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When a flat plate is withdrawn from a liquid bath, a liquid film will adhere to it. Knowledge of the thickness of this film and the amount of liquid withdrawn (the flux) are important to many practical applications, some examples of which are coating of photographic films (2), metal coating (hot tinning, enamelling, etc.), and lubrication of moving machine parts ( 1 ) . Another example requiring the knowledge of fluid dynamics of liquid films is the drainage of paints after deposition on a vertical support during painting.In many cases, the fluids involved in drainage and withdrawal operations are non-Newtonian. Although the equivalent Newtonian case has been studied extensively, both theoretically (2, 10, 16) and experimentally (2, 12, 1 5 ) , the results cannot, in general, be applied to nonNewtonian fluids. This article presents the results of a study for non-Newtonian fluids. Both drainage and withdrawal cases are considered theoretically. Experiments are reported for the withdrawal case only.For this analysis, the fluids were assumed to be simple non-Newtonian fluids without elastic properties or memory. In the drainage case, the fluid used in the theoretical description was the three-constant Ellis model. For the withdrawal case, the power law fluid was employed. Although the influence of elastic properties was not investigated theoretically, a viscoelastic fluid [ carboxymethyl cellulose (CMC) solution] was tested. This test was made to estimate deviations both from the simple theory and from the experiments with inelastic fluids, such as aqueous Carbopol solutions. ASSUMPTIONS

A theory of the amount of liquid entrained by cylinders upon withdrawal from liquid baths is derived for a wide range of cylinder radii. The theory is based on matching curvatures for static and dynamic menisci. Predicted values are expressed as the effect of the dimensionless , wire radius (Goucher number) and dimensionless withdrawal speed (capillary number) on the dimensionless flux.The theory was verified experimentally for all wire radii by removing short cylinders from oily fluids and with other information. The fluids used included kerosene, mineral oil, motor oil, and glycerine, with viscosities from 2 to 500 centipoise; Goucher numbers ranged from 0.05 to 1.2. Deviations, which were noted at high capillary numbers where velocity gradients become' appreciable, indicated that the theory is a plug flow or low speed theory. Also discussed are differences found with water and the conditions under which films coalesced into droplets.The first known general description of cylinder withdrawal has only recently been published (6). Conditions and restrictions on the problem are given in that paper, but in the interest of clarity are summarized here.The problem of concern is the vertical constant-speed withdrawal of cylinders from uiescent baths of laminar Navier-Stokes fluids. It is alsoenited to cases where the liquid-gas interface in the meniscus is free of waves, ripples, or other instabilities. These conditions are not very restrictive; many of the important coating, pickling, and lubrication applications are described by this motion. Figure 1 is a sketch of the withdrawal process from a free surface GH. When a wire of radius R is withdrawn at a velocity uw from a wetting liquid having constant fluid properties, the entrained film has a constant external radius so at a point A some distance above the liquid meniscus.The problem is the determination of the quantity of the uid entrained per unit time Q, which is given by the 7 re ation where u is the vertical velocity of the fluid element having a radial position r. Here the flu Q is a function of the film radius so and thereby a function of withdrawal speed, c linder radii, gravity, and fluid properties. These purposes (6) by two groups: the dimensionless witdrawal speed Nca and the dimensionless wire radius N G~.By defining a dimensionless flux 3, the problem can be restated as one of predicting the functional relationship denoted by Equation (2). k indepen CY ent variables are best characterized for desiHere T is related to Q by dehing a flux radius s a n d from it the dimensionless flux. ThusDavid White is with Cambxidge University Cambrid e. England. John A. Tallmadge is with Drexel Institute, Phhadelphia, f'ennsylvania. This flux radius xis that which would result from solidifying the film; it is also the thickness of a hypothetical film in which the velocity profile is uniform and the flux is identical to the actual film. In practice S i s often almost as large as so. LITERATUREThe three flow regimes for c linder withdrawal (6) that is, where Nco f 0.04;...

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