Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T03:03:43.944Z Has data issue: false hasContentIssue false
  • English
  • Français

A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid

Published online by Cambridge University Press:  26 April 2006

Y. J. Liu ,
T. Y. Liao  and
D. D. Joseph
Y. J. Liu
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
T. Y. Liao
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

When an air bubble rises in a viscoelastic fluid there is a critical capillary number for cusping and jump in velocity: when the capillary number is below critical, which is about 1 in our data, there is no cusp at the tail of a (smooth) air bubble. For larger volumes, a two-dimensional cusp, sharp in one view and broad in the orthogonal view, is in evidence. Measurements suggest that the cusp tip is in the generic form y = ax2/3 satisfied by analytic cusps. The intervals of volumes for which dramatic changes in air bubble shape take place is very small and the two to ten fold increase in the rise velocity which accompanies the small change of volume could be modelled as a discontinuity. A second drag transition and an orientational transition occurred when U/c > 1 where U is the rise velocity of an air bubble and c is the shear wave speed. For U/c < 1, U is proportional to d2, where d is the equivalent diameter for a sphere of diameter d having the same volume, and when U/c > 1 then U is proportional to d and the Deborah number does not change with U. Moreover the bubble shapes when U/c < 1 are overall prolate (with or without a cusped tail) with the long side parallel to gravity, in contrast to the oblate shapes which are always observed in Newtonian fluids and in viscoelastic fluids with U/c > 1 when inertia is dominant. The formation of cusps occurs in all kinds of columns of different sizes and shapes. Cusping is generic but the orientation of the broad edge with respect to the sidewalls is an issue. There is no preferred orientation in columns with round cross-sections, or in the case of walls far away from the rising bubble. In columns with rectangular cross-sections, three relatively stable configurations can be observed: the cusp can be observed in the wide window and the broad edge in the narrow window; the cusp can be observed in the narrow window and the broad edge in the wide window or, less frequently, the broad edge lies along a diagonal. These orientational and drag alternatives are directly analogous to those which are observed in the settling of long or broad solid bodies (Liu & Joseph 1993).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 304 , 10 December 1995 , pp. 321 - 342
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

