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A spherical-cap bubble moving at terminal velocity in a viscous liquid

Published online by Cambridge University Press:  02 May 2007

JAMES Q. FENG
JAMES Q. FENG*
Affiliation:
Cardiovascular R & D, Boston Scientific Corporation, Three Scimed Place C-150, Maple Grove, MN 55311, [email protected]
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Abstract

The nonlinear Navier–Stokes equations governing steady, laminar, axisymmetric flow past a deformable bubble are solved by the Galerkin finite-element method simultaneously with a set of elliptic partial differential equations governing boundary-fitted mesh. For Reynolds number 20 ≤ Re ≤ 500, numerical solutions of spherical-cap bubbles are obtained at capillary number Ca = 1. Increasing Ca to 2 leads to a highly curved, cusp-like bubble rim that seems to correspond to skirt formation. The computed steady, axisymmetric spherical-cap bubbles with closed, laminar wakes compare reasonably with the available experimental results, especially for Re ≤ 100. By exploring the parameter space (for Re ≤ 200), a sufficient condition for steady axisymmetric solutions of bubbles with the spherical-cap shape is found to be roughly Ca > 0.4. The basic characteristics of spherical-cap bubbles of Ca ≥ 0.5, for a given Re ≥ 50, are found to be almost independent of the value of Ca (or Weber number We ≡ Re Ca). At a fixed Re ≥ 50, continuation by increasing Ca (or We) from a spherical bubble solution cannot lead to solutions of spherical-cap bubbles, but rather to a turning point at We slightly greater than 10 where the solution branch folds back to reduced values of Ca (or We). Yet continuation by reducing Ca (or We) from a spherical-cap bubble solution cannot arrive at a spherical bubble solution for Re ≥ 50, but rather at solutions with bubbles having more complicated shapes such as a sombrero, etc. Without thorough examinations of the solution stability, multiple steady axisymmetric solutions are shown to exist in the parameter space for a given set of parameters.

