Steady thermocapillary migration of a droplet in a uniform temperature gradient combined with a radiation energy source at large Marangoni numbers

Steady thermocapillary droplet migration in a uniform temperature gradient combined with a radiation energy source at large Reynolds and Marangoni numbers is studied. To reach a terminal quasisteady process, the magnitude of the radiation energy source is required to preserve the conservative integral thermal flux across the surface. Under a quasisteady state assumption, an analytical result for the steady thermocapillary migration of a droplet at large Reynolds and Marangoni numbers is derived by using the method of matched asymptotic expansions. It is shown that the thermocapillary droplet migration speed increases as the Marangoni number increases, while a radiation energy source with a sine square dependence is provided.

Figure 1

A schematic diagram of thermocapillary droplet migration under the combined actions of a temperature gradient $G$ and a radiation energy source $S=\mathrm{\Omega }{sin}^2\theta$ in an axisymmetric spherical coordinate system moving with a droplet velocity $V_{\infty }$.

Figure 2

A schematic diagram of potential flows and boundary layer flows of thermocapillary droplet migration at large Re numbers. Solid line: The interface of the droplet. Dashed/long-dashed lines: The interface between potential flow (white/green zone) and boundary layer flow (blue/yellow zone) in the continuous flow/within the droplet. Dashed-dotted lines: Streamlines of the potential flows inside and outside the droplet.

Figure 3

Function $g_0$ vs ${\eta }^{\prime }$ determined from Eq. (51).