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Solution selection of axisymmetric Taylor bubbles

Published online by Cambridge University Press:  22 March 2018

A. Doak*
Department of Mathematics, University College London, London WC1E 6BT, UK
J.-M. Vanden-Broeck
Department of Mathematics, University College London, London WC1E 6BT, UK
Email address for correspondence:


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.

JFM Papers
© 2018 Cambridge University Press 

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Advanpix2017 Multiprecision computing toolbox for MATLAB. Scholar
Blyth, M. G. & Părău, E. I. 2014 Solitary waves on a ferrofluid jet. J. Fluid Mech. 750, 401420.Google Scholar
Brennen, C.1966 Cavitation and other free surface phenomena. PhD thesis, University of Oxford.Google Scholar
Brennen, C. 1969 A numerical solution of axisymmetric cavity flows. J. Fluid Mech. 37 (4), 671688.Google Scholar
Christodoulides, P. & Dias, F. 2010 Impact of a falling jet. J. Fluid Mech. 657, 2235.Google Scholar
Collins, R. 1965 A simple model of the plane gas bubble in a finite liquid. J. Fluid Mech. 22 (4), 763771.Google Scholar
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.Google Scholar
Dumitrescu, D. T. 1943 Strömung an einer Luftblase im senkrechten Rohr. Z. Angew. Math. Mech. 23 (3), 139149.Google Scholar
Garabedian, P. R. 1957 On steady-state bubbles generated by Taylor instability. Proc. R. Soc. Lond. A 241 (1226), 423431.Google Scholar
Garabedian, P. R. 1985 A remark about pointed bubbles. Commun. Appl. Maths 38 (5), 609612.Google Scholar
Jeppson, R. W. 1970 Inverse formulation and finite difference solution for flow from a circular orifice. J. Fluid Mech. 40 (1), 215223.Google Scholar
Levine, H. & Yang, Y. 1990 A rising bubble in a tube. Phys. Fluids A 2 (4), 542546.Google Scholar
Maneri, C. C. & Zuber, N. 1974 An experimental study of plane bubbles rising at inclination. Intl J. Multiphase Flow 1 (5), 623645.Google Scholar
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.Google Scholar
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.Google Scholar
Southwell, R. V. 1946 Relaxation Methods in Theoretical Physics. Oxford Clarendon Press.Google Scholar
Vanden-Broeck, J. M. 1984a Bubbles rising in a tube and jets falling from a nozzle. Phys. Fluids 27 (5), 10901093.Google Scholar
Vanden-Broeck, J.-M. 1984b Rising bubbles in a two-dimensional tube with surface tension. Phys. Fluids 27 (11), 26042607.Google Scholar
Vanden-Broeck, J.-M. 1986 Pointed bubbles rising in a two-dimensional tube. Phys. Fluids 29, 13431344.Google Scholar
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.Google Scholar
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.Google Scholar
Vanden-Broeck, J.-M. 2010 Gravity-Capillary Free-Surface Flows. Cambridge University Press.Google Scholar
Vanden-Broeck, J. M., Miloh, T. & Spivack, B. 1998 Axisymmetric capillary waves. Wave Motion 27 (3), 245256.Google Scholar
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.Google Scholar
Woods, L. C. 1951 A new relaxation treatment of flow with axial symmetry. Q. J. Mech. Appl. Maths 4 (3), 358370.Google Scholar
Woods, L. C. 1953 The relaxation treatment of singular points in Poisson’s equation. Q. J. Mech. Appl. Maths 6 (2), 163185.Google Scholar
Yang, Y. & Levine, H. 1992 Spherical cap bubbles. J. Fluid Mech. 235, 7387.Google Scholar
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.Google Scholar