On the rising motion of a drop in stratified fluids

Abstract

The rising dynamics of a deformable drop in a linearly stratified fluid is numerically obtained using a finite-volume/front-tracking method. Our results show that the drag coefficient of a spherical drop in a stratified fluid enhances as \documentclass[12pt]{minimal}\begin{document}$C_{d,s}/C_{d,h}-1\sim Fr_d^{-2.86}$\end{document}Cd,s/Cd,h−1∼Frd−2.86 for drop Froude numbers in the range of 4 < Frd < 16. The role of the deformability of the drop on the temporal evolution of the motion is investigated along with stratification and inertial effects. We also present the important role of stratification on the transient rising motion of the drop. It is shown that a drop can levitate in the presence of a vertical density gradient. The drop undergoes a fading oscillatory motion around its neutrally buoyant position except for high viscosity ratio drops where the oscillation occurs around a density level lighter than the neutral buoyancy level. In addition, a detailed characterization of the flow signature of a rising drop in a linearly stratified fluid including the buoyancy induced vortices and the resultant buoyant jet is presented.

Rising motion of a swarm of drops in a linearly stratified fluid

“…In driving Eq. (3), we have assumed that the thermal diffusivity and conductivity coefficients in the dispersed and continuous phases are uniform and equal (Bayareh et al, 2013). Eqs.…”

“…In order for the swarm to reach a steady state condition, we set q f 0 b f ¼ q d 0 b d , so that the drop density reduces as it enters warmer fluid layers. As a result, the spatial variation of the temperature inside and outside the drops become identical and drops eventually reach a statistically steady rise velocity (Bayareh et al, 2013). The rise Reynolds number Re W ¼ W s d=m f is calculated a posteriori based on the statistically steady-state average slip velocity of the swarm of drops W s .…”

“…The migration of a swarm of drops in an incompressible, linearly stratified fluid is governed by the following equations (Bayareh et al, 2013):…”

“…However, the size of bubbles/drops in aquatic environments are generally much smaller than the length scales of density gradients in the water column and thus a more realistic physical model in the natural environment is represented by a linear stratification. It has been recently found by Bayareh et al (2013) that the presence of a linear density gradient results in a notable drag enhancement of a rising drop and subsequently extends the travel time of the drop in the water column by up to 30%.…”

“…The rising motion of bubbles in oceans and lakes (Bayareh et al, 2013), bubble mixers used for aeration of lakes and reservoirs (Hill et al, 2008) and motion of drops during oil spills (Blumer et al, 1971) are few examples of important processes that are being affected by density stratification. Oil spills can cause extensive hazards to marine and wildlife habitats as well as fishing and tourism industry (Juhasz, 2012).…”