Ann. Rev. Fluid Mech. 1985.17:65~9Copyright 1985 by Annual Reviews Inc. All rights reserved

COATING FLOWSKenneth J. Ruschak

Research Laboratories, Eastman Kodak Company, Rochester,New York 14650

IntroductionA coating flow is a fluid flow that is useful for covering a large surface areawith one or more thin, uniform liquid layers. The liquid film is subsequentlydried or cured and often serves to protect or decorate the substrate. The filmmay also serve a more active function, such as the recording of information.

A familiar coating flow is that associated with the application of paint bybrush or roller. Although everyday painting experience may suggest thatthe problem of applying a thin liquid layer to a surface is a trivial one, this isnot the case. In industrial applications, the layer thickness may have to bevery small and at the same time highly accurate. Moreover, a wide range oftheologies are encountered, and often the properties of the liquid are notadjustable for the purpose of coating. For productivity reasons a high speedof application may be required, and several discrete layers may have to beapplied simultaneously. As a result of such demands, attempts to use aspecific coating process for a given application frequently fail. The liquidlayer may not be continuous, and if it is, longitudinal or transverse waves orstreaks may be observed. It is also common for air to be entrained by thesubstrate along with the liquid.

The theoretical fluid-mechanics problem is first to identify a prospectivesteady, two-dimensional, film-forming flow, given the liquid properties, thecoating thickness, and the application speeds of interest. This step iscomplicated by the fact that coating flows are free-boundary flows andtherefore inherently nonlinear; that is, the region in space occupied by theflowing liquid is not known at the start but is in fact part of the solution tothe hydrodynamic equations. Once a flow field is found, its stability todisturbances must be evaluated. Finally, it is necessary to predict whetherthe flow will effectively displace air from the surface to be coated. Stated inthis way, the theoretical coating problem is anything but trivial, and the fact

650066-4189/85/0115-0065502.00

Ann. Rev. Fluid Mech. 1985. 17: 65--89 Copyright 1985 by Annual Reviews Inc. All rights reserved

COATING FLOWS

Kenneth J. Ruschak

Research Laboratories, Eastman Kodak Company, Rochester, New York 14650

Introduction A coating flow is a fluid flow that is useful for covering a large surface area with one or more thin, uniform liquid layers. The liquid film is subsequently dried or cured and often serves to protect or decorate the substrate. The film may also serve a more active function, such as the recording of information.

A familiar coating flow is that associated with the application of paint by brush or roller. Although everyday painting experience may suggest that the problem of applying a thin liquid layer to a surface is a trivial one, this is not the case. In industrial applications, the layer thickness may have to be very small and at the same time highly accurate. Moreover, a wide range of rheologies are encountered, and often the properties of the liquid are not adjustable for the purpose of coating. For productivity reasons a high speed of application may be required, and several discrete layers may have to be applied simultaneously. As a result of such demands, attempts to use a specific coating process for a given application frequently fail. The liquid layer may not be continuous, and if it is, longitudinal or transverse waves or streaks may be observed. It is also common for air to be entrained by the substrate along with the liquid.

The theoretical fluid-mechanics problem is first to identify a prospective steady, two-dimensional, film-forming flow, given the liquid properties, the coating thickness, and the application speeds of interest. This step is complicated by the fact that coating flows are free-boundary flows and therefore inherently nonlinear; that is, the region in space occupied by the

flowing liquid is not known at the start but is in fact part of the solution to the hydrodynamic equations. Once a flow field is found, its stability to disturbances must be evaluated. Finally, it is necessary to predict whether the flow will effectively displace air from the surface to be coated. Stated in this way, the theoretical coating problem is anything but trivial, and the fact

65 0066--4189/85/0115-0065$02.00

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66 RUSCHAK

that coating technology has developed largely as an art can perhaps bemore readily appreciated.

