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Abstract We develop a mathematical model of heat and mass transfer in a configuration which involves a spherical droplet levitating near a flat liquid layer heated from below. Analytical solutions for vapor concentration in air and the temperature distributions both inside the droplet and in moist air around it are coupled to the numerical solution for heat transfer in the liquid layer. In the limit of weak evaporation, the liquid layer surface is cooled locally due to the presence of the droplet, while the effect is reversed for strong evaporation. The latter case is also characterized by higher temperature near the bottom of the droplet and stronger temperature gradients in the droplet itself, an unexpected conclusion given the high liquid-to-air thermal conductivity ratio. The observations are explained in terms of interplay between geometric and thermal effects of the presence of the droplet. A simple analytical criterion is formulated to determine the condition when the droplet presence has no effect on the layer temperature; remarkably, the condition does not depend on the distance between the droplet and the layer surface. The calculation of the evaporation rate leads to determination of the flow around the droplet, treated in the Stokes flow approximation, as well as the levitation height.
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Heat transfer, vapour diffusion and Stefan flow around levitating droplets near a heated liquid surface
We consider a slowly condensing droplet levitating near the surface of an evaporating layer, and develop a mathematical model to describe diffusion, heat transfer and fluid flow in the system. The method of separation of variables in bipolar coordinates is used to obtain the series expansions for temperature, vapour concentration and the Stokes stream function. This framework allows us to determine temperature profiles and condensation rates at the surface of the droplet, and to calculate the upward force that allows the droplet to levitate. Somewhat counter-intuitively, condensation is found to be the strongest near the bottom of the droplet, which faces the hot liquid layer. The experimentally observed deviations from the classical law predicting the square of the radius to grow linearly in time are explained by the model. A spatially non-uniform phase change rate results in a contribution to the force not considered in previous studies, and comparable to droplet weight and the upward force calculated from the Stokes drag law. The levitation conditions are formulated accordingly, resulting in the prediction of levitation height as a function of droplet size without any fitting parameters. A simple criterion is formulated to define the parameter ranges in which levitation is possible. The results are in good agreement with the experimental data except that the model tends to slightly underpredict the levitation height.
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- Award ID(s):
- 2009741
- PAR ID:
- 10436468
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 964
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2023.351
