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The coating dynamics of a drying paint film with a yield stress is studied. The liquid is modeled as a binary mixture with one volatile component, solvent, and one nonvolatile component, resin. When the solvent has a different surface tension than the resin, solvent evaporation can lead to the creation of surface tension gradients which can potentially overcome the yield stress and dramatically affect the flow history. Using the lubrication approximations to derive the flux of the liquid film parallel to the substrate, we find that the presence of the yield stress causes several distinct flow regimes. The total flux of each of these regimes is summed, and using the continuity equation we derive an evolution equation giving the height of the free surface as a function of the distance along the substrate and time. The resulting equations are discretized and solved numerically using finite differences. High order derivative is treated implicitly, allowing for large time steps and reducing the computational requirements. We find that the presence of a yield stress greatly affects the leveling behavior of the coating. Critical yield stresses exist that can cause maximal leveling of the coating film.

Spin coating is frequently used by the coating industry to achieve a very uniform final coating layer on a given substrate. Most of the research into this area has focused on flat substrates, but in this work we use scaling arguments and perturbation methods to derive the lubrication form of the equations governing the fluid motion of a thin liquid film on a curved, rotating, axisymmetric substrate. Though the substrate must be axisymmetric, the coating need not be. Though the slope of the substrate must be continuous, the curvature of the substrate need not be. One application for this work is the spin coating of food and beverage cans, most of which have a curved bottom due to structural reasons. Using an implicit finite difference scheme, we use our derivation to develop a numerical model to simulate the spin coating of the interior of a model soup can. We assume that the coating is initially uniform and model how centrifugal forces drive the coating outward past a series of axisymmetric undulations on the can bottom.

We consider the effect of a magnetic field generated by a current flowing in the axial direction, in a cylindrical conducting medium on the evolution of a thin magnetic liquid film coating the outside of the cylinder. We first derive the new governing equations when a body force, due to the interaction of the magnetic dipoles in the liquid with the magnetic field due to the current in the cylinder, is included with surface tension and gravitational forces. The equations are discretized and solved numerically using an alternating direction implicit algorithm. Simulations demonstrate that the transition from a uniform coating to the formation of undulations to a final configuration of distinct drops follows a similar evolution for a wide range of cylinder radii and magnetic field strengths. Magnetic forces generally oppose the effects of gravity, and consequently the drainage toward the bottom of the cylinder is slowed, the characteristic time for drop formation is delayed, and the final coating is not conf...

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.

Following the application of a liquid coating to a curved substrate, surface tension forces will act to redistribute the coating layer. The coating will thin at outside corners and thicken at inside corners as the free surface contracts to minimize the surface energy. If the coating is a multicomponent liquid with a volatile component, the dynamics of the thinning process may be quite complex. Compositional changes in the bulk liquid during drying and convection of surfactant may cause surface tension gradient or Marangoni effects. In addition there may be viscosity variations due to concentration changes and the liquid may exhibit shear-thinning rheology. A numerical model has been developed, based on the lubrication approximations, for predicting the time-evolution of the coating layer thickness of a complex liquid on a curved substrate. Substrate geometry is modeled as a time-independent overpressure distribution and the model includes such effects as evaporation, convection, and diffusion of solvent in the bulk liquid, and convection and diffusion of a soluble surfactant. For a given starting profile and substrate geometry, the temporal and spatial evolution of the free surface, bulk composition, surfactant concentration, surface tension, and layer-averaged viscosity are calculated until the drying process is complete. We show that convection of surfactant away from outside corners may slow the thinning in these regions. In addition, solvent evaporation may lead to Marangoni forces in the corner region, causing a “rebound” effect. Surface tension forces will initially displace liquid from corner regions, but the thinning will produce surface tension gradients which act to pull liquid back to the corner region. If the evaporation time scale is suitably matched to the time scale for flow induced by surface tension gradients, corner defects in the final dry coating layer can be substantially mitigated.

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.

The motion of the interface between two fluids in a Hele–Shaw cell for the case of a cell oriented with the plates vertical is considered. The bottom edge of the cell may be at any angle to the horizontal and only density differences between the two fluids drive the flow. A boundary integral technique is employed to numerically predict the motion of the interface and numerical simulations are compared with experimental results. Unlike pressure‐driven Hele–Shaw flow, where the simplified equations predict only qualitative features of the displacement profiles, here the agreement is quite good, in general. Theory predicts and experiment confirms that the displacement profiles are not a function of fluid viscosity.

