Numerical simulation of a film coating flow at low capillary numbers

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 CaCa. 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 CaCa and Morton number mm). Moreover, the steady state is found only for a range of parameters, CaCa and mm, and wavy states can produce fluctuations which can reach 10% of the mean value of the film thickness.