 The Marangoni number for a simple liquid of viscosity formula_2 with a surface tension change formula_3 over a distance formula_4 parallel to the surface, can be estimated as follows.Note that we assume that formula_4 is the only length scale in the problem, which in practice implies that the liquid be at least formula_4 deep.The transport rate is usually estimated using the equations of Stokes flow, where the fluid velocity is obtained by equating the stress gradient to the viscous dissipation.