous flow in the presence of surface tension. These solutions can describe a bubble in a linear shear flow as well as an expanding or contracting bubble in an otherwise quiescent flow. In the case of expanding bubbles, the solutions have a simple behaviour in the sense that for essentially arbitrary initial shapes the bubble its asymptote is expanding circle. Contracting bubbles, on the other hand, can develop narrow structures (‘near-cusps’) on the interface and may undergo ‘breakup’ before all the bubble fluid is completely removed. The mathematical structure underlying the existence of these exact solutions is also investigated.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 301 , 25 October 1995 , pp. 325 - 344
Copyright
© 1995 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

Acrivos, A. 1983 The breakup of small drops and bubbles in shear flows. Ann. NY Acad. Sci. 404, 1–11.Google Scholar
Antanovskii, L. K. 1994a Quasi-steady deformation of a two-dimensional bubble placed within a potential flow. Meccanica-J. Ital. Assoc. Theor. Appl. Mech. 29, 27–42.Google Scholar
Antanovskii, L. K. 1994b A plane inviscid incompressible bubble placed within a creeping viscous flow: formation of a cusped bubble. Eur. J. Mech. B Fluids 13, 491–509.Google Scholar
Buckmaster, J. D. & Flaherty, J. E. 1973 The bursting of two-dimensional drops in slow viscous flow. J. Fluid Mech. 60, 625–639.Google Scholar
Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.
Gradshteyn, I. S. & Ryzhik, I. M. 1980 Table of Integral, Series and Products. Academic Press.
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.Google Scholar
Hopper, R. W. 1991 Plane Stokes flow driven by capillarity on a free surface, Part 2. Further developments. J. Fluid Mech. 230, 355–364.Google Scholar
Hopper, R. W. 1992 Stokes flow of a cylinder and a half-space driven by capillarity. J. Fluid Mech. 243, 171–181.Google Scholar
Hopper, R. W. 1993 Capillarity-driven plane Stokes flow exterior to a parabola. Q. J. Mech. Appl. Maths 46, 193–210.Google Scholar
Howison, S. D. 1992 Complex variable methods in Hele-Shaw moving boundary problems. Eur. J. Appl. Maths 3, 209–224.Google Scholar
Howison, S. D. & Richardson, S. 1994 Cusp development in free boundaries, and two-dimensional slow viscous flows. Eur. J. Appl. Maths (submitted).Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.
Langlois, W. E. 1964 Slow Viscous Flow. Macmillan.
Muskhelishvili, N. I. 1953 Some Basic Problems in the Mathematical Theory of Elasticity (transl. J. R. M. RacGok). Groningen: P. Noordhoff.
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45–66.Google Scholar
Richardson, S. 1968 Two-dimensional bubbles in slow-viscous flow. J. Fluid Mech. 33, 475–493.Google Scholar
Richardson, S. 1973 Two-dimensional bubbles in slow viscous flow. Part 2. J. Fluid Mech. 58, 115–127.Google Scholar
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. Appl. Maths 3, 193–207.Google Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Ann. Rev. Fluid Mech. 26, 65–102.Google Scholar
Tanveer, S. 1987 Analytic theory for the selection of symmetric Saffman-Taylor finger. Phys. Fluids 30, 1589–1605.Google Scholar
Tanveer, S. 1993 Evolution of Hele-Shaw interface for small surface tension. Phil. Trans. R. Soc. Lond. A 343, 155–204.Google Scholar
Tanveer, S. & Vasconcelos, G. L. 1994 Bubble breakup in two-dimensional Stokes flow. Phys. Rev. Lett. 73, 2845–2848.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146, 501–523.Google Scholar
Vorst, G. A. L. van de 1994 Modelling and numerical simulation of viscous sintering PhD Thesis, Eindhoven University of Technology.