Numerical simulation of a film coating flow at low capillary numbers

Abstract

The drag-out problem in film coating has long been the subject of academic study and the question is of great interest for industrial processes. Landau’s theoretical analysis [Landau L, Levich B. Dragging of a liquid by moving a plate. Acta Physicochim URSS 1942;17:42–54] provides a formula which permits the prediction of the final film thickness but only for a steady flow at small capillary numbers Ca . Unfortunately, certain more recent experimental results [Kizito J, Kamotani Y, Ostrach S. Experimental free coating flows at high capillary and Reynolds number. Exp Fluids 1999;27:235–43] have shown the formula cannot be used to predict the mean value of the thickness for several regimes at low capillary numbers. The aim of this paper is to develop a reliable numerical code to correctly predict the hydrodynamics field for this configuration. This numerical simulation requires an unsteady Navier–Stokes code using the ALE formulation and a semi-implicit front tracking method for the moving free surface. The code, developed here, uses the finite element environment Freefem++ and provides quantitative results which fit the experimental data. It also shows that the problem requires the description of two dimensionless parameters (capillary number Ca and Morton number m ). Moreover, the steady state is found only for a range of parameters, Ca and m , and wavy states can produce fluctuations which can reach 10% of the mean value of the film thickness.