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The drag-out problem in film coating

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10.1063/1.2079927

### 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classical coating flow problem of determining the asymptotic film thickness (and hence the load) on a flat plate being withdrawn vertically from an infinitely deep bath is examined via a numerical solution of the steady-state Navier-Stokes equations.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 is computed as a function of the capillary number and, for , 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 , asymptotes to 0.582, well below the value of postulated by Deryagin and Levi [Film Coating Theory (Focal, London, 1964)]. For finite Reynolds numbers, where is a dimensionless number involving only the gravitational acceleration and the properties of the fluid, is found to remain essentially independent of at a given , but only up to a critical capillary number , dependent on , 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 from the vertical. Here, the corresponding dimensionless flow rate depends on both and , and its maximum is found to increase monotonically with and to become equal to when exceeds a critical angle , where the plate is inclined midway to the horizontal with its coating surface on the topside.

© 2005 American Institute of Physics

Received 05 August 2005
Accepted 18 August 2005
Published online 17 October 2005

Acknowledgments: During the course of this work, two of the authors (B.J. and A.A.) were supported in part by the Engineering Research Program of the Office of Basic Energy Sciences at the Department of Energy under Grant No. DE-FG02-03ER46068; One of the authors (A.M.) acknowledges support by a Heisenberg scholarship (DFG MU 1626/3-1) and the DFG research center Matheon, Berlin. Two of the authors (B.J. and A.A.) are also grateful to Dr. Steven Weinstein of the Kodak Company and Professor E. John Hinch of Cambridge University for their constructive comments on an earlier version of the paper.

Article outline:

I. INTRODUCTION

II. VERTICAL WITHDRAWAL WITHIN THE CREEPING FLOW REGIME (Re=0 OR,EQUIVALENTLY,*m*=0)

III. VERTICAL WITHDRAWAL BEYOND THE CREEPING FLOW REGIME ( OR, EQUIVALENTLY, *m*>0)

IV. INCLINED WITHDRAWAL FOR CREEPING FLOWS ( OR, ALTERNATIVELY, )

V. CONCLUSIONS

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2005-10-17

2014-04-23

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