Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers

Abstract

We present the results of numerical simulations of the three-dimensional thermocapillary motion of deformable viscous drops under the influence of a constant temperature gradient within a second liquid medium. In particular, we examine the effects of shape deformations and convective transport of momentum and energy on the migration velocity of the drop. A numerical method based on a continuum model for the fluid–fluid interface is used to account for finite drop deformations. An oct-tree adaptive grid refinement scheme is integrated into the numerical method in order to track the interface without the need for interface reconstruction. Interface deformations arising from the convection of energy at small Reynolds numbers are found to be negligible. On the other hand, deformations of the drop shape due to inertial effects, though small in magnitude, are found to retard the motion of the drop. The steady drop shapes are found to resemble oblate or prolate spheroids without fore and aft symmetry, with the direction of elongation of the drop depending on the value of the density ratio between the two phases. As in the case of a gas bubble, convection of energy is shown to retard the thermocapillary motion of a viscous drop, as the isotherms get wrapped around the front surface of the drop and effectively reduce the surface temperature gradient which drives the motion. The effect of inertia on the mobility of viscous drops is found to be weaker than that in the case of gas bubbles.

Thermocapillary migration of nondeformable drops

“…It is clear that the scaled migration velocities slightly increase with increasing values of Re 1 , which is similar to the results in Ref. 16, and these increasing rates of final speeds are higher for larger values of Ma 1 . The Ma 1 = 100 runs have not been calculated long enough to reach their final states, but we can roughly draw the similar conclusions.…”

“…Numerical simulations on the three-dimensional thermocapillary motion of deformable viscous drops were reported by Haj-Harir et al 16 They used the level-set method to catch the interface and found that the strong heat convection may retard the thermocapillary motion of the drop be- cause the isotherms get wrapped around the front surface of the drop and reduce the surface temperature gradient. Nas 17,18 calculated the thermocapillary interaction of multidrops with the front-tracking finite-difference methods.…”

Thermocapillary migration and interaction of drops: two non-merging drops in an aligned arrangement

“…Since the YGB theory was published, there have been many studies on this problem using theoretical, experimental and numerical tools (see Balasubramaniam & Chai 1987;Chen & Lee 1992;Treuner et al 1996;Haj-Hariri, Shi & Borhan 1997;Welch 1998;Wozniak et al 2001;Sun & Hu 2002Nas, Muradoglu & Tryggvason 2006;Borcia & Bestehorn 2007;Liu, Zhang & Valocchi 2012;Liu et al 2013). Most early findings have been summarized and discussed in the review book by ; see also Subramanian, Balasubramaniam & Wozniak (2002).…”

Large deformation of red blood cell ghosts in a simple shear flow

“…Front tracking requires the solution of Poisson's equation on the entire fluid grid at every time step, and is thus more costly to implement. Recently, HajHariri et al 28 have developed an adaptive grid refinement scheme in combination with a front tracking method to study thermocapillary motion of drops where the ratio of internal to external fluid viscosity varied from 10 −3 to 10 1 . The immersed boundary method has also been used to simulate the flow of suspensions of solid particles, 29 and the deposition of nondeforming particles on solid substrates in the simulation of biofilm processes.…”