Skip to main content
×
×
Home

Solution selection of axisymmetric Taylor bubbles

  • A. Doak (a1) and J.-M. Vanden-Broeck (a1)
Abstract

A finite difference scheme is proposed to solve the problem of axisymmetric Taylor bubbles rising at a constant velocity in a tube. A method to remove singularities from the numerical scheme is presented, allowing accurate computation of the bubbles with the inclusion of both gravity and surface tension. This paper confirms the long-held belief that the solution space of the axisymmetric Taylor bubble for small surface tension is qualitatively similar to that of the plane Taylor bubble. Furthermore, evidence suggesting that the solution selection mechanism associated with plane bubbles also occurs in the axisymmetric case is presented.

Copyright
Corresponding author
Email address for correspondence: alexander.doak.11@ucl.ac.uk
References
Hide All
Advanpix2017 Multiprecision computing toolbox for MATLAB. www.advanpix.com.
Blyth, M. G. & Părău, E. I. 2014 Solitary waves on a ferrofluid jet. J. Fluid Mech. 750, 401420.
Brennen, C.1966 Cavitation and other free surface phenomena. PhD thesis, University of Oxford.
Brennen, C. 1969 A numerical solution of axisymmetric cavity flows. J. Fluid Mech. 37 (4), 671688.
Christodoulides, P. & Dias, F. 2010 Impact of a falling jet. J. Fluid Mech. 657, 2235.
Collins, R. 1965 A simple model of the plane gas bubble in a finite liquid. J. Fluid Mech. 22 (4), 763771.
Davies, R. M. & Taylor, G. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. A 200 (1062), 375390.
Dumitrescu, D. T. 1943 Strömung an einer Luftblase im senkrechten Rohr. Z. Angew. Math. Mech. 23 (3), 139149.
Garabedian, P. R. 1957 On steady-state bubbles generated by Taylor instability. Proc. R. Soc. Lond. A 241 (1226), 423431.
Garabedian, P. R. 1985 A remark about pointed bubbles. Commun. Appl. Maths 38 (5), 609612.
Jeppson, R. W. 1970 Inverse formulation and finite difference solution for flow from a circular orifice. J. Fluid Mech. 40 (1), 215223.
Levine, H. & Yang, Y. 1990 A rising bubble in a tube. Phys. Fluids A 2 (4), 542546.
Maneri, C. C. & Zuber, N. 1974 An experimental study of plane bubbles rising at inclination. Intl J. Multiphase Flow 1 (5), 623645.
McLean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele Shaw cell. J. Fluid Mech. 102, 455469.
Modi, V. 1985 Comment on ‘bubbles rising in a tube and jets falling from a nozzle’ [Phys. Fluids 27, 1090 (1984)]. Phys. Fluids 28 (11), 34323433.
Southwell, R. V. 1946 Relaxation Methods in Theoretical Physics. Oxford Clarendon Press.
Vanden-Broeck, J. M. 1984a Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids 27 (5), 10901093.
Vanden-Broeck, J.-M. 1984b Rising bubbles in a two-dimensional tube with surface tension. Phys. Fluids 27 (11), 26042607.
Vanden-Broeck, J.-M. 1986 Pointed bubbles rising in a two-dimensional tube. Phys. Fluids 29, 13431344.
Vanden-Broeck, J.-M. 1991 Axisymmetric jet falling from a vertical nozzle and bubble rising in a tube of circular cross section. Phys. Fluids A 3 (2), 258262.
Vanden-Broeck, J.-M. 1992 Rising bubble in a two-dimensional tube: asymptotic behavior for small values of the surface tension. Phys. Fluids A 4 (11), 23322334.
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.
Vanden-Broeck, J. M., Miloh, T. & Spivack, B. 1998 Axisymmetric capillary waves. Wave Motion 27 (3), 245256.
Viana, F., Pardo, R., Yanez, R., Trallero, J. L. & Joseph, D. D. 2003 Universal correlation for the rise velocity of long gas bubbles in round pipes. J. Fluid Mech. 494, 379398.
Woods, L. C. 1951 A new relaxation treatment of flow with axial symmetry. Q. J. Mech. Appl. Maths 4 (3), 358370.
Woods, L. C. 1953 The relaxation treatment of singular points in Poisson’s equation. Q. J. Mech. Appl. Maths 6 (2), 163185.
Yang, Y. & Levine, H. 1992 Spherical cap bubbles. J. Fluid Mech. 235, 7387.
Zukoski, E. E. 1966 Influence of viscosity, surface tension, and inclination angle on motion of long bubbles in closed tubes. J. Fluid Mech. 25 (4), 821837.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 7
Total number of PDF views: 99 *
Loading metrics...

Abstract views

Total abstract views: 178 *
Loading metrics...

* Views captured on Cambridge Core between 22nd March 2018 - 21st August 2018. This data will be updated every 24 hours.