Two bubbles migrating along their line of centers under the influence of an imposed thermal gradient are considered in the quasi-static limit. Results are reported for representative values of the governing parameters

Meyyappan, M.

Subramanian, R. S.

Wilcox, W. R.

English

NASA Technical Reports Server

Interaction effects in th~i--c-ntllarg bubble aigration 
H. Heyyappan, William R. Wilcox, and R. Shankar S u b r a ~ n i m  
Department of Chemical Engineerina, Clarkson College of Techmlogv 
Potsdam, Ne-3 vork 13676 
Abstract 
The interaction effects between two bubbles migrating along their line of centers under 
the influence of an imposed thermal gradient are considered in the quasi-static limit. Re- 
sults are reported for representative values of the governing parameters. 
A gas bubble, when placed in a liquid with a non-uniform temperature field, will move in 
the dire 'ion of the temperature gradient. Such motion is a consequence cf the dependence 
of surf;-e tension on temperature. The temperature pradient at the bubble surface qives 
rise to a surface tension gradient. The resulting tanqential stress at the interface will 
cause motion of the neighboring liquid in the direction of increasina surface tension, or 
toward the colder pole of the bubble. By reaction, the bubble will propel itself toward 
the warmer regions of the liquid. The thermocapillary motion of isolated bubbles has been 
the subject of much study. Both theoretical descriptions1 ', and experimental observations 
in general agreement with theory1 * ' ' have been reported. 
Under conditions of creeping motion, the buoyant rise velocity of r gas bubble is propor- 
tional to the square of its diameter whereas its thernocapillary velocity (for negligible 
convective enerpy transport) is proportional to the diameter. Thus, it may be expected that 
on earth, very small bubbles will be influenced significantly by temperature aradients. In 
contrast, in reduced gravity processing, capillarity induced motion of bubbles and droplets 
will be important over a wide range of bubble sizes. Thus, there 1s renewed interest in 
this subject. At Clarkson, for instance, theoretical models of thermocapillary motion are 
being d e v e l ~ ~ e d ~ " - ~ ~  and a reduced gravity experiment has been flown in collaboration with 
Uestinghouse Research and Development Center on a NASA rocket in their SPAR (Space Pro- 
cessing Applicatl m s  Rocket) Pf7Sfam11. Mcre rocket experiments as well as Space Shuttle 
experiments are being planned 
The problem of the isolated bubble moving in a thermal yradient has received considerahle 
attention as mentioned earlier. However, in many practical applications, interactions with 
neighboring bubbles and boundaries will be important. We have begun addressing such prob- 
lems theoretically in the quasi-static limit. work on huhble miqration normal to a % 
fluid or solid surface is reported in ( 8 ) .  Elsewhere in this ~roceedin~s"', a brief disc 
sion of the problem of thermocapillary motion of bubbles inside drops is presented in the 
context of applications to the containerless processing of glasses and other materials in 
free fall. The axially symmetric interaction problems for two uneaual hubbles has been 
solved using bispherical coordinates, and a complete treatment is beinu reported elsewhere" 
Details will be given in (16). Here, only a brief summary of the results will be presented. 
When two bubbles, possibly of unequal diameters, are placed in a liquid possessina a 
linear temperature field such that the line of centers of the bubbles is parallel to the 
temperature gradient, the resulting problem is axially symmetric. Both bubbles will move in 
the direction of the temperature gradient. The problem, in general, is unsteady. However, 
when the Reynolds and'Marangoni numbers are very small, the auasi-static assumption" may 
be made in order to ohtain first order results. This involves ianoring the unsteady accumu- 
lation terms as well as the convective transport terms in the equations of conservation of 
momentum and energy. In physical terms, in part, it is assumed that the vorticity and 
temperature distributions for a given spacing of the bubbles relax quickly to their steady 
representations for that configuration compared to the time scale for the bubbles to move an 
appreciable distance and change the configurati~n. Further, the convective transport terms 
