Fig. 3: Marangoni stresses and their dependence on capillary number in thin liquid films.
From: Mapping out the interplay between surfactant induced forces in thin liquid films
a Schematic of Van der Waals (ΠVdW) and electrostatic (Πelec) normal disjoining pressures, and Marangoni tangential stresses (σMa) acting on a draining thin liquid film. Blue arrows indicate the drainage flow of the bulk fluid (light blue), while surfactants are shown at the air-liquid interface. b Schematic of the parameters used in the Marangoni local stress relation Eq. (13), in the CBF limit. The surface velocity is denoted as us. c Maximum Marangoni stress (σMax-Ma) as a function of the Capillary number for the NBD-PC (red) and NBD-PG (blue). d Non-dimensionalized values versus Ca. e, f Experimental data (red) and analytical relation predictions based on the right term of Eq. (13) (green) for NBD-PC (e) and NBD-PG f), respectively. In (e, f), the results are rescaled relative to the value at ΔP = 50 Pa, using σMax-Ma(ΔP) − σMax-Ma(ΔP = 50 Pa). Error bars represent the 95% confidence interval, calculated using Student’s t-distribution and appropriate error propagation for combined data sets. The shaded areas in (e, f) connect consecutive error-bar extrema and serve as a visual guide for comparing experimental data with the analytical relationship.