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Papers
Information
Journal of Fluid Mechanics , Volume 579 , 25 May 2007 , pp. 347 - 371
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bhaga, D. & Weber, M. E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 61–85.CrossRefGoogle Scholar
Bonometti, T. & Magnaudet, J. 2006 Transition from spherical cap to toroidal bubbles. Phys. Fluids 18, 052102.CrossRefGoogle Scholar
Boudouvis, A. G. 1987 Mechanisms of surface instabilities and pattern formation in ferromagnetic liquids. PhD thesis, University of Minnesota. Available from University Microfilms International, Ann Arbor, MI 48106.Google Scholar
Bozzi, L. A., Feng, J. Q., Scott, T. C. & Pearlstein, A. J. 1997 Steady axisymmetric motion of deformable drops falling or rising through a homoviscous fluid in a tube at intermediate Reynolds number. J. Fluid Mech. 336, 1–32.CrossRefGoogle Scholar
Brignell, A. S. 1973 The deformation of a liquid drop at small Reynolds number. Q. J. Mech. Appl. Maths 26, 99–107.CrossRefGoogle Scholar
Carvalho, M. S. 1996 Roll coating flows in rigid and deformable gaps. PhD thesis, University of Minnesota.Google Scholar
Chan, T. F. 1984 Newton-like pseudo-arclength methods for computing simple turning points. SIAM J. Sci. Statist. Comput. 5, 135–148.CrossRefGoogle Scholar
Chen, L., Garimella, S. V., Reizes, J. A. & Leonardi, E. 1999 The development of a bubble rising in a viscous liquid. J. Fluid Mech. 387, 61–96.CrossRefGoogle Scholar
Chen, K. S., Schunk, P. R. & Sackinger, P. A. 1995 Finite-element analyses of blade and slot coating flows using a Newton–Raphson pseudo-solid domain mapping technique and unstructured grid. In Proc. TAPPI Conference.Google Scholar
Christodoulou, K. N. 1989 Computational physics of slide coating flow. PhD thesis, University of Minnesota.Google Scholar
Christodoulou, K. N. & Scriven, L. E. 1992 Discretization of free surface flows and other moving boundary problems. J. Comput. Phys. 99, 39–55.CrossRefGoogle Scholar
Christov, C. I. & Volkov, P. K. 1985 Numerical investigation of the steady viscous flow past a stationary deformable bubble. J. Fluid Mech. 158, 341–364.CrossRefGoogle Scholar
Clift, R. C., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.Google Scholar
Collins, R. 1966 A second approximation for velocity of a large gas bubble rising in an infinite liquid. J. Fluid Mech. 25, 469–480.CrossRefGoogle Scholar
Davies, R. M. & Taylor, G. I. 1950 The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Soc. Lond. A 200, 375–390.Google Scholar
Feng, J. Q. 1998 Computational analysis of slot coating on a tensioned web. AIChE J. 44, 2137–2143.CrossRefGoogle Scholar
Feng, J. Q. 2000 Contact behavior of spherical elastic particles: A computational study of particle adhesion and deformation. Colloids Surf. A 172, 175–198.CrossRefGoogle Scholar
Feng, J. Q. & Basaran, O. A. 1994 Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall. J. Fluid Mech. 275, 351–378.CrossRefGoogle Scholar
Feng, J. Q. & Scriven, L. E. 1992 Electrostatic effects in continuous liquid coating. Paper 38d in Intl. Symposium on Coating Science and Technology. AIChE National Meeting. New Orleans, Louisiana.Google Scholar
Grace, J. R., Wairegi, T. & Brophy, J. 1978 Break-up of drops and bubbles in stagnant media. Can. J. Chem. Engng 56, 3–8.CrossRefGoogle Scholar
Haberman, W. L. & Morton, R. K. 1953 An experimental investigation of the drag and shape of air bubbles in various liquids. David Taylor Model Basin Rep. 802.Google Scholar
Haberman, W. L. & Sayre, R. M. 1958 Motion of rigid and fluid spheres in stationary and moving liquids inside cylindrical tubes. David Taylor Model Basin Rep. 1143.Google Scholar
Harper, J. F. 1972 The motion of bubbles and drops through liquids. Adv. Appl. Mech. 12, 59–129.CrossRefGoogle Scholar
Hnat, J. G. & Buckmaster, J. D. 1976 Spherical cap bubbles and skirt formation. Phys. Fluids 19, 182–194.CrossRefGoogle Scholar
Hood, P. 1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Meth. Engng 10, 379–399. (corrigendum 11, 1055 (1977)).CrossRefGoogle Scholar
Huyakorn, P. S., Taylor, C., Lee, R. L. & Gresho, P. M. 1978 A comparison of various mixed interpolation finite elements in the velocity–pressure formulation of the Navier–Stokes equations. Comput. Fluids 6, 25–35.CrossRefGoogle Scholar
Iooss, G. & Joseph, D. D. 1990 Elementary Stability and Bifurcation Theory, 2nd edn. Springer.CrossRefGoogle Scholar
Joseph, D. D. 2003 Rise velocity of a spherical-cap bubble. J. Fluid Mech. 488, 213–223.CrossRefGoogle Scholar
Keller, H. B. 1977 Numerical solution of bifurcation and nonlinear eigenvalue problems. In Applications of Bifurcation Theory (ed. Rabinovich, P. H.), pp. 359–384. Academic.Google Scholar
Koh, C. J. & Leal, L. G. 1990 An experimental investigation on the stability of viscous drops translating through a quiescent fluid. Phys. Fluids A 2, 2103–2109.CrossRefGoogle Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
Miksis, M. J., Vanden-Broeck, J.-M. & Keller, J. B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.CrossRefGoogle Scholar
Miksis, M. J., Vanden-Broeck, J.-M. & Keller, J. B. 1982 Rising bubbles. J. Fluid Mech. 123, 31–41.CrossRefGoogle Scholar
Moore, D. W. 1959 The rise of a gas bubble in a viscous liquid. J. Fluid Mech. 6, 113–130.CrossRefGoogle Scholar
Moore, D. W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.CrossRefGoogle Scholar
Musson, L. C. 2001 Two-layer slot coating. PhD thesis, University of Minnesota.Google Scholar
Ohta, M., Imura, T., Yoshida, Y. & Sussman, M. 2005 A computational study of the effect of initial bubble condition on the motion of a gas bubble rising in viscous liquids. Intl J. Multiphase Flow 31, 223–237.CrossRefGoogle Scholar
Ortega, J. M. & Rheinboldt, W. C. 1970 Iterative Solution of Nonlinear Equations in Several Variables. Academic.Google Scholar
Parlange, J. Y. 1969 Spherical-cap bubbles with laminar wakes. J. Fluid Mech. 37, 257–263.CrossRefGoogle Scholar
Pozrikidis, C. 1990 The instability of a moving viscous drop. J. Fluid Mech. 210, 1–21.CrossRefGoogle Scholar
Riks, R. 1976 The application of Newton's method to the problem of elastic stability. J. Appl. Mech. 39, 1060–1065.CrossRefGoogle Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid. J. Fluid Mech. 148, 19–35.CrossRefGoogle Scholar
Sackinger, P. A., Schunk, P. R. & Rao, R. R. 1996 A Newton–Raphson pseudo-solid domain mapping technique for free and moving boundary problems: A finite element implementation. J. Comput. Phys. 125, 83–103.CrossRefGoogle Scholar
de Santos, J. M. 1991 Two-phase cocurrent downflow through constricted passages. PhD thesis, University of Minnesota.Google Scholar
Slaughter, I. & Wraith, A. E. 1968 The wake of a large gas bubble. Chem. Eng. Sci. 23, 932.CrossRefGoogle Scholar
Stewart, G. W. 1981 On the implicit deflation of nearly singular systems of linear equations. SIAM J. Sci. Statist. Comput. 2, 136–140.CrossRefGoogle Scholar
Strang, G. & Fix, G. J. 1973 An Analysis of the Finite Element Method. Prentice-Hall.CrossRefGoogle Scholar
Sugiyama, K., Takagi, S. & Matsumoto, Y. 2001 Multi-scale analysis of bubbly flows. Comput. Methods Appl. Mech. Engng 191, 689–704.CrossRefGoogle Scholar
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 166–176.CrossRefGoogle Scholar
Thompson, J. F., Warsi, Z. U. A. & Mastin, C. W. 1985 Numerical Grid Generation. Elsevier.Google Scholar
Tryggvason, G., Bunner, B. Esmaeeli, A., Juric, D. Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y.-J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.CrossRefGoogle Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. The Parabolic Press.Google Scholar
Wegener, P. P. & Parlange, J.-Y. 1973 Spherical-cap bubbles. Annu. Rev. Fluid Mech. 5, 79–100.CrossRefGoogle Scholar