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Transient and steady state of a rising bubble in a viscoelastic fluid

Published online by Cambridge University Press:  08 October 2007

SHRIRAM B. PILLAPAKKAM ,
PUSHPENDRA SINGH ,
DENIS BLACKMORE  and
NADINE AUBRY
SHRIRAM B. PILLAPAKKAM
Affiliation:
Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA
PUSHPENDRA SINGH
Affiliation:
Department of Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102, USA
DENIS BLACKMORE
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
NADINE AUBRY
Affiliation:
Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA
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Abstract

A finite element code based on the level-set method is used to perform direct numerical simulations (DNS) of the transient and steady-state motion of bubbles rising in a viscoelastic liquid modelled by the Oldroyd-B constitutive equation. The role of the governing dimensionless parameters, the capillary number (Ca), the Deborah number (De) and the polymer concentration parameter c, in both the rising speed and the deformation of the bubbles is studied. Simulations show that there exists a critical bubble volume at which there is a sharp increase in the terminal velocity with increasing bubble volume, similar to the behaviour observed in experiments, and that the shape of both the bubble and its wake structure changes fundamentally at that critical volume value. The bubbles with volumes smaller than the critical volume are prolate shaped while those with volumes larger than the critical volume have cusp-like trailing ends. In the latter situation, we show that there is a net force in the upward direction because the surface tension no longer integrates to zero. In addition, the structure of the wake of a bubble with a volume smaller than the critical volume is similar to that of a bubble rising in a Newtonian fluid, whereas the wake structure of a bubble with a volume larger than the critical value is strikingly different. Specifically, in addition to the vortex ring located at the equator of the bubble similar to the one present for a Newtonian fluid, a vortex ring is also present in the wake of a larger bubble, with a circulation of opposite sign, thus corresponding to the formation of a negative wake. This not only coincides with the appearance of a cusp-like trailing end of the rising bubble but also propels the bubble, the direction of the fluid velocity behind the bubble being in the opposite direction to that of the bubble. These DNS results are in agreement with experiments.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 589 , 25 October 2007 , pp. 215 - 252
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Arigo, M. T. & McKinley, G. H. 1998 An experimental investigation of negative wakes behind spheres settling in a shear-thinning viscoelastic fluid. Rheol. Acta 37 (4), 307–327.Google Scholar
Astarita, G. & Apuzzo, G. 1965 Motion of gas bubbles in non-Newtonian liquids, AIChE J. 11 (5), 815–820.CrossRefGoogle Scholar
Barnett, S. M., Humphrey, A. E. & Litt, M. 1966 Bubble motion and mass transfer in non-Newtonian fluid AIChE J. 12 (2), 253–259.CrossRefGoogle Scholar
Belmonte, A. 2000 Self oscillations of a cusped bubble rising through a micellar solution. Rheol. Acta 39, 554–559.Google Scholar
Bisgaard, C. 1983 Velocity fields around spheres and bubbles investigated by laser-Doppler anemometry. J. Non-Newtonian Fluid Mech. 12, 283–302.CrossRefGoogle Scholar
Blackmore, D. & Ting, L. 1985 Surface integral of its mean curvature vector. SIAM Rev. 27–4.Google Scholar
Bristeau, M. O., Glowinski, R. & Periaux, J. 1987 Numerical methods for Navier-Stokes equations. Application to the simulation of compressible and incompressible flows. Comput. Phys. Rep. 6, 73.CrossRefGoogle Scholar
Calderbank, P. H., Johnson, D. S. & Loudon, J. 1970 Velocity fields around spheres and bubbles investigated by laser-doppler anemometry. Chem. Engng. Sci. 25, 235–256.CrossRefGoogle Scholar
Chen, S. & Rothstein, J. P. 2004 Flow of a wormlike micelle solution past a falling sphere. J. Non-Newtonian Fluid Mech. 116, 205–234.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.Google Scholar
Funfschilling, D. & Li, H. Z. 2001 Flow of Non-Newtonian fluids around bubbles: PIV measurements and birefringence visualization, Chem. Engng Sci. 56, 1137–1141.Google Scholar
