*The Drag-Out Problem in Film Coating*. B. Jin and A. Acrivos. *Physics of Fluids*, 17:103603, 2005.

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classical coating flow problem of determining the asymptotic film thickness (and hence the load) on a flat plate being withdrawn vertically from an infinitely deep bath is examined via a numerical solution of the steady-state Navier-Stokes equations.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-->[infinity], 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[equivalent]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 alpha from the vertical. Here, the corresponding dimensionless flow rate qalpha[equivalent]q(cos alpha)1/2 depends on both Ca and alpha, and its maximum is found to increase monotonically with alpha and to become equal to 2/3 when alpha exceeds a critical angle alphac(~pi/4), where the plate is inclined midway to the horizontal with its coating surface on the topside

`@ARTICLE { jin_acrivos_05,`

AUTHOR = { B. Jin and A. Acrivos },

TITLE = { The Drag-Out Problem in Film Coating },

YEAR = { 2005 },

JOURNAL = { Physics of Fluids },

VOLUME = { 17 },

PAGES = { 103603 },

ABSTRACT = { 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classical coating flow problem of determining the asymptotic film thickness (and hence the load) on a flat plate being withdrawn vertically from an infinitely deep bath is examined via a numerical solution of the steady-state Navier-Stokes equations.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-->[infinity], 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[equivalent]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 alpha from the vertical. Here, the corresponding dimensionless flow rate qalpha[equivalent]q(cos alpha)1/2 depends on both Ca and alpha, and its maximum is found to increase monotonically with alpha and to become equal to 2/3 when alpha exceeds a critical angle alphac(~pi/4), where the plate is inclined midway to the horizontal with its coating surface on the topside },

}