Steady thermocapillary droplet migration under thermal radiation with a uniform flux

Thermocapillary migration of a droplet under thermal radiation with a uniform flux is numerically investigated and theoretically analyzed. By using the front-tracking method, it is observed that thermocapillary droplet migration at small Reynolds numbers and moderate Marangoni numbers reaches a steady process. The steady migration velocity decreases as Marangoni number increases. The time-evolution behavior of temperature fields in the steady migration process is found to be a quadratic function, in which the linear rise of the steady state temperatures with the relative time is a main characteristics. The quadratic function behavior of the temperature fields is further used to derive the steady energy equations. From the steady momentum and energy equations, an analytical result at small Reynolds number and zero Marangoni number is determined by using the method of matched asymptotic expansions. The steady migration velocity decreases as Reynolds number increases, which is in qualitatively agreement with the numerical result.