  Thermocapillary migration of a planar non-deformable droplet in flow fields
with two uniform temperature gradients at moderate and large Marangoni numbers
is studied numerically by using the front-tracking method. It is observed that
the thermocapillary motion of planar droplets in the uniform temperature
gradients is steady at moderate Marangoni numbers, but unsteady at large
Marangoni numbers. The instantaneous migration velocity at a fixed migration
distance decreases with increasing Marangoni numbers. The simulation results of
the thermocapillary droplet migration at large Marangoni numbers are found in
qualitative agreement with those of experimental investigations. Moreover, the
results concerned with steady and unsteady migration processes are further
confirmed by comparing the variations of temperature fields inside and outside
the droplet. It is evident that at large Marangoni numbers the weak transport
of thermal energy from outside of the droplet into inside cannot satisfy the
condition of steady migration process, which implies that the advection around
the droplet is a more significant mechanism for heat transfer across/around the
droplet at large Ma numbers. Furthermore, from the condition of overall
steady-state energy balance in the flow domain, the thermal flux across its
surface is studied for a steady thermocapillary droplet migration in a flow
field with uniform temperature gradient. By using the asymptotic expansion
method, a nonconservative integral thermal flux across the surface is
identified in the steady thermocapillary droplet migration at large Marangoni
numbers. This nonconservative flux may well result from the invalid assumption
of quasi-steady state, which indicates that the thermocapillary droplet
migration at large Marangoni numbers cannot reach steady state and is thus a
unsteady process.
