# Terminal states of thermocapillary migration of a planar droplet at moderate and large Marangoni numbers

## TL;DR

This study analyzes the terminal states of thermocapillary droplet migration at various Marangoni numbers using analytical and numerical methods, revealing the dominant physical mechanisms and surface tension effects.

## Contribution

It provides a detailed analysis of the terminal states and physical mechanisms of thermocapillary droplet migration at different Marangoni numbers, combining analytical and numerical approaches.

## Key findings

- Steady/unsteady terminal states are identified at different Marangoni numbers.
- Surface tension effects vary between the top and bottom surfaces of the droplet.
- Thermal convection dominates over conduction at large Marangoni numbers.

## Abstract

In this paper, thermocapillary migration of a planar droplet at moderate and large Marangoni numbers is investigated analytically and numerically. By using the dimension-analysis method, the thermal diffusion time scale is determined as the controlling one of the thermocapillary droplet migration system. During this time, the whole thermocapillary migration process is fully developed. By using the front-tracking method, the steady/unsteady states as the terminal ones at moderate/large Marangoni numbers are captured in a longer time scale than the thermal diffusion time scale. In the terminal states, the instantaneous velocity fields in the unsteady migration process at large Marangoni numbers have the forms of the steady ones at moderate Marangoni numbers. However, in view of the former instantaneous temperature fields, the surface tension of the top surface of the droplet gradually becomes the main component of the driving force on the droplet after the inflection point appears. It is different from that the surface tension of the bottom surface of the droplet is the main component of the driving force on the droplet for the latter ones. The physical mechanism of thermocapillary droplet migration can be described as the significance of the thermal convection around the droplet is higher than/just as the thermal conduction across the droplet at large/moderate Marangoni numbers.