DIP COATING AT LARGE CAPILLARY NUMBERS AN INITIAL VALUE PROBLEM DARDO MARQUES, VINCENTE COSTANZAt and RAMON L CERRO$ Institute de Desarrollo Tecnologlco para la Industna Quumca (INTEC), CONICET, Uruversldad del LItoral, Santiago de1 Ester0 2654, 3000, Santa Fe, Argentma (Recewed 14 July 1975, accepted 7 Apnl 1977) Abstract-A computatIonal-theoretical approach was used to determme the flow profiles m a thm film adhered to a sohd surface which 1s contmuously removed from a hqmd bath The case under study assumes large capillary numbers, Ncu, meanmg that merttal forces are predommant over capdlary forces Results are used to prove that under these condltlons, the equations of motion loose their capacity to feed mformatlon upstream, thus the problem becomes an mltta[ value problem Independent of the condltlons downstream of the hquld bath Many different velocity profiles were fed as nutml condltlons, and computations were carried using an lmphcd finite-difference method untd either an asymptotic solution for large distances from the liquid bath was reached or computations showed a physically unacceptablp behavtor Those uutlal velocity profiles which could be carried forward, showed that a dehcate balance between VISCOUS, gravItatIonal, and Inertial forces was necessary m order for the velocity profiles to be contmuously accelerated Moreover, It IS possible to explam on macroscoplcal basis the fact that for lame removal speeds the drmenslonless flow rate of the hquld adhered to the solrd surface becomes independent of the_Caplllary nimber, N,, Startmg from the classlcal Landau-Levlch[l] theory of dip-coating, most of the theoretical and experlmental work on the subJect was designed to show the depen- dence of the coating phenomena on the Capillary number defined as N =ccuw ca u* (1) For small values of the NC,, the proposed theories agreed, wlthm experlmental error, with the experimental values For large values of the NC,, none of the pro- posed theones was able to predict a quahtatlve change m behavior when experiments showed that the coated film tl-uckness became a function of the removal speed, such tihat the dlmenslonless film thickness became mdepen- dent of the CapdIary number, Q = 2 ($)“= = constant for large NC, (2) W Experimental evidence mdlcated that the maxImum attamable value of Q was found around Q = 0 56 (see Table 1) although recent determinations showed values somewhat larger than this value Morey [2] found Q = 0 53, a value of Q = 0 58 can be obtamed from Gutfinger and Tallmadge’s [4] determmatlons, and van Rossum 131 found a maxlmum attamable value of Q = 0 54 More recently, Groenveld [9] found values of Q = 0 56, Soroka and Tallmadge[6] found Q = 0 61 and Spiers et al [7] found their maximum value of Q = 0 63 Heurlstlc ar- guments were introduced to explain this behavior, al- though as yet no satisfactory theory has been developed tPresent address Chemical Engmeermg Department, Prm- ceton Umversity, Pnnceton, NJ 08540, U S A *To whom correspondence should be addressed Deryagm and Levt[S] comparmg the thickness of a fiIm drammg on a vertical flat plate with the terminal film thickness, deduced a maxlmum value of Q = 0 67 which corresponds to the maximum attamable value of Q given by the contmuity equation and a parabolic profile Groenveld [9] assummg a parabohc velocity profile at the stagnation point and also at the mflectlon pomt of the streamhnes, and addmg the arbitrary condltlon that the film thickness at the tnflectlon point 1s the arlthmetlc mean of the film thickness at the stagnation point and the final film thickness, deduced that the only possible com- bmatlon of flow rate and wall velocity 1s given by the relationship Q = 0 56 Where T IS the dimensionless film thickness defined as l The purpose of this paper IS to show that the maxl- mum expenmentally attamable value of Q 1s a product of a delicate balance between VISCOUS,inertial and gravl- Table 1 Maxlmum dlmenslonless flow rates Reference Q InBX Morey [2] 0 53 van Rossum [3] 0 54 Gutfinger and Talimadge [4] 0 58 Groenveld [5] 0 56 Soroka et al 161 061 Spiers et al [7] 0 63 Maxunum theoretical value [Deryagm and Levi [S]], Qmax= 0 67 a7