Experimental study of substrate roughness and surfactant effects on the Landau-Levich law

In this work we present an experimental study of deviations from the classical Landau-Levich law in the problem of dip coating. Among the examined causes leading to deviations are the nature of the liquid-gas and liquid-solid interfaces. The thickness of the coating film created by withdrawal of a plate from a bath was measured gravimetrically over a wide range of capillary numbers for both smooth and well-characterized rough substrates, and for clean and surfactant interface cases. In view of the dependence of the lifetime of a film on the type of liquid and substrate, and liquid-gas and liquid-solid interfaces, we characterized the range of measurability of the film thickness in the parameter space defined by the withdrawal capillary number, the surfactant concentration, and substrate roughness size. We then study experimentally the effect of a film thickening due to the presence of surfactants. Our recent theory based on a purely hydrodynamic role of the surface active substance suggests that there is a sorption-controlled coating regime in which Marangoni effects should lead to film thinning. However, our experiments conducted in this regime demonstrate film thickening, calling into question the conventional wisdom, which is that Marangoni stresses (as accounted by the conventional interfacial boundary conditions) lead to film thickening. Next we examine the effect of well-characterized substrate roughness on the coated film thickness, which also reveals its influence on wetting-related processes and an effective boundary condition at the wall. In particular, it is found that roughness results in a significant thickening of the film relative to that on a smooth substrate and a different power of capillary number than the classical Landau-Levich law

Dip coating in the presence of a substrate-liquid interaction potential

In this work we investigate theoretically the Landau-Levich problem of dip coating in the presence of a strong interaction potential normal to the substrate. This study is motivated by dip coating at very low capillary numbers when the deposited film thickness is less than 1 µm and such interaction forces become important. The objective of this work is to demonstrate that in the presence of an extra body force the solution procedure differs significantly from the classical one and leads to substantial deviations from the Landau-Levich law for the entrained film thickness. In particular, attractive potentials produce film thickening and the resulting film thickness is independent of speed to lowest order. Repulsive potentials bring about more complicated behavior and lead either to films whose thickness is also independent of speed, or to a modification of the leading order constant in the classical Ca^(2/3) law. Demonstration of these effects is given for a model potential. The analysis is generally applicable to many physical situations when there is an interaction between a coating liquid and a substrate, e.g., dip coating of ferromagnetic liquids on magnetic substrates, or dip coating of liquids carrying charges

Surfactant effects in the Landau–Levich problem

In this work we study the classical Landau–Levich problem of dip-coating. While in the clean interface case and in the limit of low capillary numbers it admits an asymptotic solution, its full study has not been conducted. With the help of an efficient numerical algorithm, based on a boundary-integral formulation and the appropriate set of interfacial and inflow boundary conditions, we first study the film thickness behaviour for a clean interface problem. Next, the same algorithm allows us to investigate the response of this system to the presence of soluble surface active matter, which leads to clarification of its role in the flow dynamics. The main conclusion is that pure hydrodynamical modelling of surfactant effects predicts film thinning and therefore is not sufficient to explain the film thickening observed in many experiments

On upstream influence in supersonic flows

The general problem of propagation of three-dimensional disturbances in viscous supersonic flows is considered in the framework of characteristic analysis. Unlike previous results for linear disturbances we deduce a condition determining nonlinear characteristic surfaces which is exact and therefore allows both qualitative and quantitative studies of the speed of propagation as a function of various physical phenomena. These include negative and adverse pressure gradients, and effects of wall cooling and suction–blowing, which are studied in this work as an illustration of the general theory

Fluids in art: "The water's language was a wondrous one, some narrative on a recurrent subject ..."

Artists spent a great deal of time studying anatomy for precise rendering of the human body as well as light, shadows, and perspective for convincing representation of the three-dimensional world. But in many paintings, they also had to depict fluids in their static and dynamic states -- a subject they could not study thoroughly, which led to a number of glaring misrepresentations or deliberate deceits

Nonlinear Schr\"{o}dinger equation in cylindrical coordinates

Nonlinear Schr\"{o}dinger equation was originally derived in nonlinear optics as a model for beam propagation, which naturally requires its application in cylindrical coordinates. However, the derivation was done in the Cartesian coordinates with the Laplacian $\Delta_{\perp} = \partial_{x}^{2} + \partial_{y}^{2}$ transverse to the beam $z$-direction tacitly assumed to be covariant. As we show, first, with a simple example and, next, with a systematic derivation in cylindrical coordinates, $\Delta_{\perp} = \partial_{r}^{2} + \frac{1}{r} \partial_{r}$ must be amended with a potential $V(r)=-\frac{1}{r^{2}}$, which leads to a Gross-Pitaevskii equation instead. Hence, the beam dynamics and collapse must be revisited