# Terminal thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers

## TL;DR

This study investigates the steady-state assumptions in thermocapillary droplet migration at small Reynolds and large Marangoni numbers, revealing that steady solutions do not exist and the process remains unsteady.

## Contribution

It provides a detailed analysis of the momentum and energy balances, showing the nonexistence of steady temperature solutions under certain conditions.

## Key findings

- Total momentum is conserved at small Reynolds numbers.
- A nonconservative thermal flux prevents steady temperature solutions.
- The droplet migration process remains unsteady under studied conditions.

## Abstract

In this paper, the overall steady-state momentum and energy balances in the thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers are investigated to confirm the quasi-steady state assumption of the system. The droplet is assumed to have a slight axisymmetric deformation from a sphere shape. It is shown that under the quasi-steady state assumption, the total momentum of the thermocapillary droplet migration system at small Reynolds numbers is conservative. The general solution of the steady momentum equations can be determined with its parameters depending on the temperature fields. However, a nonconservative integral thermal flux across the interface for the steady thermocapillary migration of the droplet at small Reynolds numbers and large Marangoni numbers is identified. The nonconservative integral thermal flux indicates that no solutions of the temperature fields exist for the steady energy equations. The terminal thermocapillary migration of the droplet at small Reynolds numbers and large Marangoni numbers cannot reach a steady state and is thus in an unsteady process.