Thermocapillary migration of a planar droplet at moderate and large   Marangoni numbers
  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. 
Zuo-Bing Wu and Wen-Rui Hu
physics.flu-dyn physics.space-ph