Olga M. Lavrenteva

Thermocapillary migration of bubbles: convective effects at low Péclet number

The effect of a weak convective heat transfer on the thermocapillary interaction of two bubbles migrating in an externally imposed temperature gradient is examined. It is shown that, for short and moderate separation distances, the corrections to the individual migration velocities of the bubbles are of O ( Pe ), where Pe is Peclet number. For separation distances larger than O ( Pe −1/2 ) the correction is of O ( Pe 2 ) as previously found for an isolated drop. The perturbations to the bubble velocities have opposite signs: the motion of the leading bubble is enhanced while the motion of the trailing one is retarded. A newly found feature is that equal-sized bubbles, which otherwise would move with equal velocities, acquire a relative motion apart from each other under the influence of convection. For slightly unequal bubbles there are three different regimes of large-time asymptotic behaviour: attraction up to the collision, infinite growth of the separation distance, and a steady migration with equal velocities, the steady motion separation distance being a function of the parameters of the problem. Sufficient conditions for the realization of each regime are given in terms of the Peclet number, initial separation and radii ratio.

SPONTANEOUS THERMOCAPILLARY INTERACTION OF DROPS, BUBBLES AND PARTICLES : UNSTEADY CONVECTIVE EFFECTS AT LOW PECLET NUMBERS

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 v...

Thermocapillary motion of hybrid drops

Hybrid drops are composed of two or more immiscible phases. They occur in various natural and technological processes and environments, e.g., the atmosphere, liquid membranes, and liquid bilayers and direct-contact heat and phase separation processes. In this work we have studied the dynamics of an axisymmetric hybrid drop in an infinite viscous domain. The drop moves under the influence of the Marangoni effect due to an arbitrarily imposed temperature field. In the classical case of a constant temperature gradient, it was found that in general the motion is induced in the direction of the temperature gradient. However, under certain physical conditions and drop geometry, motion against the temperature gradient is obtained.

Stationary regimes of axisymmetric thermal wake interaction of two buoyant drops at low Reynolds and high Peclet number

Axisymmetric motion of a leading fluid drop and a trailing gas bubble (or thermally nonconducting drop) in a viscous fluid under the combined effect of gravity and thermocapillarity is considered under the assumption of negligible inertia effects and of nondeformable interfaces. The ambient fluid far from the inclusions is isothermal and the temperature of the leading particle differs from that of the continuous medium. At large Peclet number, thermal boundary layers are present along the fluid-liquid and the gas-liquid interfaces, and thermal wakes are formed downstream from the particles. The interaction of the thermal wake, shed from the leading inclusion, with the thermal boundary layer on the surface of the trailing one causes a nonuniform temperature distribution on the surface of the latter. The induced Marangoni flow results in the change of the flow pattern, the velocity of both particles, and the equilibrium separation distance. In the present paper, the influence of the Marangoni effect on the ...

Deformation and breakup of a non-Newtonian slender drop in an extensional flow: inertial effects and stability

We consider the deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid, subject to an axisymmetric extensional flow, and the influence of inertia in the continuous phase. The non-Newtonian fluid inside the drop is described by the simple power-law model and the unsteady deformation of the drop is represented by a single partial differential equation. The steady-state problem is governed by four parameters: the capillary number; the viscosity ratio; the external Reynolds number; and the exponent characterizing the power-law model for the non-Newtonian drop. For Newtonian drops, as inertia increases, drop breakup is facilitated. However, for shear thinning drops, the influence of increasing inertia results first in preventing and then in facilitating drop breakup. Multiple stationary solutions were also found and a stability analysis has been performed in order to distinguish between stable and unstable stationary states.

The leading effect of fluid inertia on the motion of rigid bodies at low Reynolds number

We investigate the influence of fluid inertia on the motion of a finite assemblage of solid spherical particles in slowly changing uniform flow at small Reynolds number, Re, and moderate Strouhal number, Sl. We show that the first effect of fluid inertia on particle velocities for times much larger than the viscous time scales as rootSl Re given that the Stokeslet associated with the disturbance flow field changes with time. Our theory predicts that the correction to the particle motion from that predicted by the zero-Re theory has the form of a Basset integral. As a particular example, we calculate the Basset integral for the case of two unequal particles approaching (receding) with a constant velocity along the line of their centres. On the other hand, when the Stokeslet strength is independent of time, the first effect of fluid inertia reduces to a higher order of magnitude and scales as Re. This condition is fulfilled, for example, in the classical problem of sedimentation of particles in a constant gravity field.

The weakly inertial settling of particles in a viscous fluid

In this paper we investigate the influence of fluid inertia on the settling of a finite assemblage of solid spherical particles in a constant gravity field at small Reynolds number, Re. We show that the first effect of fluid inertia on particle velocities scales as Re, for times much larger than the viscous–relaxation time. In this case the Eulerian acceleration terms associated with the unsteadiness of the stresslet in the far velocity field and the entire local fluid inertia (acceleration and advective terms) contribute at O(Re). As a particular example, Oseen velocities are calculated of two spheres falling along the line of their centres. The inertia–induced relative motion between the particles is in excellent agreement with previous experimental results.

Axisymmetric thermal wake interaction of two drops in a gravity field at low Reynolds and high Peclet numbers

The axisymmetric motion of two drops in a viscous fluid under combined effect of gravity and thermocapillarity is considered. The ambient fluid is isothermal far from the drops and the temperature of the leading drop exceeds that of the continuous media. At large Peclet number, thermal boundary layers are present along the interfaces, and thermal wakes are formed downstream of the drops. The interaction of the thermal wake, shedded from the leading drop, with the boundary layer of the trailing one was shown to influence significantly the temperature distribution on the surface of the latter. The induced change in the speed of the trailing drop is comparable in magnitude with its speed when isolated even for large separation distance between the drops where the hydrodynamic interaction is negligible. In the extreme case of very large Marangoni effect the direction of the trailing drop motion can be reversed.

Spontaneous thermocapillary interaction of drops: Effect of surface deformation at nonzero capillary number

The axisymmetric spontaneous thermocapillary induced motion and deformation of two viscous droplets caused by interfacial mass transfer is considered. Convective transport and inertial effects are neglected. The case of slightly deformable drops at close proximity was analyzed making use of lubrication approximation and tangent sphere coordinates. The effects of the drop-to-medium viscosity ratio and drops size ratio on the motion is illustrated. For cases with significant deformations, solute concentration and velocity fields are computed using boundary-integral techniques for Laplace and Stokes equations, respectively. Numerical results on axisymmetric drop motion, deformations and temporal evolution of the separation between the drops are presented for equal viscosities of the drops and the ambient fluid.

Spontaneous thermocapillary interaction of drops: Unsteady convective effects at high Peclet numbers

Initial nonequilibrium distribution of temperature or surfactant concentration between suspended drops and the continuous fluid in a suspension results in an unsteady-state heat/mass transfer between the phases. Nonuniformities of temperature or solute concentration, which arise as a natural result of local geometrical inhomogeneities in the suspension, produce temperature/concentration gradients along the interfaces that, in turn, generate thermocapillary fluid motion along the interfaces and migration of drops toward or apart from each other. Asymptotic analysis of the process is carried out for large Peclet numbers of the dispersed phase. The dynamics of drops is studied and the approach time is estimated for the limiting cases of small and large Peclet numbers of the continuous phase.