R. Shankar Subramanian

Thermocapillary motion of a liquid drop on a horizontal solid surface.

The motion of drops of decane on horizontal poly(dimethylsiloxane) (PDMS)-coated glass surfaces resulting from a temperature gradient on the surface is studied experimentally, and a theoretical description of the thermocapillary motion of spherical-cap drops on a horizontal solid surface obtained using the lubrication approximation also is presented. The drop size and the applied temperature gradient are varied in the experiments, and the measured velocities of the drops are compared with predictions from the model. The scalings of the velocity with drop size and with the applied temperature gradient are predicted correctly by the theoretical model, even though the actual velocities are smaller than those predicted. The influence of contact angle hysteresis, which leads to a critical drop size below which drops do not move, is found to be minimal. Unlike in previous studies (Chen, J. Z.; Troian, S. M.; Darhuber, A. A.; Wagner, S. J. Appl. Phys. 2005, 97, 014906; Brzoska, J. B.; Brochard-Wyart, F.; Rondelez, F. Langmuir 1993, 9, 2220), this small critical drop size appears to be independent of the applied temperature gradient. Results also are presented on the deformation of the contact lines of the moving drops in the form of an aspect ratio, and correlated with the temperature difference across the footprints of the drops and the capillary number.

The migration of liquid drops in a vertical temperature gradient

Abstract Results are reported from experiments conducted on liquid drops migrating in a vertical temperature gradient. Drops of pure ethyl salicylate, which were observed to sink in diethylene glycol under isothermal conditions, were seen to migrate upward toward warmer regions in the experiments. Control experiments, in which small amounts of Triton X-100, a common surface-active chemical, were added to both phases, revealed that the drops sank in the presence of the surfactant even in steep upward temperature gradients. This provides good confirmation that the observed migration is of thermocapillary origin. The thermocapillary contribution to the drop velocities was found to scale correctly with both the drop radius and the applied temperature gradient as predicted by Young et al. (1). However, the interfacial tension gradient inferred from the data was substantially smaller than that obtained from equilibrium interfacial tension measurements reported by Lacy et al. (12). A tentative reason for this discrepancy is suggested.

The migration of a drop in a uniform temperature gradient at large Marangoni numbers

The steady thermocapillary motion of a spherical drop in a uniform temperature gradient is treated in the situation where convective transport of energy is predominant in the drop phase as well as in the continuous phase, i.e., when the Marangoni numbers are large. It is assumed that the Reynolds numbers in both phases are large as well; to leading order, the velocity fields are given by a potential flow field in the continuous phase and Hill’s vortex inside the drop. The migration velocity of the drop is obtained by equating the rate at which work is done by the thermocapillary stress to the rate of viscous dissipation of energy. The analysis deals with an asymptotic situation wherein convective transport of energy dominates with conduction playing a role only where essential. This leads to thin thermal boundary layers both outside and within the drop. The method of matched asymptotic expansions is employed to solve the conjugate heat transfer problem in the two phases. It is shown that the demand for en...

THERMOCAPILLARY MIGRATION OF BUBBLES AND DROPS AT MODERATE VALUES OF THE MARANGONI NUMBER IN REDUCED GRAVITY

Experiments were performed on the motion of isolated drops and bubbles in a Dow‐Corning silicone oil under the action of an applied temperature gradient in a reduced gravity environment aboard the NASA Space Shuttle in orbit. Images of the interior of the test cell during these experiments were recorded on cine film and later analyzed to obtain data on the migration velocity as a function of size and the applied temperature gradient. The data are presented in scaled form. Predictions are available in the case of gas bubbles, and it is found that the scaled velocity decreases with increasing Marangoni number qualitatively as expected even though there are quantitative discrepancies. The scaled velocity also appears to approach a theoretical asymptote predicted in the limit of large values of the Marangoni number for Stokes motion. Finally, sample results from a preliminary experiment on a pair of drops are presented. They display the remarkable feature that a small drop which leads a large drop in a temper...

