Slow migration of a gas bubble in a thermal gradient

Abstract

The steady migration velocity of a gas bubble placed in a liquid with a linear temperature field in the absence of gravity is obtained for small Marangoni Numbers using a matched asymptotic expansion procedure for solving the governing equations. A result good to O(N&,) is obtained, and in the limiting case of zero Marangoni Number, the results of Young, Goldstein and Block are recovered. R. S. SUBRAMANIAN Deportment of Chemical EngineeringClorkron College of Technology Potsdom, NY 13676 SCOPEWith the advent of orbital facilities for experimentation such as Spacelab to be flown aboard the Space Shuttle, and "Space Processing" applications, the migration of droplets and bubbles in a continuous phase due to forces other than buoyancy will become a subject of considerable interest.A theoretical development is presented for the description of bubble migration in a free fall environment due to interfacial tension gradients generated on the bubble surface by a temperature gradient in the surrounding liquid. The objective is to obtain a result for the migration velocity as a function of system parameters for a class of systems characteristic of glasses suggested for Space Processing. These systems, because of their high viscosities, possess very low Reynolds Numbers SO that the inertial terms in the equations of motion may be ignored* However, the Prandtl can be large; therefore, the convective transport terms in the energy equation are not usually negligible. CONCLUSIONS AND SIGNIFICANCEThe principal result of this work is Eq. 53 for the scaled migration velocity of the bubble. In addition to providing a quantitative expression for the buhhle velocity, this result identifies the influence of convective transport of energy in the system. The effect of such transport is to reduce the migration velocity over the value which would prevail were conduction to be the only mechanism. In the limit of zero Marangoni Number, when convective transport is negligible, the result derived here reduces to that of Young, Goldstein and Block utility in the description of bubble migration in space processing applications. Also, applications may be anticipated in the areas of bubble elimination in glass furnaces and in sealing operations where large thermal gradients are encountered. It is expected that this work will haveThe migration of a gas bubble (or a droplet in general) d u e to forces other than buoyancy in a surrounding fluid medium is a subject which so far has received only a small amount of attention. However, with the advent of the Space Shuttle, and the opportunity it provides for the conduct of experiments in a free fall environment, there is a need for developing improved descriptions of this process. In space experiments, it is expected that many occasions will arise where liquid bodies containing droplets of a second fluid, either liquid or gaseous, will be encountered. An example is in the manufacture of spaceprocessed glasses.It has been suggested that the containerless environment available in orbit can be used to make...

Thermocapillary migration of a planar droplet at moderate and large Marangoni numbers

“…They gave an analytical prediction on its migration speed in the limit case of zero Reynolds (Re) and zero Marangoni (Ma) numbers, which is called the YGB model. Since then, the thermocapillary migration of a bubble has been studied extensively by a series of theoretical analyses [3][4][5][6], numerical simulations [7][8][9][10] and experimental investigations [11]. In the mean time, several numerical techniques for treating the two-phase flow, such as the front-tracking method [12,13] and the level-set method [14], have also been developed, which may provide effective techniques to directly investigate thermocapillary migration processes of bubbles or droplets [15][16][17][18], interfacial mass transfer [19,20] and interfacial flows with soluble surfactants [21,22].…”

Thermocapillary Migration of Deformable Bubbles at Moderate to Large Marangoni Number in Microgravity

“…Thompson et al (1980) extended this expansion to the next higher order term and determined a nonzero correction to the migration velocity at O(Re 2 ). Using the method of matched asymptotic expansions, Subramanian (1981Subramanian ( , 1983 provided a correction to the migration velocity, in which the Marangoni number was used as a perturbation parameter. It was shown to be useful only for Ma < 0.5.…”

Numerical simulation of drop Marangoni migration under microgravity

“…The motion of liquid drops and gas bubbles driven by the temperature gradient is called thermocapillary migration or Marangoni migration. Since the pioneering work of Young et al [1], who derived the mathematical formula for the migration velocity of a spherical drop in a constant temperature gradient at small Reynolds and Peclet numbers by omitting the inertia force and convective energy transport, some succeeding work in this ÿeld has been performed experimentally, theoretically and numerically [2][3][4][5][6][7][8][9][10].…”

“…In the limiting case of a gas bubble, which was analyzed by Balasubramaniam et al [7], the thermal conductivity of the gas in the bubble is considered to be negligible. Subramanian [8,9] investigated the motion of a gas bubble, taking the e ect of convective transport of energy as a small perturbation, and extended the analysis to the case of a uid drop, which accounted for the transport process in both phases. Recently Balasubramaniam et al [10] furthered their theoretical work to the migration of a drop in a uniform temperature gradient at large Marangoni numbers.…”