Glowinski, R., Tallec, P., Ravachol, M. & Tsikkinis, V. 1992 In Finite Elements in Fluids, (ed. Chung, T. J.), vol. 8, chap. 7. Hemisphere.Google Scholar
Glowinski, R. & Pironneau, O. 1992 Finite element methods for Navier-Stokes equations. Annu. Rev. Fluid Mech. 24, 167.Google Scholar
Glowinski, R., Pan, T. W., Hesla, T. I. & Joseph, D. D. 1999 A distributed Lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flows 25 (5), 755.CrossRefGoogle Scholar
Handzy, N. Z. & Belmonte, A. 2004 Oscillatory motion of rising bubbles in wormlike micellar fluid with different microstructures. Phys. Rev. Lett. 92, 124501.Google Scholar
Happel, J. & Brenner, H. 1981 Low Reynolds Number Hydrodynamics: with special applications to particulate media. Springer.Google Scholar
Hassagar, O. 1979 Negative wake behind bubbles in non-Newtonian liquids. Nature 279, 402–403.Google Scholar
Herrera-Velarde, J. R. Zenit, R., Chehata, D., Mena, B. 2003 The flow of non-Newtonian fluids around bubbles and its connection to the jump discontinuity. J. non-Newtonian Fluid Mech. 111, 199–209.Google Scholar
Jayaraman, A. & Belmonte, A. 2003 Oscillations of a solid sphere falling through a wormlike micellar fluid. Phys. Rev. E 67, 65301.Google ScholarPubMed
Jeong, J. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 1–22.CrossRefGoogle Scholar
Joseph, D. D., Nelson, J., Renardy, M. & Renardy, Y. 1991 Two-dimensional cusped interfaces. J. Fluid Mech. 223, 383–409.CrossRefGoogle Scholar
King, M. J. & Walters, N.D. 1972 The unsteady motion of a sphere in an elastico-viscous liquid. J. Phys. D: Appl. Phys. 5, 141–150.Google Scholar
Leal, L. G., Skoog, J. & Acrivos, A. 1971 On the motion of gas bubbles in a viscoelastic fluid. Can. J. Chem. Engng 49, 569–575.CrossRefGoogle Scholar
Liu, Y. J., Liao, T. Y. & Joseph, D. D. 1995 A two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid. J. Fluid Mech. 304, 321–342.CrossRefGoogle Scholar
Marchuk, G. I. 1990 Splitting and alternate direction methods. In Handbook of Numerical Analysis (ed. Ciarlet, P. G. & Lions, J. L.), Vol. 1, P. 197. North-Holland.Google Scholar
McKinley, G. H. 2001 Transport Processes in Bubbles, Drops and Particles. (ed. Chhabra, R. P. & DeKee, D.), 2nd edn, Taylor Francis.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 5, 1315–1332.Google Scholar
Philippoff, W. 1937 Viscosity characteristics of rubber solution. Rubber Chem. Tech. 10, 76.Google Scholar
Pillapakkam, S. B. & Singh, P. 2001 A level-set method for computing solutions to viscoelastic two-phase flow. J. Comput. Phys. 174, 552–578.Google Scholar
Ramaswamy, S. & Leal, L. G. 1999 The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid. J. Non-Newtonian Fluid Mech. 85, 127.Google Scholar
Rodrigue, D., Chhabra, R. P. & Chan Man Fong, C. F. 1996 An experimental study of the effect of surfactants on the free rise velocity of gas bubbles. J. Non-Newtonian Fluid Mech. 66, 213–232.CrossRefGoogle Scholar
Rodrigue, D., Chhabra, R. P. & Fong, C. F. C. M. 1998 Bubble velocities: further developments on the jump and discontinuity. J. Non-Newtonian Fluid Mech. 79, 45–55.Google Scholar
Shikhmurzeav, Y. D. 1998 On cusped interfaces. J. Fluid Mech. 359, 131–328.Google Scholar
Singh, P. & Joseph, D. D. 2000 Sedimentation of a sphere near a vertical wall in an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 94, 179–203.CrossRefGoogle Scholar
Singh, P., Joseph, D. D., Hesla, T. I., Glowinski, R. & Pan, T.W. 2000 A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows. J. Non-Newtonian Fluid Mech. 91, 165.Google Scholar
Singh, P. & Leal, L.G. 1993 Finite element simulation of the start-up problem for a viscoelastic problem in an eccentric cylinder geometry using third-order upwind scheme Theor. Comput. Fluid Dyn. 5, 107–137.Google Scholar
Singh, P. & Leal, L. G. 1994 Computational studies of dumbbell model fluids in a co-rotating two-roll mill. J. Rheol. 38, 485–517.Google Scholar
Singh, P. & Leal, L. G. 1995 Viscoelastic flows with corner singularities. J. Non-Newtonian Fluid Mech. 58, 279–313.Google Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146.CrossRefGoogle Scholar
Wagner, A. J., Giraud, L. & Scott, C. E. 2000 Simulations of a cusped bubble rising in a viscoelastic fluid with a new numerical method. Comput. Phys. Comm. 129, 227–231.CrossRefGoogle Scholar
Zheng, R. & Pan-Thien, N. 1992 A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid. Rheol. Acta 31, 323–332.Google Scholar