Abstract
The boundary integral method is applied to model the initial motion of two-dimensional or cylindrical deformable gas bubbles in an inviscid, incompressible fluid. Following the success of recent boundary integral studies to predict the qualitative behaviour of a single gas bubble, this numerical study is extended to consider the interaction of several bubbles. Surface tension, relative initial position and volume are all found to be important factors affecting the bubble interaction, jet formation, “trapping” of fluid between bubbles and bubble shedding. As well as computing the evolution of the bubble surfaces, consideration of the pressure fields and resulting instantaneous streamlines is given.
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References
J.K. Walter and J.F. Davidson, The initial motion of a gas bubble in an inviscid liquid. Part I. The two-dimensional bubble. J. Fluid Mech. 12 (1962) pp. 408–417.
R.T. Baumel, S.K. Burley, D.F. Freeman, J.K. Gammel and J. Nuttall, The rise of a cylindrical bubble in an inviscid liquid. Can J. Phys. 60 (1982) pp. 997–1007.
G.R. Baker and D.W. Moore, The rise and distortion of a two-dimensional bubble in an inviscid liquid. Phys. Fluids A, 1 (9) (1989) 1451–1459.
J.R. Blake, P.B. Robinson, A. Shima andY. Tomita, Interaction of two cavitation bubbles with a rigid boundary. J. Fluid Mech. 255 (1993) 707–721.
S.O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multifluid flows. J. Comp. Phys. 100 (1) (1992) 25–37.
R. Clift, T.J.R. Grace and M.E. Weber, Bubbles, Drops, and Particles. Academic Press, 1978.
D. Bhaga and M.E. Weber, Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105 (1981) 61–85.
J.W.B. Kok, Dynamics of a pair of gas bubbles moving through fluid. Part I. Theory. European J. Mechanics B/Fluids, 12(4) (1993) 252–540.
J.W.B. Kok, Dynamics of a pair of gas bubbles moving through fluid. Part II. Experiment. European J. Mechanics B/Fluids, 12(4) (1993) 541–560.
P.C. Duineveld, Bouncing and coalescence of two bubbles in water. PhD thesis, The Netherlands, Twente University, (1994).
V. Kumaran and D.L. Koch. The rate of coalescence in a suspension of high Reynolds number, low Weber number bubbles. Phys. Fluids 5 (1993) 1135–1140.
L.van Wijngaarden. The mean rise velocity of pairwise interacting bubbles in liquid. J. Fluid Mech. 251 (1993) 55–78.
M. Manga and H.A. Stone, Buoyancy-driven interactions between two deformable viscous drops. J. Fluid Mech. 256 (1993) 647–683.
J.F. Harper, The motion of bubbles and drops through liquids. Adv. Appl. Mech. 12 (1972) 59–129.
G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
M.J. Lighthill, An informal Introduction to Fluid Mechanics, Oxford University Press, 1986.
N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, (1953).
J.R. Blake, B.B. Taib and G. Doherty, Transient cavities near boundaries. Part II. Free surface. J. Fluid Mech. 181 (1987) 197–212.
J.P. Best and A. Kucera, A numerical investigation of nonspherical rebounding bubbles. J. Fluid Mech. 245 (1992) 137–154.
M.S. Longuet-Higgins, On the forming of sharp corners at a free surface. Proc. Roy. Soc. Lonc. A 371 (1980) 453–478.
H.N. O guz and A. Prosperetti, Surface tension effects in the contact of liquid surfaces. J. Fluid Mech. 203 (1989) 149–171.
G.K. Batchelor, The stability of a large gas rubble rising through liquid. J. Fluid Mech. 184 (1987) 399–422.
M.A. Jaswon and G.T. Symm, Integral Equation Methods in Potential Theory and Elastostatics. Academic Press (1977).
A. Esmaeeli, E. Ervin and G. Tryggsvason, Numerical simulations of rising bubbles. In J.R. Blake, J.M. Boulton-Stone and N.H. Thomas, editors, Bubble dynamics and interface phenomena, Kluwer (1994).
J.M. Boulton-Stone, A comparison of boundary integral methods for studying the motion of a two-dimensional bubble in an infinite fluid. Comp. Meths. in Appl. Mech. and Engng. 102 (1993) 213–234.
J.M. Boulton-Stone and J.R. Blake, Gas bubbles bursting at a free surface. J. Fluid Mech. 254 (1993) 437–466.
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Robinson, P.B., Boulton-Stone, J.M. & Blake, J.R. Application of the boundary integral method to the interaction of rising two-dimensional deformable gas bubbles. J Eng Math 29, 393–412 (1995). https://doi.org/10.1007/BF00043975
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DOI: https://doi.org/10.1007/BF00043975