by Peter Llewellyn Evans
A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering
Fall 2000
Coating is a common operation in manufacturing, in which defects may be expensive. For the thin layers and viscous fluids typically encountered in coating applications, the lubrication approximation may be used to simplify the governing equations of fluid motion. The resulting partial differential equations may then be solved numerically. Numerical simulation is a powerful tool for investigating coating defects. Here three specific problems are considered.
First we consider flow of a thin viscous liquid on a horizontal cylinder rotating about its axis. A liquid layer on such a cylinder drains to the underside and drips off under the influence of gravity, but may be maintained by the effects of rotation and surface tension. The model developed includes the effects of gravity, cylinder rotation, and surface tension, and flow in the axial direction. Numerical simulations are used to model the coating in one and two dimensions.
In one dimension, we find a family of steady solutions exists. These range from a pendant droplet, hanging near the cylinder underside at very slow speeds, to solutions which are nearly symmetric in a horizontal plane through the cylinder center at higher speeds.
Using two-dimensional simulations, we find that axial flow is initially minor. The coating forms a ridge extending along the cylinder axis. An axial instability develops, causing the ridge to break up, and leading to drop formation at low rotation rates. For large cylinders, the break-up may result in drops forming, which accumulate mass as they drain toward the cylinder underside, forming fingers. Experiments verify the basic features of these two-dimensional solutions.
Next we consider the effect of surface tension gradients arising from a surfactant on the surface of a two-component coating. Surfactants are believed to cause defects known as ``craters'' in coatings. We present a mathematical model for formation of such defects. The model coating dries as one component evaporates. One-dimensional and axisymmetric numerical simulations using the model are performed. Two candidate crater production mechanisms are evaluated: an initial release of concentrated surfactant, and a steady surfactant source. The effects of changes in properties are examined. The model produces craters with features similar to those seen in practice.
Drying rate has a large influence on crater diameter and depth, by limiting the extent to which flow may occur within a given time. Reduction of the paint viscosity increase during drying causes increased flow rates, leading to larger craters. A pre-existing layer of surfactant on the paint surface sharply reduces the extent of cratering. Surfactant diffusion also tends to reduce the severity of cratering by alleviating surface tension gradients. In some cases, a simplified form of the drying model may be used to quickly approximate the results of the full model.
Finally the tendency of a two-component viscous mixture to separate when it is forced through a narrow cell is considered. A model for the flow of such a mixture, which allows for forcing by an applied pressure difference, or by dragging of the upper and lower confining surfaces is presented. For motion driven by an applied pressure difference, we find that the work required for flow of a given volumetric flux is a minimum when the two components are distributed uniformly everywhere. This result requires 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, we demonstrate that a uniform mixture does not minimize the required force, indicating that a separated mixture is energetically preferred. This may provide the basis for useful separation devices.