Acharya, A., Mashelkar, R. A. & Ulbrecht, J. 1977 Mechanics of bubble motion and deformation in non-Newtonian media. Chem. Engng Sci. 32, 863–872.Google Scholar
Astarita, G. & Apuzzo, G. 1965 Motion of gas bubbles in non-Newtonian liquids. AIChE J. 11, 815–819.Google Scholar
Barnett, S. M., Humpherey, A. E. & Litt, M. 1966 Bubble motion and mass transfer in non-Newtonian fluids. AIChE J. 12, 253.Google Scholar
Bhaga, D. & Weber, M. E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 61–85.Google Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids, 2nd Edn, Vol. I.John Wiley & Sons.
Bisgaard, C. 1983 Velocity fields around spheres and bubbles investigated by laser-Doppler anemometry. J. Non-Newtonian Fluid Mech. 12, 283–302.Google Scholar
Bisgaard, C. & Hassager, O. 1982 An experimental investigation of velocity fields around spheres and bubbles moving in non-Newtonian liquids. Rheol. Ada 21, 537–539.Google Scholar
Calderbank, P. H. 1967 Review series No. 3 - Gas absorption from bubbles. Trans. Inst. Chem. Engrs 45, 209.Google Scholar
Calderbank, P. H., Johnson, D. S. L. & Loudon, J. 1970 Mechanics and mass transfer of single bubbles in free rise through some Newtonian and non-Newtonian liquids. Chem. Engng Sci. 25, 235–256.Google Scholar
Chilcott, M. D. & Rallison, J. M. 1988 Creeping flow of dilute polymer solutions past cylinders and spheres. J. Non-Newtonian Fluid Mech. 29, 381–432.Google Scholar
Clift, R., Grace, J. R. & Weber, M. 1978 Bubbles, Drops and Particles. Academic.
Coutanceeau, M. & Hajjam, M. 1982 Viscoelastic effect on the behavior of an air bubble rising axially in a tube. Appl. Sci. Res. 38, 199–207.Google Scholar
De Kée, D., Carreau, P. J. & Mordarski, J. 1986 Bubble velocity and coalescence in viscoelastic liquids. Chem. Engng Sci. 41, 2273–2283.Google Scholar
Fararouri, A. & Kintner, R. C. 1961 Flow and shape of drops in non-Newtonian fluids. Trans. Soc. Rheol. 5, 369–380.Google Scholar
Feng, J., Joseph, D. D., Glowtnski, R. & Pan, T. W. 1995 A three-dimensional computation of the force and moment on an ellipsoid settling slowly through a viscoelastic fluid. J. Fluid Mech. 283, 1–16.Google Scholar
Garner, F. H., Matrus, K. B. & Jensen, V. G. 1957 Distortion of fluid drops in the Stokesian region. Nature 180, 331.Google Scholar
Gordon, R. J. & Balakrishnan, C. 1972 Vortex inhibition: a new viscoelastic effect with importance in drag reduction and polymer characterization. J. Appl. Polymer Sci. 16, 1629–1639.Google Scholar
Hassager, O. 1979 Negative wake behind bubbles in non-Newtonian liquids. Nature 279, 402–403.Google Scholar
Hassager, O. 1985 The motion of viscoelastic fluids around spheres and bubbles. In Viscoelasticity and Rheology (ed. A. S. Lodge, M. Renardy & J. A. Nohel). Academic.
Jeong, J. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 1–22.Google Scholar
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.
Joseph, D. D. 1992 Understanding cusped interfaces. J. Non-Newtonian Fluid Mech. 44, 127–148.Google Scholar
Joseph, D. D. & Feng, J. 1995 The negative wake in a second-order fluid. J. Non-Newtonian Fluid Mech. 57, 313–320.Google Scholar
Joseph, D. D. & Liu, Y. J. 1993 Orientation of long bodies falling in a viscoelastic liquid. J. Rheol. 37, 961–983.Google Scholar
Joseph, D. D. & Liu, Y. J. 1995 Motion of particles settling in a viscoelastic fluid. Proc. 2nd Intl Conf. on Multiphase Flow, Kyoto, Japan, April 3-7 (ed. A. Serizawa, T. Fukano & J. Bataille).
Joseph, D. D., Liu, Y. J., Poletto, M. & Feng, J. 1994 Aggregation and dispersion of spheres falling in viscoelastic liquids. J. Non-Newtonian Fluid Mech. 54, 45–86.Google Scholar
Joseph, D. D., Nelson, J., Renardy, M. & Renardy, Y. Y. 1991 Two-dimensional cusped interfaces. J. Fluid Mech. 223, 383–409.Google Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Springer.
Leal, L. G., Skoog, J. & Acrivos, A. 1971 On the motion of gas bubbles in a viscoelastic liquid. Can. J. Chem. Engng 49, 569–575.Google Scholar
Liu, Y. J. & Joseph, D. D. 1993 Sedimentation of particles in polymer solutions. J. Fluid Mech. 255, 565–595.Google Scholar
Mhatre, M. V. & Kintner, R. C. 1959 Fall of liquid drops through pseudoplastic liquids. Ind. Engng Chem. 51, 865.Google Scholar
Noh, D. S., Kang, I. S. & Leal, L. G. 1993 Numerical solutions for the deformation of a bubble rising in dilute polymeric fluids. Phys. Fluids A 5, 1315–1332.Google Scholar
Philippoff, W. 1937 The viscosity characteristics of rubber solutions. Rubber Chem. Tech. 10, 76.Google Scholar
Sigli, S. & Coutanceau, M. 1977 Effect of finite boundaries on the slow laminar isothermal flow of a viscoelastic fluid around a spherical obstacle. J. Non-Newtonian Fluid Mech. 2, 1–22.Google Scholar
Warshay, M. E., Bogusz, E., Johnson, M. & Kintner, R. C. 1959 Ultimate velocity of drops in stationary liquid media. Can. J. Chem. Engng 37, 29.Google Scholar
Zana, E. 1975 PhD thesis, California Institute of Technology.
Zana, E. & Leal, L. G. 1978 The dynamics and dissolution of gas bubbles in a viscoelastic fluid. Intl J. Multiphase Flow 4, 237.Google Scholar