Dewynne, J.

Lacey, A.A.

Myers, T.G.

Wilson, S.K.

English

Mathematics in Industry Information Service Eprints Archive

Bubble Bursting in Molten Glass
Pilkingtons
Bubbles generated in glass making should, hopefully, all burst through the free sur-
face whilst the glass is still molten. Pilkingtons find that certain bubbles, approximately
one in a thousand, can approach the surface and sit underneath it for up to an hour.
They would like to understand the mechanism behind this process. In this short report
we suggest that the presence of surface tension gradients provide a plausible explanation
for this phenomenon.
film thickness
free
~
bubble
Figure 1: Possible effect of surface tension gradients
When a bubble approaches a free surface a thin liquid film forms above the bubble as
shown in Figure 1. This film drains under the effects of gravity, surface tension and
possibly surface tension gradients before the bubble eventually breaks through. Figure
2 shows how gravity and surface tension effects may compete to prolong this draining
process.
gravity flow surface tension + gravity
Figure 2: Geometry of the problem
Typically, as the film drains, the thickness reaches a minimum in a ring away from the
bubble centre line, as shown in Figure 1. This variation in thickness provides a possible
mechanism for surface tension variation since 802, which is known to affect surface
tension, diffuses from the bubble to the surface. The non-uniform film thickness means
that the 802 concentration will also be non-uniform at the surface giving rise to surface
tension gradients.
1
Assuming, for simplicity, that the surface of the bubble can be modelled as a rigid
boundary and applying lubrication theory to the thin film of molten glass above the
bubble leads to the following equation for the leading order film thickness h = h(r, t)
1 0 { [h3 (2 ) M 2os] }ht+-- r - C(O"\7h)r-Bhr --h - =0,r or 3p 2p or (1)
where s = s(r, t) denotes the leading order concentration of S02' The non-dimensional
Capillary number C, Bond number Band Marangoni number M are given by
C = O"o<P
PoU'
B = pgR2<53
po TT'
M = !10"<5
poU
where uo, p, 0"0 and !10" denote characteristic values of viscosity, density, surface tension
and surface tension variation respectively, R is the bubble radius and 9 the acceleration
due to gravity. The parameter <5~ 10-2 denotes the aspect ratio of the film while U
denotes the as yet unspecified velocity scale. Setting C = 1, B = 1 and M = 1 in turn
yields the appropriate surface tension, gravity and surface tension gradient velocity scales
for the flow; the corresponding timescales are given by
~ = poho '" 104s
Uc 0"0<53
~ = poho '" 106s
UB pgR2<53
ho _ poho 103
UM - <5!10" > S •
Tc
TB (2)
TM =
The value for TM is a lower bound since there was no accurate estimate for !10". If the
fluid were to drain purely by gravity, i.e. with no surface effects, then the gravity time
scale would be O( <52)less then the present value making it '" 102s. The true draining time
scale must lie somewhere between the pure gravity case and the lowest value obtained
from (2).
A more accurate model of the thin film would treat the surface of the bubble as a
free (rather than rigid) boundary. This makes the analysis somewhat more complicated
since now in order to close the leading order problem we must consider the first order
approximation. For a thin, two-dimensional, symmetric sheet this leads to the following
equations for the leading order film thickness h = h(x, t) and velocity u = u(x, t);
ht+(uh)x = 0
4(phux)x+Ch(O"hxx)x+Bhsino.-Msx = 0,
(3)
(4)
where s = s(x, t) again denotes the leading order distribution of S02. These equations
are valid in a segment of bubble in which 0., the angle of the segment with respect to
horizontal, does not vary significantly. Since they are higher order corrections to (1) they
do not significantly affect the time scales.
This work is based on surface tension forces moving fluid in the opposite direction to
gravity forces and thus retarding the draining. A simple check on the time scales shows
that this can make the bursting time last from between a minute (with pure gravity) up
to a few hours (with surface tension and surface tension gradients). Clearly then this
provides a plausible mechanism for long lasting bubbles. It is possible that other factors
2
not considered here may also act to increase bursting times. For example a locally high
viscosity, due to an inhomogeneity in the glass would have a similar effect.
T.G. Myers, S.K. Wilson, J. Dewynne, A.A. Lacey
3


Bubble Bursting in Molten Glass

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