When a paint film is applied to the underside of a horizontal substrate, instabilities will lead to the formation of pendant drops. For sufficiently thick coatings, the drops may even detach from the coating before the coating dries. But even for thin paint films, with sufficiently high evaporation rates and viscosity, drop formation will lead to coating irregularities with thick regions (corresponding to the center of the drop) adjacent to much thinner regions (corresponding to troughs between drops.) In this paper we present a mathematical and numerical model based on the lubrication approximations for the drying of paint films on the underside of horizontal substrates. The paint is modeled as a multicomponent liquid with one non-volatile and one volatile component, termed the ``resin'' and ``solvent'' respectively. Our model includes the effects of surface tension and gravitational forces, and we show that surface tension gradient effects due to solvent evaporation must be considered to correctly model the post-application flow of certain paints. Simulations are given for an array of parameters, and they show that surface tension gradients can dramatically affect the post-application behavior of a model paint film, generally causing the suppression and reversal in drop growth, consequently leading to a more even final coating layer.

Our recently‐published linear analysis [Schwartz et al., Langmuir 11, 3690 (1995)] demonstrated that an initially rippled thin layer of Newtonian liquid with uniformly distributed surfactant may level in unexpected ways. While the presence of surfactant will, in general, slow the rate of leveling compared to that of a perfectly clean system, there was shown to exist a realistic parameter range where increasing, rather than reducing, the amount of surfactant present will hasten leveling. Here, for the two‐dimensional problem, we investigate the importance of nonlinearity though numerical solution of (i) the unsteady lubrication form of the evolution equations with surfactant, and (ii) finite‐element solution of the exact governing equations for slow viscous flow. Confirmation of the linear results is demonstrated and quantitative discrepancy only appears for large‐amplitude and short‐wavelength ripples. Surface tension gradient driven flow explains the anomalies; for moderate surfactants, the surface quick...

The lubrication form of the equations governing the flow of a thin liquid film on a horizontal right circular cylinder is derived. The equations are discretized and solved numerically using an alternating-direction implicit algorithm. Simulations demonstrate that the transition from a uniform coating to a final configuration of distinct drops follows a similar evolution for a wide range of cylinder radii. Initially gravity-driven drainage from the top and sides of the cylinder dampens the formation of any axial disturbances; only when this drainage slows do longitudinal waves begin to develop along the bottom of the cylinder. These waves grow rapidly and a series of alternating primary and satellite drops form during the transition from a linear to a nonlinear wave growth regime. This is followed by a slow drainage between adjacent drops as the drop pattern approaches an equilibrium state where surface tension forces exactly balance gravitational forces in each discrete drop. For large cylinder radii, these drops are localized on the bottom of the cylinder, while, for sufficiently small cylinder radii, these drops may wrap around the entire circumference of the cylinder. Integral measures of the evolving coating profile, such as the total energy and viscous dissipation rate, clearly show these growth phases. The equilibrium shape of large-amplitude pendant drops and the maximum sustainable drop volume for various cylinders are also considered.

When a thin liquid film is applied to the surface of a horizontal cylinder, gravity will cause a drainage of liquid from the top and sides of the cylinder towards the cylinder bottom. If surfactant is present on the surface of the film, this will cause a convection of surfactant resulting in a higher concentration of surfactant on the cylinder bottom compared to the top and sides of the cylinder. The result is a surface tension gradient, which is equivalent to a surface shear stress, and acts to oppose the drainage of the coating layer due to gravity. For sufficiently small cylinders, this cannot only slow the drainage but reverse the flow, causing a net flux of liquid upward from the bottom of the cylinder towards the top of the cylinder. If this flux is sufficiently strong, a “collar” of liquid forms around the cylinder. In this paper, we develop a mathematical model, based on the lubrication approximations, of the gravitational, surface tension, and surface tension gradient forces, and their effects on the evolution of a thin liquid film coating a horizontal circular cylinder. Using finite differences and an alternating direction implicit technique, numerical simulations show that even for comparatively weak surfactants, surface tension gradient effects greatly affect the flow history and must be included to accurately model the evolution of the film. They cannot only slow the drainage of liquid towards a pendant drop on the bottom of the cylinder, but reverse the flux, resulting in a thicker coating on the top of the cylinder compared to the surfactant-free case. Results from the simulation are presented over a wide range of the dimensionless parameters which characterize the problem.

In this work, we consider the evolution of a single axisymmetric droplet as it grows from an initially uniform thin liquid film on the underside of a solid, horizontal substrate. We consider the effects of an insoluble surfactant on the free surface, the concentration of which changes due to convection and diffusion. Employing the lubrication approximations, we derive a linear solution valid in the initial stages of drop growth, and a full nonlinear solution, which we solve numerically using finite differences. Both a linear analysis and a numerical solution show that for sufficiently thin films, corresponding to low Bond numbers, diffusion of surfactant can effectively negate the effects of surfactant convection, and the drop evolves as if there is no surfactant on the free surface. For sufficiently thick films, corresponding to high Bond numbers, gravitational forces are stronger than surface tension gradient forces, and again the droplet evolves as if there is no surfactant present. For intermediate Bo...