Growth and translation of a liquid-vapour compound drop in a second liquid. Part 1. Fluid mechanics

Abstract

The fluid dynamics associated with a compound drop consisting of a vapour bubble, partly surrounded by its own liquid in another immiscible liquid is considered. The fluid motion is analysed in the limit of Stokes flow and at the same time the surface tension forces are considered to be large enough to allow the interfaces to have uniform curvature. The flow field consists of translation and growth that can arise from change of phase.An exact analytical solution for the axisymmetric flow field is obtained. The important results of physical interest are the drag force and the flow behaviour. In the case without growth, the drag force lies between the bubble and the solid-sphere limits for a sphere of the same volume as the total liquid and vapour dispersed phase. The maximum drag force is observed when the liquid and vapour volumes are nearly the same. This is the effect of weak circulation due to the smaller available space as compared with a spherical drop. With growth this effect appears to be enhanced. The flow streamlines exhibit secondary vortices in the dispersed phase when there is growth. The velocity field and the drag results here are applied to the heat transfer problem for the compound drop in Part 2 of this two-part series.

Thermocapillary motion of hybrid drops

“…͑1͒-͑6͒, with a constant U ϱ , is axisymmetric. We follow the approach used by Vuong and Sadhal 13 and Oğuz. 14 Instead of solving for the velocity and pressure distributions we choose to use the Stokes stream function, with…”

“…One of the goals of the current study is to provide an alternative approach in which all the tangential boundary conditions are satisfied. Another limiting case was examined by Vuong and Sadhal 13 who studied the fluid dynamics associated with a compound drop consisting of a vapor bubble, partially surrounded by its own liquid immersed in another immiscible liquid. The exact solution of this compound drop system, describing the growth and translation of an axisymmetric liquid-vapor drop, was found.…”

“…Their solution was valid for any contact angle but was limited by the assumption that one of the dispersed phases is gaseous, i.e., inviscid. Oğuz 14 extended the methods of Vuong and Sadhal 13 to the case when both parts of a compound drop are viscous and applied them to compute the drag on such a hybrid drop in a uniform flow. In these works 13,14 the authors used the toroidal coordinate system which is highly useful in the treatment of fluid flow problems involving lens-shaped axisymmetric bodies as was shown by Payne and Pell.…”

“…Oğuz 14 extended the methods of Vuong and Sadhal 13 to the case when both parts of a compound drop are viscous and applied them to compute the drag on such a hybrid drop in a uniform flow. In these works 13,14 the authors used the toroidal coordinate system which is highly useful in the treatment of fluid flow problems involving lens-shaped axisymmetric bodies as was shown by Payne and Pell. 15 Their computations were restricted to O͑1͒ values of the viscosity ratio.…”

Boiling of small droplets

“…They find the rate of heat transfer to the droplet and compare it to experimental results obtained by photographing droplets boiling while rising through water. Tochitani and coworkers [9,10] also develop an analytical model of heat transfer to a boiling droplet using a geometric model of the droplet similar to that of Sideman and Taitel, but assume that the flow around the Other researchers examine boiling droplets using more sophisticated models of the geometry of the bubble and droplet [12,13]. Raina and Grover [14] study the effects of sloshing of the droplet around the bubble.…”