Kenneth J. Ruschak

Limiting flow in a pre-metered coating device

Abstract A limiting case of flow in a pre-metered coating device is analyzed using a singular perturbation method. The ability of this device to coat a range of film thicknesses at the same coating speed is traced to the freedom of the two free surfaces to adopt any curvature within certain limits. The observed limitations on operating conditions for this coater are explained by the failure of the desired steady-state to exist for all values of the operating parameters. The parameter bounds within which the device can coat a uniform liquid layer are predicted quantitatively. Some analyses of dip coating are reviewed in the light of the present work.

The Effect of Applied Pressure on the Shape of a Two-Dimensional Liquid Curtain Falling under the Influence of Gravity

The shape of a two-dimensional liquid curtain issuing from a slot and falling under the influence of gravity is predicted theoretically and verified experimentally for cases where a pressure is applied to the curtain. A set of approximate equations is derived which governs the location of the curtain for a liquid having surface tension σ, density ρ, volumetric flow per unit width Q , and local free-fall velocity V . These equations possess a singularity at the point where the local Weber number, We = ρ QV /2σ, is equal to 1. Despite the fact that previous work on the stability of two-dimensional curtains shows that curtains having locations where We It is found that the singularity can be eliminated from the governing equations if the curtain assumes a definite direction as it leaves the slot. By contrast, if the curtain leaves the slot such that We > 1, there is no such restriction, and experimentally it is found that the curtain leaves parallel to the slot walls. The theoretical predictions of the curtain shapes are in agreement with those measured experimentally for all Weber numbers investigated.

Boundary conditions at a liquid/air interface in lubrication flows

A difficulty in applying the lubrication approximation to flows where a liquid/air interface forms lies in supplying boundary conditions at the point of formation of the interface that are consistent with the lubrication approximation. The method of matched asymptotic expansions is applied to the flow between partially submerged, counter-rotating rollers, a representative problem from this class, and the lubrication approximation is found to generate the first term of an outer expansion of the problem solution. The first term of an inner expansion describes the two-dimensional flow in the vicinity of the interface, and approximate results are found by the finite-element method. Matching between the inner and outer solutions determines boundary conditions on the pressure and the pressure gradient at the point of formation of the interface which allow the solution to the outer, lubrication flow to be completed.

Wetting: Static and Dynamic Contact Lines

Wetting is basic to coating. Initially air contacts the solid, and during coating the liquid displaces the air from the moving solid surface so that none is visible in the coated film. Thus, coating is a process of dynamic wetting. For uniform coating, the wetting line must remain straight and advance steadily. At sufficiently high speeds, however, the wetting line becomes segmented and unsteady as a thin air film forms between the solid and liquid. The air film disrupts the uniformity of the coated film, and often air bubbles appear in the coating. Dynamic wetting failure limits coating speed.

A three-dimensional linear stability analysis for two-dimensional free boundary flows by the finite-element method

Abstract The stability of two-dimensional, steady flows of Newtonian liquid with free boundaries affected by surface tension to small, three-dimensional disturbances is calculated by the finite-element method. The only simplification made is the neglect of the time partial derivative in each component of the Navier-Stokes equation, which leads to a major reduction in the order of the resulting matrix eigenvalue problem. The disturbances introduced vary sinusoidally in the third dimension and exponentially in time, and as a result the two-dimensional finite-element grid used for the base flow calculations can be used also for the stability calculations. Stability predictions for a stationary layer and for air displacing liquid in a Hele-Shaw cell show that the method can be effective.

