Thermocapillary motion of a slender viscous droplet in a channel

We extend the previously developed low-capillary-number asymptotic theory of thermocapillary motion of a long bubble and a moderately viscous droplet in a channel [S. K. Wilson, "The effect of an axial temperature gradient on the steady motion of a large droplet in a tube," J. Eng. Math.29, 205 (1995)10.1007/BF00042854; A. Mazouchi and G. M. Homsy, "Thermocapillary migration of long bubbles in cylindrical capillary tubes," Phys. Fluids12, 542 (2000)10.1063/1.870260] toward droplets with an arbitrary viscosity. A generalized modified Landau-Levich-Bretherton equation, governing the thickness of the carrier liquid film entrained between the droplet and the channel wall in the transition region between constant thickness film and constant curvature cap, is solved numerically. The resulting droplet velocity is determined applying the mass balance and it is a function of two dimensionless parameters, the modified capillary number, δσ*, equal to the surface tension variance over a distance of channel half-width scaled with the mean surface tension, and the inner-to-outer liquid viscosity ratio, λ. It is found that the droplet speed decreases with the increase in droplet viscosity, as expected, while this retardation becomes more operative upon the increase in δσ*.