Effects of Marangoni numbers on thermocapillary drop migration: constant for quasi-steady state?

TL;DR

This paper investigates how Marangoni numbers influence thermocapillary drop migration, revealing that steady-state assumptions hold at small Marangoni numbers but not at large ones, where migration becomes unsteady.

Contribution

It provides a thermal flux analysis across the drop surface, distinguishing steady and unsteady migration regimes based on Marangoni numbers.

Findings

Conservative thermal flux at small Marangoni numbers supports quasi-steady state.

Nonconservative flux at large Marangoni numbers indicates unsteady migration.

Steady state is invalid at high Marangoni (Reynolds) numbers.

Abstract

The overall {\it steady}-state energy balance with two phases in a flow domain requires that the change in energy of the domain is equal to the difference between the total energy entering the domain and that leaving the domain. From the condition, the integral thermal flux across the surface is studied for a {\it steady} thermocapillary drop migration in a flow field with uniform temperature gradient at small and large Marangoni (Reynolds) numbers. The drop is assumed to have only a slight axisymmetric deformation from a sphere. It is identified that a conservative/nonconservative integral thermal flux across the surface in the {\it steady} thermocapillary drop migration at small/large Marangoni (Reynolds) numbers. The conservative flux confirms the assumption of {\it quasi-steady} state in the thermocapillary drop migration at small Marangoni (Reynolds) numbers. The nonconservative flux may well result from the invalid assumption of {\it quasi-steady} state, which indicates that the thermocapillary drop migration at large Marangoni (Reynolds) numbers cannot reach {\it steady} state and is thus a {\it unsteady} process.