Spontaneous thermocapillary interaction of drops, bubbles and particles: Unsteady convective effects at low Peclet numbers

Abstract

Mass and heat transfer between two adjacent droplets and the surrounding viscous fluid induce local variations in the surface properties of the drops. These may result in a self-induced surface flow and a subsequent motion of the droplets toward or away from each other. Previous studies of this spontaneous thermocapillary interaction were conducted under the limiting assumptions that inertia, convective effects, and interfacial deformation were negligible. In the present paper the effect of convective transport on the spontaneous interaction of droplets at small nonzero Peclet numbers is examined. It is shown that at large separation distances the motion maintains its quasi-steady nature and the correction to the approach velocity is of O(Pe). When the droplets are at closer proximity the temporal changes of the domain are dominant. They result in the appearance of a Basset type history term in the expansion of concentration field and, hence, in the force balance equation. The correction to the approach velocity is of O(Pe1/2) and it depends on the initial position and the evolution in time of the interaction process.

On the locomotion of a drop, induced by the internal secretion of surfactant

“…For small Pe, it is reasonable to anticipate that the solution will be close to the quasi-steady one discussed above, and search for a small correction. Below we construct a leading order correction term following Lavrenteva et al (1999) and Leshansky et al (2001). Our study of non-stationary convection transport at high Peclet in a special case when the resistance to mass transfer is concentrated mostly in the ambient fluid is performed following the methods of Lavrenteva and Nir (2001).…”

Unsteady forced convection heat/mass transfer around two spheres in tandem at low Reynolds numbers

“…In almost all articles dedicated to this problem, the effects of the convective transport were neglected and the heat transfer was modeled by the Laplace equation. The ef- refers to the downstream sphere fects of convective transport were studied only in [15][16][17][18]. Different types of approximations (boundary layer, for example) were used in [15][16][17][18] to solve the heat/mass balance equations.…”

“…The ef- refers to the downstream sphere fects of convective transport were studied only in [15][16][17][18]. Different types of approximations (boundary layer, for example) were used in [15][16][17][18] to solve the heat/mass balance equations. The momentum and forced convection heat transfer around three spheres in-line have been investigated numerically in [19] (unconfined spheres) and [20] (the spheres are placed at the axis of a tube).…”

Effect of non-homogeneous surface viscosity on the Marangoni migration of a droplet in viscous fluid

“…We solved the case were O(Pe (1) ) Pe (2) = Pe for the problem 'b.' Our considerations follow Lavrenteva et al [25], where a similar convection problem is solved using the matched asymptotic expansions method for the spontaneous thermo-capillary interactions of two drops.…”

“…Following a well-established procedure [23][24][25], we construct inner and outer expansions of the fields outside the drop. The inner and outer concentration fields are denoted by c (1,2) and H , respectively.…”