During the electrolytic evolution of oxygen bubbles forming on a vertically
oriented transparent tin oxide electrode, bubbles were found to be mutually attractive
[1, 2]. The mechanism of the aggregation had never been explained satisfactorily until
Guelcher et al. [3] attributed it to thermocapillary flow. The gradient of surface tension of the
liquid at the bubble’s surface, which was established because of reaction heat and ohmic heat loss at
the electrode wall, drives flow of the liquid adjacent to each bubble; the bubble "pumps" fluid
along its surface away from the wall. Fluid flows toward the bubble to conserve mass and entrains
nearby bubbles in the flow pattern. The same logic would apply when two bubbles of
equal size are adjacent to each other on a warm wall. Each bubble drives thermocapillary flow
and hence entrains the other in its flow pattern, which drives the aggregation. Our
objective here is to perform experiments where the temperature gradient at the wall is well known and
controlled. The theory can be quantitatively tested by studying aggregation of bubble pairs
of equal size, and by varying system parameters such as temperature gradient, bubble size and fluid
viscosity. The results are then compared with the theory in a quantitatively rigorous manner.
We demonstrate that the theory without adjustable parameters is capable of quantitatively
modeling the rate of aggregation of two bubbles.
The equations governing the thermocapillary flow around a single stationary
bubble on a heated or cooled wall in a semi-infinite domain were solved. Both Reynolds
number and Marangoni number were much less than unity. The critical result is that liquid
in the vicinity of a warm wall flows toward a stationary collector bubble (Figure 1). Consequently
the thermocapillary flow around the stationary bubble entrains another bubble toward
itself. The bubbles undergo hindered translation parallel to the wall with velocity U while
the fluid flow field is described with u. Two velocities were equated by using a wall hindrance
parameter q: U = qu (1) which shows the velocity of bubble is proportional to the entraining velocity.
The hindrance parameter q can experimentally be measured independently. q can also be
calculated by solving the equations of motion for a bubble translating parallel to a solid wall.
The experimental cell is cylindrical with an ID of 10 cm and consists of a 1 cm
deep main cell filled with silicone oil and flanked by two thermal reservoirs. The upper
thermal reservoir was heated and the lower thermal reservoir was cooled so that the bubbles
aggregate. Two types of silicone oil ( h = 0.02 and 0.50 Pa s) were used. Two equal sized air bubbles
were injected into the cell with a syringe. The center-to-center distance of bubbles was observed
through a microscope. Bubble radius ranged from 0.40 mm to 0.65 mm and the temperature
gradients along with the cell ranged from 1400 to 5000 K/m.
The bubbles aggregated when heat flows from the wall to the fluid. The
velocities of bubbles were in the range of 1 – 10 mm/s. The separation r decreased more
quickly when the temperature gradient was higher, bubble size was larger, and the oil viscosity
was lower. r decreased more rapidly as the bubbles approached each other. Figure 2 is a plot
of scaled data by appropriate time scale and bubble radius. Dimensionless time was arbitrarily
set to be zero when the dimensionless center-to-center distance between the bubbles was 4. All
the bubble trajectories fall onto one line, especially in the range of dimensionless
distance from 4 to 3. This means the relative movement of the bubble pair is proportional to the
temperature gradient and bubble size and it is inversely proportional to the viscosity of the oil. This
result strongly suggests that the thermocapillary flow-based aggregation mechanism is correct.
A value of q can be estimated by fitting the scaled data to Eq. [1]. A best fit
value of q was obtained as q = 0.26 with a standard deviation of 0.03. Independent
experimental results for q for a 0.5 mm radius bubble, give values of q in the range 0.11 to 0.23. The
value of q obtained from solving the equations of motion reveals q has values in the range
0.23-0.30. Since the full scale of possible values of q is zero to one, the maximum deviation of
independently determined values of q from the best fit value was 15% of this full scale. Thus reasonable
quantitative agreement between theory and experiment has been obtained.
This work was supported by the NASA Microgravity Program, grant
NAG3-2159. We acknowledge the use of a macro for the image software written by Professor Darrell Velegol
of the Pennsylvania State University.
Figure 2 Scaled experimental bubble trajectories trajectory. The ordinate shows
the center-to center
distance of the bubbles scaled by the bubble radius, and the abscissa shows the
dimensionless time. Solid lines are the bubble trajectory calculated from Eq. 1.
Kasumi, H., Solomentsev, Y.E., Guelcher, S.A., Anderson, J.L., Sides, P.J.,, Transition from Pool to Flow Boiling - The Effect of Reduced Gravity, Proceedings of the Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 1656-1673, August 9, 2000.