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Small-amplitude perturbations of shape for a nearly spherical bubble in an inviscid straining flow (steady shapes and oscillatory motion)

Published online by Cambridge University Press:  21 April 2006

I. S. Kang  and
L. G. Leal
I. S. Kang
Affiliation:
Department of Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
L. G. Leal
Affiliation:
Department of Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
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Abstract

The method of domain perturbations is used to study the problem of a nearly spherical bubble in an inviscid, axisymmetric straining flow. Steady-state shapes and axisymmetric oscillatory motions are considered. The steady-state solutions suggest the existence of a limit point at a critical Weber number, beyond which no solution exists on the steady-state solution branch which includes the spherical equilibrium state in the absence of flow (e.g. the critical value of 1.73 is estimated from the third-order solution). In addition, the first-order steady-state shape exhibits a maximum radius at θ = ⅙π which clearly indicates the barrel-like shape that was found earlier via numerical finite-deformation theories for higher Weber numbers. The oscillatory motion of a nearly spherical bubble is considered in two different ways. First, a small perturbation to a spherical base state is studied with the ad hoc assumption that the steady-state shape is spherical for the complete Weber-number range of interest. This analysis shows that the frequency of oscillation decreases as Weber number increases, and that a spherical bubble shape is unstable if Weber number is larger than 4.62. Secondly, the correct steady-state shape up to O(W) is included to obtain a rigorous asymptotic formula for the frequency change at small Weber number. This asymptotic analysis also shows that the frequency decreases as Weber number increases; for example, in the case of the principal mode (n = 2), ω2 = ω00(1−0.31W), where ω0 is the oscillation frequency of a bubble in a quiescent fluid.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 187 , February 1988 , pp. 231 - 266
Copyright
© 1988 Cambridge University Press

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References

Annamalai, P., Trinh, E. & Wang, T. G. 1985 Experimental study of the oscillations of the rotating drop. J. Fluid Mech. 158, 317–327.Google Scholar
Busse, F. H. 1984 Oscillations of a rotating liquid drop. J. Fluid Mech. 142, 1–8.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.
Kang, I. S. & Leal, L. G. 1987 Numerical solution of unsteady free-boundary problems at finite Reynolds number - I. Finite-difference scheme and its application to the deformation of a bubble in a uniaxial straining flow. Phys. Fluids 30, 1929–1940.Google Scholar
Lamb, H. 1932 Hydrodynamics. 6th edn. Cambridge University Press.
Marston, P. L. 1980 Shape oscillation and static deformation of drops and bubbles driven by modulated radiation stress: theory. J. Acoust. Soc. Am. 67, 15–26.Google Scholar
Miksis, M. 1981 A bubble in an axially symmetric shear flow. Phys. Fluids 24, 1229–1231.Google Scholar
Miller, C. A. & Scriven, L. E. 1968 The oscillations of a fluid droplet immersed in another fluid. J. Fluid Mech. 32, 417–435.Google Scholar
Natarajan, R. & Brown, R. A. 1987 The role of three-dimensional shapes in the break-up of charged drops. Proc. R. Soc. Lond. A 410, 209–227.Google Scholar
Prosperetti, A. 1977 Viscous effects on perturbed spherical flows. Q. Appl. Maths 34, 339–352.Google Scholar
Rayleigh, J. W. S. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond, 29, 71–97.Google Scholar
Reid, W. H. 1960 The oscillations of a viscous liquid drop. Q. Appl. Maths 18, 86–89.Google Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solutions of free-boundary problems in fluid mechanics. Part 3. Bubble deformation in an axisymmetric straining flow. J. Fluid Mech. 148, 37–43.Google Scholar
Subramanyam, S. V. 1969 A note on the damping and oscillations of a fluid drop moving in another fluid. J. Fluid Mech. 37, 715–725.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1983 Nonlinear oscillations of inviscid drops and bubbles. J. Fluid Mech. 127, 519–537.Google Scholar
Tsamopoulos, J. A. & Brown, R. A. 1984 Resonant oscillations of inviscid charged drops. J. Fluid Mech. 147, 373–395.Google Scholar
Vanden-Broeck, J.-M. & Keller, J. B. 1980 Bubble or drop distortion in a straining flow in two dimensions. Phys. Fluids 23, 1491–1495.Google Scholar