Moving towards the cold region or the hot region? Thermocapillary migration of a droplet attached on a horizontal substrate

Abstract

We study computationally thermocapillary migration of a two-dimensional droplet attached on a horizontal substrate with a constant temperature gradient. A level-set approach is employed to track the droplet interface and a Navier slip boundary condition is imposed to alleviate a stress singularity at the moving contact lines. The present numerical model allows us to consider droplets with large contact angles and to take into account effects of the fluid outside the droplet, both have not been well studied so far. In the limits of a zero contact angle hysteresis and a small viscosity ratio of the fluids outside and inside the droplet (μout/μin ⩽ 0.1), we find the droplet finally migrates towards the cold region, and both the steady migration speed and the velocity field inside the droplet obtained from numerical simulation agree very well with the lubrication theory of Ford and Nadim [“Thermocapillary migration of an attached drop on a solid surface,” Phys. Fluids 6, 3183–3185 (1994)] when the contact angles are small (⩽45°). Beyond this regime, increasing the contact angles leads to increased deviations between numerical simulation and the lubrication theory, and the steady migration speed of the droplet towards the cold side decreases with the contact angles. The simulation results show that the droplet could fall in a motionless regime when its contact angles are around 100° even without any contact angle hysteresis. It is very interesting to find that a droplet with even larger contact angles migrates towards the hot region in a steady speed. We also find the transition of the migration direction of a droplet could strongly depend on the viscosity ratio. With increasing the viscosity of the external fluid, the transition could happen at much smaller values of contact angles. We summarize the results in a phase diagram and discuss the effects of other system parameters, including the contact angle hysteresis, the effective Marangoni number, the Prandtl number, and the slip length, on thermocapillary migration of the droplet.

Validation of the S‐CLSVOF method with the density‐scaled balanced continuum surface force model in multiphase systems coupled with thermocapillary flows

“…Thermocapillary flows can be computed by introducing the tangential surface tension force term into the CSF model in simulating multiphase systems . For instance, multiphase flow techniques (VOF, front tracking, and phase field) related to thermocapillary flow investigations include thermocapillary flow propulsion of bubbles (drops) , flow instability , and interface deformation by a thermocapillary flow in a cavity . Details can be found in Refs.…”

Non-isothermal droplet spreading/dewetting and its reversal

“…We have confirmed that the phase diagram is robust when 0.5 Ma 40, and that the contact line moves in an opposite direction when using a cold instead of a hot wall. This flow diagram is surprisingly similar to that for the direction of thermocapillary translation of an entire droplet on a non-uniformly heated/cooled substrate (Sui 2014). In the subsequent subsections we investigate the origin of the spreading/dewetting reversal, not addressed in prior work.…”

“…An explicit test from some asymptotic analysis is not available for the problem simulated though, due to differences in boundary conditions used in lubrication theory and the contact-line model in the simulations. A direct validation test for a non-isothermal system with contact lines has been conducted for another system, a cylindrical droplet on a substrate that is kept at a non-uniform temperature, see Sui (2014).…”

Atomic-scale thermocapillary flow in focused ion beam milling

“…An alternative approach would be to anticipate that the flow is indeed Marangoni and thus use the Marangoni velocity scale V M . [50][51][52]16,53 Here, it is most simply defined as…”