Drift instability in the motion of a fluid droplet with a chemically   reactive surface driven by Marangoni flow
  We theoretically derive the amplitude equations for a self-propelled droplet driven by Marangoni flow. As advective flow driven by surface tension gradient is enhanced, the stationary state becomes unstable and the droplet starts to move. The velocity of the droplet is determined from a cubic nonlinear term in the amplitude equations. The obtained critical point and the characteristic velocity are well supported by numerical simulations. 
Natsuhiko Yoshinaga, Ken H. Nagai, Yutaka Sumino, Hiroyuki Kitahata
cond-mat.soft