Mathematical models for coating processes

Abstract

The flow is considered of a Newtonian fluid, of viscosity η and surface tension T, in the narrow gap between a pair of rollers of radii R₁ and R₂, whose peripheral speeds are constant and equal to U₁ and U₂ respectively. The objective is to determine the coating thickness h₁^∞ on the upper roller as a function of the non-dimensional parameters H₀/R, ηU/T and U₁/U₂, where H₀ is the minimum gap thickness, U = ½(U₁ + U₂), and 2R⁻¹ = R₁⁻¹ + R₂⁻¹.

Using lubrication theory to provide an adequate description of the fluid flow, two mathematical models are derived whose essential difference lies in the specification of the boundary conditions. In the separation model it is assumed that the pressure distribution will terminate at a position which is both a stagnation point and a point of separation, whereas the Reynolds model incorporates the classical Reynolds conditions. In each case, theoretical predictions for the non-dimensional coating thickness, h₁^∞/H₀ as a function of U₁/U₂ are found to compare well with experiment. However, theory does suggest that the two models are applicable to different and complementary regions of parameter space, and hence together they may form a basis for further investigations into the various features of coating processes.