The equations describing the temporal evolution of a thin, Newtonian, viscous liquid layer are extended to include the effect of substrate curvature. It is demonstrated that, subject to the standard assumptions required for the validity of lubrication theory, the surface curvature is equivalent to an applied time-independent overpressure distribution. Within the mathematical model, a variety of substrate shapes, possessing both 'inside' and 'outside' corners, are shown to be equivalent. Starting with an initially uniform coating layer, the evolving coating profile is calculated for substrates with piecewise constant curvature. Ultimately, surface tension forces drive the solutions to stable minimum-energy configurations. For small time, the surface profile history, for a substrate with a single curvature discontinuity, is given as the self-similar solution to a linear fourth-order diffusive equation. Using a Fourier transform, the solution to the linear problem is found as a convergent infinite series. This Green's function generates the general solution to the linearized problem for arbitrary substrate shapes. Calculated solutions to the non-linear problem are suggestive of coating defects observed in industrial applications.

It is demonstrated that, for the slow advance of a viscous liquid onto a previously dry substrate, the well-known moving contact line paradox is alleviated for liquids exhibiting power-law shear-thinning behavior. In contrast to previous models that allow contact-line motion, it is no longer necessary to abandon the no-slip condition at the substrate in the vicinity of the contact point. While the stress is still unbounded at the contact point, the equations of motion are shown to be integrable. A three-constant Ellis viscosity model is employed that allows a low-shear Newtonian viscosity, and may thus be used to model essentially Newtonian flows where shear thinning only becomes important in the immediate vicinity of the contact point. Calculations are presented for the model problem of the progression of a uniform coating layer down a vertical substrate using the lubrication approximations. The relationship between viscous heating and shear-thinning rheology is also explored.

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Mathematical and numerical modeling of drying coating layers is of interest to both industrial and academic communities. Compositional changes that occur during the drying process make the implementation of practical and efficient numerical models rather difficult. In this paper we present a three-dimensional mathematical and numerical model based on the lubrication approximation for the flow of drying paint films on horizontal substrates. The paint is modeled as a multicomponent liquid with one nonvolatile and one volatile component, termed the “resin” and the “solvent” respectively. Our model includes the effects of surface tension and gravitational forces as well as surface tension gradient effects which arise due to solvent evaporation. The dependence of viscosity, diffusivity, and evaporation rate on resin concentration is also incorporated in the model. A closed-form linearized solution has been found for coating layers that are of almost uniform thickness. The numerical solution agrees closely with the linear solution in the appropriate limit. A model simulation demonstrates the effect of surface tension gradients due to compositional changes in a three-dimensional flow field, and we suggest methods by which these gradients may be used to obtain a more uniform final coating layer.

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