The drag-out problem in film coating

The classical coating flow problem of determining the asymptotic film thickness (and hence the load) on a flat plate being withdrawn vertically from an infinitely deep bath is examined via a numerical solution of the steady-state Navier-Stokes equations. Under creeping flow conditions, the dimensionless load q is computed as a function of the capillary number Ca and, for Ca<0.4, is found to agree with Wilson’s extension [J. Eng. Math. 16, 209 (1982)] of Levich’s well-known expression. On the other hand, for Ca→∞, q asymptotes to 0.582, well below the value of 2∕3 postulated by Deryagin and Levi [Film Coating Theory (Focal, London, 1964)]. For finite Reynolds numbers Re≡mCa3∕2, where m is a dimensionless number involving only the gravitational acceleration g and the properties of the fluid, q is found to remain essentially independent of m at a given Ca, but only up to a critical capillary number Ca*, dependent on m, beyond which our numerical scheme failed. Analogous results, but only for creeping flows, are presented for the case where the plate is inclined at an angle α from the vertical. Here, the corresponding dimensionless flow rate qα≡q(cosα)1∕2 depends on both Ca and α, and its maximum is found to increase monotonically with α and to become equal to 2∕3 when α exceeds a critical angle αc(∼π∕4), where the plate is inclined midway to the horizontal with its coating surface on the topside.