The dozens of coating devices in use attest to the practical difficulty of thecoating problem. Higgins (1965) and Booth (1970) have described many these. Even a cursory review of this material makes it evident that basicscience has not had a great deal to do with the development of thetechnology. The outline by Kaulakis (1974) of the development of coatingtechnology in the paper industry conveys much the same message.

This is not to say that useful research on coating flows has not been done.Rather, as in many other areas of technology, largely empirical develop-ments have outpaced scientific capabilities and understanding. Theoreticaland experimental studies have led to many insights into the structure andstability of film-forming flows. For the most part, however, flow geometrieshave been highly simplified and the mathematical analyses quite limited intheir range of applicability. Methods for computer simulation of coatingflows under very general conditions have been maturing rapidly, however,and there is reason to expect that scientific studies will have a greater effecton the development of the technology in the future.

Nearly Rectilinear Flow FieldsConsider the steady, two-dimensional flow of a Newtonian liquid ofviscosity/~, density p, and surface tension tr. Frequently the flow fields thathave been studied in connection with coating flows exhibit nearly parallelstreamlines. In such cases the governing equations can be substantiallysimplified. As will be seen, however, there has been no general agreement onwhich terms are properly discarded, and frequently the limitations of theresults have not been clearly set forth. It therefore seems worthwhile topresent the equations scaled for nearly rectilinear or quasi-one-dimensionalflow. Higgins et al. (1977) and Higgins & Scriven (1979) have developed equations and solution strategies more generally.

Suppose that the streamlines are all nearly parallel to the X axis, with asmall characteristic slope 3. The characteristic length in the Y direction, T,will be a measure of the film thickness, and it follows that the characteristiclength in the X direction is T/6. The characteristic speed in the X direction,S, is usually the speed of the substrate. For a nontrivial result, the two termsin the continuity equation must be of the same size; this implies that thecharacteristic speed in the Y direction is 6S. In strictly rectilinear flow thestreamwise pressure gradient is balanced by the transverse gradient of theshear stress. The scaling for the pressure in nearly rectilinear flow, I~S/6T,follows when the same two terms are balanced. With x, y as the dimension-less coordinates, u, v as the corresponding dimensionless velocity compo-nents, and p as the dimensionless pressure, the momentum and continuity

66 RUSCHAK

that coating technology has developed largely as an art can perhaps be more readily appreciated.

The dozens of coating devices in use attest to the practical difficulty of the coating problem. Higgins (1965) and Booth (1970) have described many of these. Even a cursory review of this material makes it evident that basic science has not had a great deal to do with the development of the technology. The outline by Kaulakis (1974) of the development of coating technology in the paper industry conveys much the same message.

This is not to say that useful research on coating flows has not been done. Rather, as in many other areas of technology, largely empirical developments have outpaced scientific capabilities and understanding. Theoretical and experimental studies have led to many insights into the structure and stability of film-forming flows. For the most part, however, flow geometries have been highly simplified and the mathematical analyses quite limited in their range of applicability. Methods for computer simulation of coating

flows under very general conditions have been maturing rapidly, however, and there is reason to expect that scientific studies will have a greater effect on the development of the technology in the future.

Nearly Rectilinear Flow Fields Consider the steady, two-dimensional flow of a Newtonian liquid of viscosity p" density p, and surface tension (J'. Frequently the flow fi elds that have been studied in connection with coating flows exhibit nearly parallel streamlines. In such cases the governing equations can be substantially simplifi ed. As will be seen, however, there has been no general agreement on which terms are properly discarded, and frequently the limitations of the results have not been clearly set forth. It therefore seems worthwhile to present the equations scaled for nearly rectilinear or quasi-one-dimensional

flow. Higgins et al. (1977) and Higgins & Scriven (1979) have developed the equations and solution strategies more generally.