Dip coating on a planar non-vertical substrate in the limit of negligible surface tension

Abstract The problem of dip coating of a planar non-vertical substrate is considered for negligible surface tension effects. As in the problem of vertical withdrawal (Cerro and Scriven, J. Fluid Mech. 208 (1980) 40), a singularity arises in the approximate steady-state equation governing the shape of the air–liquid interface; the ultimate thickness of the entrained film on the substrate follows directly from the elimination of this singularity. The removable singularity corresponds to a critical point that divides regions where waves propagate upstream and downstream (subcritical flow) and waves only propagate downstream (supercritical flow). These waves move at the speed of characteristics of the linearized time-dependent hyperbolic equation governing the film flow. The range of flows and film thicknesses that may be achieved by dip coating in an inclined substrate configuration are examined; these film thicknesses are self-metered, in that it they are determined by the substrate speed, substrate angle, and fluid properties. We also examine the parameter space to find flow conditions where the final uniform thickness of the film is not determined by a critical point. In these cases, the film thickness is set by some other means, such as in premetered die coating where both the flow rate and substrate speed are imposed. Examination of the asymptotic behavior of the film downstream shows that some of the possible uniform film thicknesses are not attainable. While there may be two possible uniform film thickness solutions for an imposed volumetric flow rate, there can be at most one supercritical and one subcritical flow solution.

Thin-Film Flow at Moderate Reynolds Number

Viscous, laminar, gravitationally-driven flow of a thin film over a round-crested weir is analyzed for moderate Reynolds numbers. A previous analysis of this flow utilized a momentum integral approach with a semiparabolic velocity profile to obtain an equation for the film thickness (Ruschak, K. J., and Weinstein, S. J.). In this work, a viscous boundary layer is introduced in the manner of Haugen. As in the previous analysis of Ruschak and Weinstein, the approximate equations have a critical point that provides an internal boundary condition for a bounded solution

Viscous Thin-Film Flow Over a Round-Crested Weir

Gravity-driven flow over a round-crested weir is analyzed for viscous flow. An equation for the entire flow profile is obtained by simplifying the equations for slowly varying film thickness, assuming a velocity profile, and integrating across the film. Solution of the resulting first order, ordinary differential equation requires a boundary condition generated at a critical point of the flow, beyond which waves cannot propagate upstream. Results for the relationship between head and flow rate are consolidated on a dimensionless master curve represented by an empirical equation.

Developing flow of a power-law liquid film on an inclined plane

Developing flow of a liquid film along a stationary inclined wall is analyzed for a power-law constitutive equation. For films with appreciable inertia and therefore small interfacial slopes, the boundary-layer approximation may be used. The boundary-layer equations are solved numerically through the von Mises transformation that gives a partial differential equation over a semi-infinite strip and approximately by the method of von Karman and Polhausen that gives an ordinary differential equation for the film thickness, called a film equation. Film equations derived from self-similar velocity profiles fail when the film thickens and the flow undergoes a supercritical to subcritical transition; a nonremovable singularity arises at the critical point, the location of the flow transition. A film equation is developed that accommodates this transition. Predictions exhibit a standing wave where hydrostatic pressure becomes important and opposes inertia. This thickening effect is accentuated for small angles of inclination at moderate Reynolds numbers. In the limit of small film thickness in which gravitational effects are negligible, the thickness profile is nonlinear in agreement with an independent and new similarity solution. This result contrasts with the established linear thickness profile for a Newtonian liquid. The circumstances in which the film equation gives results close to the full boundary layer equation are identified.

On the mathematical structure of thin film equations containing a critical point

Abstract In an accelerating thin liquid film, it is possible for a location to exist in which the flow field passes from subcritical flow, in which waves affect both upstream and downstream locations, to supercritical flow, in which all waves propagate downstream. At that location, referred to as the critical point, the steady-state equation governing the shape of the film exhibits a removable singularity that sets an internal boundary condition on the flow. This boundary condition provides essential mathematical structure as well as physical insight into the flow problem. In previous papers, Cerro and Scriven (1980) and Kheshgi et al. (1992) postulate that neglected higher-order terms may become important and induce an internal boundary layer in the vicinity of a critical point. In this paper, we reconsider the rapid dip coating analysis of Cerro and Scriven (1980) and demonstrate that no such boundary layer exists through the use of a regular asymptotic expansion. Thus, higher-order terms, when sensibly small based on scaling arguments, may be formally neglected in all regions of the flow, and the mathematical simplifications afforded by the film equation containing the critical point are uniformly valid. Although focused on a specific model problem here, it is likely that the mathematical structure uncovered may be generalized to other thin film flows with critical points.