Slip-induced suppression of Marangoni film thickening in surfactant-retarded Landau–Levich–Bretherton flows

We report that the well-known Marangoni film thickening in surfactant-laden Landau–Levich–Bretherton coating flow can be completely suppressed by wall slip. The analysis is made by mainly looking at how the deposited film thickness varies with the capillary number $Ca$ ($\ll 1$) and the dimensionless slip length ${\it\Lambda}={\it\lambda}/R$ ($\ll 1$) in the presence of a trace amount of insoluble surfactant, where ${\it\lambda}$ is the slip length and $R$ is the radius of the meniscus. When slip effects are weak at sufficiently large $Ca$ (but still $\ll 1$) such that $Ca\gg {\it\Lambda}^{3/2}$, the film thickness can still vary as $Ca^{2/3}$ and be thickened by surfactant as if wall slip were absent. However, when slip effects become strong by lowering $Ca$ to $Ca\ll {\it\Lambda}^{3/2}$, the film, especially when surface diffusion of surfactant is negligible, does not get thinner according to the strong-slip quadratic law reported previously (Liao et al., Phys. Rev. Lett., vol. 111, 2013, 136001; Li et al., J. Fluid Mech., vol. 741, 2014, pp. 200–227). Instead, the film behaves as if both surfactant and wall slip were absent, precisely following the no-slip $2/3$ law without surfactant. Effects of surface diffusion are also examined, revealing three distinct regimes as $Ca$ is varied from small to large values: the strong-slip quadratic scaling without surfactant, the no-slip $2/3$ scaling without surfactant and the film thickening along the no-slip $2/3$ scaling with surfactant. An experiment is also suggested to test the above findings.