Stability of a bubble expanding and translating through an inviscid ...

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364 Dinesh Khattar and B B Chakraborty where ⎡ x = ⎣ x ⎤ ⎦ y and F(x) = ⎡ ⎢ ⎣ y { 5 8 x−11/5 1 − 4p ( g 0 ρU0 2 p e x 12/5 − x 4/5 + ⎤ 2σ ) } ⎥ x 2 ⎦ . (15) p g0 R 0 We find that F(x) vanishes at the equilibrium point. Also, H and its partial derivatives are continuous at all points except when x = 0. If α is a positive real root of (13), then eq. (15) shows that x = α and y = 0 is the equilibrium point of (14). Let us choose C in (11) in such a way that at this equilibrium point, the value of H vanishes. Hence, H (α, 0) = 25 48 α−6/5 + 5p g 0 2ρU 2 0 H is minimum at the equilibrium point x = α, y = 0if ∂ 2 H ∂x 2 · ∂2 H ∂y 2 { 5 6 p eα 6/5 + 5 2 α−2/5 + 5σ 2p g0 R 0 α 4/5 } + C = 0. (16) ( ∂ 2 ) 2 − H > 0. (17) ∂x∂y The condition (17), in view of (11), gives us the condition p e x 12/5 − 2σ x 2 + 7x 4/5 + 11ρU2 0 > 0 (18) p g0 R 0 4p g0 for the existence of a minimum at the equilibrium point (α, 0). Adding the left hand side of (13) to the left hand side of (18), we find that this condition (18) becomes p e x 12/5 + 3x 4/5 + 5 ρU0 2 > 0 (19) 4 p g0 which is always satisfied as x is real and positive. Therefore, H is minimum at the equilibrium point x = α, y = 0, but H vanishes at (α, 0). Thus H is always positive near (α, 0) and is therefore positive definite in the neighborhood of the equilibrium point (α, 0). Also −F · grad H vanishes in view of (9), (10) and (14). Thus, −F · grad H is positive semidefinite. We can therefore choose H as a Liapounov function and by Liapounov’s theorem [4], we have the result that a bubble expanding and translating through an inviscid liquid is stable at its equilibrium point. References [1] Chakraborty B B, Effect of a viscous fluid flow past a spherical gas bubble on the growth of its radius, Proc. Ind. Acad. Sci. 100 (1990) 185–188
Stability of a bubble 365 [2] Chakraborty B B and Khattar Dinesh, Stability of an expanding bubble in the Rayleigh model, Proc. Ind. Acad. Sci. (Math. Sci.) 109(4) (1999) 453–456 [3] Chakraborty B B and Tuteja G S, Motion of an expanding gas bubble in a viscous liquid under gravity, Phys. Fluids A5(8) (1993) 1879–1882 [4] Glendinning Paul, Stability, Instability and Chaos (Cambridge: Cambridge University Press) (1994) [5] Noltingk B E and Neppiras E A, Proc. Phys. Soc. London Sec. B63 (1950) 674–685; 64 (1951) 1032–1038 [6] Plesset M S, The dynamics of cavitation bubbles, J. Appl. Mech. 16 (1949) 277 [7] Poritsky H, The collapse or growth of a spherical bubble or cavity in a viscous fluid, Proceedings of the first U.S. National Congress on Applied Mechanics (ed) E. Sternberg (Am. Soc. Mech. Engg., New York, 1952) 813–821 [8] Rayleigh Lord, On the pressure developed in a liquid during the collapse of a spherical void, Philos. Mag. 34 (1917) 94 [9] Turnbulll H W, Theory of Equations (London: Oliver and Boyd) (1957)
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