| Mathematical models are developed for problems in thin layer films. The lubrication approximation simplifies equations governing flow of viscous fluids in the thin layers typical in coating applications. Three specific problems are considered.; The first is flow of a thin viscous coating on a horizontal cylinder rotating about its axis. Such a layer drains to the underside due to gravity, but may be maintained by rotation and surface tension. The model developed includes the effects of gravity, cylinder rotation, and surface tension, and axial flow. Using two-dimensional numerical simulations, steady solutions are found, ranging from pendant droplets near the cylinder underside at slow speeds, to solutions which are nearly symmetric in a horizontal plane through the cylinder center at higher speeds. Three-dimensional simulations show the ridge of coating extending along the cylinder axis is unstable, leading to drop formation at low rotation rates. For large cylinders, break-up may result in drops forming, which drain toward the cylinder underside, forming fingers. Experiments verify the basic features of these three-dimensional solutions.; Next we consider the effect of surface tension gradients arising from contaminants on the surface of a two-component coating. We present a mathematical model for formation of the defects known as “craters”, attributed to surfactants, in a drying coating. Using numerical simulations, two candidate crater production mechanisms are evaluated: an initial release of concentrated surfactant, and a steady surfactant source. Effects of parameter changes, e.g., drying rate, surfactant diffusivity and pre-existing surfactant, are examined. Simulated craters are similar to those seen in practice.; Finally a model for flow of a two-component viscous mixture through a narrow cell, driven by an applied pressure difference, or by dragging of the upper and lower confining surfaces, is presented. For pressure-driven motion, the work required for a given volumetric flux is minimized by uniform mixtures, provided that the functional dependence of viscosity on concentration has a positive second derivative. For a particular cell in which the confining surfaces drag the fluid mixture, separated mixtures are energetically preferred. This may provide the basis for useful separation devices. |