The Stokes motion of a gas bubble due to interfacial tension gradients at low to moderate Marangoni numbers

Abstract The steady migration velocity of a gas bubble placed in a large liquid body possessing a uniform temperature gradient in the undisturbed state is calculated under conditions of negligible inertia. Proper account is taken of convective energy transport effects, and results are obtained by solving the energy equation using a numerical scheme. The range of utility of an expansion for the bubble velocity valid in the limit of small Marangoni numbers is extended by the used of series improvement techniques. The improved series is good for Ma ≲ 20 whereas the original series is useful for only Ma ≲ 1. Also, an excellent empirical fit of the numerical results for the bubble velocity is obtained in the limit of large values of the Marangoni number. This reslt is good for Ma ≳ 20. Streamline plots are used to demonstrate the existence of a recirculation region which moves closer to the bubble as the Marangoni number is increased. Isotherms are included to display the presence of an extended thermal wake behind the bubble whose extent increases as the Marangoni number is increased.

Thermocapillary migration of a gas bubble in an arbitrary direction with respect to a plane surface

Abstract The problem of the thermocapillary migration of a gas bubble in an unbounded fluid in the presence of a neighboring rigid plane surface is analyzed in the quasistatic limit of negligible Reynolds and Marangoni numbers. The temperature gradient in the fluid is uniform in the undisturbed state and is oriented at an arbitrary angle relative to the plane surface. The solution is constructed by superposing solutions of the corresponding problems of motion normal to the surface and motion parallel to it. Results are reported in the form of a scalar interaction parameter defined as the ratio of the speed of the bubble in the presence of the plane surface to the speed in its absence. The direction of motion of the bubble is, in general, different from that of the temperature gradient, and is given. The corresponding problem of gravity-driven migration of a gas bubble normal to a plane surface is already solved in the literature. The solution of the parallel migration problem, not reported in the past, is given here. Comparisons of the results in this case show that the plane surface exerts a weaker influence in the thermocapillary migration case. This is due to the more rapid decay, away from the bubble, of the disturbance velocity and temperature gradient fields in this case. Results presented herein show that the surface exerts the greatest influence in the case of motion normal to it, and the weakest in the case of parallel motion.

The settling of spheres in a viscoplastic fluid

Abstract Results on the variation in the steady settling velocity of spheres with a time interval between experiments in a 0.3 wt% aqueous Carbopol-941 dispersion are reported. Spheres made of stainless steel, brass, ceramics and soda lime glass ranging in nominal diameter from 1/4 to 3/8 in. were used. In each experiment, four identical spheres were permitted to settle in quick succession and the time interval between such sets was varied systematically. It was found that four spheres were sufficient to reach an asymptotic settling velocity which did not change with the addition of more spheres to the set. The dependence of the drag coefficient on a modified Reynolds number based on this asymptotic settling velocity was found to be consistent with that from prior work. The steady terminal settling velocity of the first sphere was found to decrease with increasing time intervals between experiments. The results are consistent with the hypothesis of network damage caused by shear with subsequent healing.

Thermocapillary migration of a liquid drop normal to a plane surface

Abstract The quasistatic thermocapillary migration of a liquid drop normal to a plane rigid or a free liquid surface is analyzed using bispherical coordinates. Results for the ratio of the drop speed in the vicinity of the surface to the speed when isolated are calculated and presented for both rigid and free surfaces. The results illustrate that drops of sufficiently high thermal conductivity compared to that of the continuous phase can move more rapidly in the presence of a free surface than when isolated. An explanation of this surprising observation is given. Also, representative streamlines which exhibit interesting flow patterns are presented and discussed.

Thermocapillary migration of a droplet with insoluble surfactant

Abstract The steady thermocapillary migration of a drop, in the presence of a stagnant cap of an insoluble nondiffusing surfactant, is treated theoretically. Gravitational effects are included for completeness, and analytical solutions are obtained under conditions of negligible convective transport of momentum and energy in the bulk phases. The migration velocity of the drop is calculated, and is found to reduce to the proper asymptotes in limiting cases. Typical streamline plots as well as interfacial velocity distributions are displayed. The distribution of surfactant concentration in the cap region is obtained using an ideal film model, and the conditions for the existence of the cap are discussed.

The migration of isolated gas bubbles in a vertical temperature gradient

Abstract Results from experiments on the migration of air bubbles in Dow Corning DC-200 series silicone oils in a vertical temperature gradient are reported. Even though the bubbles nearly doubled in size as they migrated toward warmer regions, the observed velocities are consistent with the quasi-static predictions of N. O. Young, J. S. Goldstein, and M. J. Block (J. Fluid Mech. 6, 350 (1959)) developed for bubbles of constant size. Results for the surface tension gradient of three silicone oils extracted from migration experiments in this study are in good agreement with the results from other techniques in earlier studies.