D. E. Weidner

Contact‐line motion of shear‐thinning liquids

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.

Modeling of coating flows on curved surfaces

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 Greens 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.

Anomalous behavior during leveling of thin coating layers with surfactant

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...

An experimental and numerical investigation of buoyancy‐driven two‐phase displacement

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.

Analysis of the flow of a thin liquid film on the surface of a rotating, curved, axisymmetric substrate

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.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...

Numerical modeling of the spray coating of spinning bodies

We derive the lubrication form of the fluid mechanical equations governing the motion of a thin liquid film on an arbitrarily curved, rotating, axisymmetric substrate. The resulting equations are discretized and then solved numerically using an efficient implicit finite difference algorithm. Non-Newtonian effects are included by incorporating an Ellis viscosity model. The primary application for this work is to model the spin coating of the interior of two-piece metal beverage cans, and we consider this problem in some depth. Specifically, we show how adjusting several parameters can eliminate one possible defect in the spin coating process: the tendency for droplets to detach from the substrate when the can is spun at high rotation rates.