are ignored compared to molecular transport. For the small hubbles in glass melts used in 
our rocket experiments, the quasi-static assumption is a good one. 
General solutions of the quasi : -atic equations in bispherical coor8inates are avail- 
able1"". They are specialized for the boundary conditions of the present problem, a 
straightforward, but tedious process1 5 .  The q~asi-static thennocapillary velocities of the 
two bubbles ther, are calculated by setting the net hydro6ynamic force on the bubbles to 
zero. 
It is convenient to discuss the results in the context of an interaction parameter Q 
https://ntrs.nasa.gov/search.jsp?R=19820015547 2020-03-21T16:53:51+00:00Z
which repreaents the ratio of the velocity cf a given bubtle in the presence of the second 
bubble to its velocity when isolated. Q will depend on the ratio of bubble radii, A 
d 1 and the scaled separation distance between the bubbles, D. (D * 1 .  Here 'dm ( A  = N T  
L 1 
is the actual separation distance, R , the radius of bubble I, and R the radius of bubble 
11. Due to tha naglect of convectivh tramport effects, the order oix;wo buhbles in the 
temperature gradiert has no influence on the result.. A representative set of values of nI 
and QII for the two bubbles as a function of scaled separation distance D is reported in 
Table I for a typ~cal value of A. X = 1.5 represents the case of bubble 11 being larger 
than bubble I. 
TABLE I 
Interaction Parameters QI and QII as a Function of D for X = 1.5 
D f i ~  O11 
2.6 1.1432 0.9662 
From the results in Table I, it may be seen that the larger of the two bubbles (11) 
moves slightly slower than it would if isolated. In contrast, the smaller bubble (I) moves 
more rapidly than it would, when isolated. Also, as the bubbles are separated further, the 
interaction becomes less important, and fi approaches unity. We might also observe that in 
view of the above behavior in the case of unequal bubbles, it is not surprising that when 
they are of equal size, both bubbles move at the same velocity that they would possess, if 
isolated. 
Acknowledgments 
This work was supported in part by the Materials Processing in Space Program Office of 
NASA, Marshall Space Flight Center, through a subcontract from Westinghouse Research and 
Developmnt Center to Clarkson College of Technolwy. Computer time for some of the cal- 
culations reported here was donated by Clarkson College of Technology. 
References 
1. Young, N. O., Goldstein, J. S., and Block, M. J., J. Fluid Mech. 6, 350 (1959). 
2. Bratukhin. Y. K., Thermocapillary Drift ~f a Viscous Fluid ~roplef, NAqA Report No. 
TT F-17093, June 1976. 
3. Thompson, R. L., Dewitt, K. J., and Labus, T. L., Chem. Eng. Comun. 5, 299 (1980). 
4 Subramanian, R. S., A.1.Ch.E.J. 27, 646 (1981). 
5. McGrew, J. L., Rehm, T. L., and mskey, R. G., App. Sci. Res. 29, 195 (1974). 
6. Hardy, S. C., J. Colloid Interface Sci. 69, 157 (1979). 
7. Shankar, N., Cole, R., and Subramanian, R. S., Int. J. of Multiphase Flow 1, 581 
(1981). 
9. Meyyappan, M., Wilcox, W. R., and Subramanian, R. S., J. Colloid Interface Sci. 9, 
199 (1981). 
9. Jayaraj, K., Cole, R., and Subramanian, R.S., J. Colloid lnterface Sci. In Press, 
1981. 
10. Shankar, N., and Subramanian, R. S., J. Fluid Mech., Submitted; 
11. Wilcox, W. R., Subramanian, R. S., Heyyappan, N., Smith, I!. D., Mattox, D. H., and 
Partlow, D. P., Final Report to NASA Contract No. NAS8-33017. 
12. Subramanian, R. S., and Cole, R., "Experiment Requirements and Implementation Plan 
(ERIP) for Physical Phenomena in Containerless Class rrocesaing." NASA Marshall Space Flight 
Center, Document 77FR010, August, 1979. 
13. Annaraalai, P., Shankar, N., Cole, R., and Suhramanian, R. S., Appl. Sci. Res., In 
Press. 
14. Shankar, N., Annirmalai, P., Cole, R., and Subramanian, R. S., This Proceedingr. 
15. Meyyappan, M., Wilcox, W. R., and Subramanian, R. S., 'The Slow Axisynunetric Motion 
of Two Bubbler in a Thermal Gradientw to be submitted for publication. 
16. Meyyappan, M., Ph.D. thesis in Chemical Engineering, in progress, Clarkson College 
of Technology. 
17. Happel, J., and Brenner, H., "Low Reynolds Number  hydrodynamic^." Prentice-Hall, 
Snglewood Cliff., N.J., 1965. 
18. Jsffery, C. B., Proc. R. Soc. A87, 1,,5 (1912). 
19. Stimson, M., and Jeffary, G. a.,~roc, R. Soc. G, 110 (1926). 


Interaction effects in thermocapillary bubble migration

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