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

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