Suppose that the streamlines are all nearly parallel to the X axis, with a small characteristic slope b. The characteristic length in the Y direction, T, will be a measure of the fi lm thickness, and it follows that the characteristic length in the X direction is T/b. The characteristic speed in the X direction, S, is usually the speed of the substrate. For a nontrivial result, the two terms in the continuity equation must be of the same size; this implies that the characteristic speed in the Y direction is bS. In strictly rectilinear flow the streamwise pressure gradient is balanced by the transverse gradient of the shear stress. The scaling for the pressure in nearly rectilinear flow, J.lS/bT,

follows when the same two terms are balanced. With x, y as the dimensionless coordinates, u, v as the corresponding dimensionless velocity components, and p as the dimensionless pressure, the momentum and continuity

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COATING FLOWS 67

equations scaled for nearly rectilinear flow are

r5(UUx + vur) = -- Px + u, + ~2uxx + cos 0, (1)rf3(UVx + vvr) = -- py 62vry + 6*Vxx + 6gsin0,

Ux+Vr = O.In these equations, r = pST/la is the Reynolds number, # = pGT2/#S is theinverse Stokes number, and the components of the gravitational vector areG cos 0, G sin 0. When 32

68 RUSCHAK

the gross shape of the meniscus is controlled by surface tension and ahydrostatic pressure field. Because the film is relatively thin, this staticmeniscus appears to be tangent to the moving substrate. If the radius ofcurvature of the static meniscus at its apparent point of tangency is R, thenthe overall scale of the flow field can be taken as R, and the supposition isthat D

COATING FLOWS 69number. Although these restrictions are quite severe, the equation never-theless helps to explain some of the basic characteristics of various coatingflows, as will now be seen.

Premetered Coating FlowsIn precision coating it is not desirable for the coating thickness to dependupon the viscosity of the liquid or other parameters that are difficult tocontrol precisely. In premetered coating flows, the coating thickness is anindependent parameter that can be varied within limits. Slide and curtaincoating are examples of premetered coating flows.

Consider the simple slot coater shown in Figure 2 under the samelimiting conditions as in the analysis of Landau & Levich (Ruschak 1976).More general geometries have been analyzed by Higgins & Scriven (1980).The two static menisci that must be considered are the coating meniscus,which forms the film, and the wetting meniscus, which displaces air from thesubstrate. It is presumed that these menisci are so small (B

70 RUSCHAK

V. 1979), as shown in Figure 2. The complex subject of dynamic wetting is.covered in the final section of this review; here, the dynamic contact angle ispresumed to be known.

The geo, metrical restrictions on the radius of curvature of the coatingmeniscus are summarized in Figure 3, which was drawn for a dynamiccontact angle of 120. The coating meniscus is free to adopt any radius ofcurvature within a range of values that depends Upon the imposed pres-sure differential. When the flow rate is imposed at a given speed, thusdetermining the coating thickness D, the coating meniscus adopts theradius of curvature necessary to satisfy Equation (3), as long as this radiussatisfies the inequalities depicted in Figure 3. If any of the inequalities isviolated, then coating a uniform layer under the specified conditions is not

APE4-

Z-

I-cos ~

o I

/ -R-I-cos~

-I~Fioure 3 Possible values for the radius of curvature R of the coating meniscus for the slot

coater at small capillary numbers.

70 RUSCHAK

V. 1979), as shown in Fig ure 2. The complex subject of dynamic wetting is covere.d in the final section of this review; here, the dynamic contact ang le is

presumed to be known. The g emetrical restrictions on the radius of curvature of the coating

meniscus are summarized in Fig ure 3, which was drawn for a dynamic contact angle of 120. The coating meniscus is free to adopt any radius of curvature within a rang e of values that depends upon the imposed pressure differential. When the flow rate is imposed at a g iven speed, thus determining the coating thickness D, the coating meniscus adopts the radius of curvature necessary to satisfy Equation (3), as long as this radius satisfies the inequalities depicted in Fig ure 3. If any of the inequa-lities is violated, then coating a uniform layer under the specified conditions is not

4

3

2

I-cos S

PE (J

04-------.----------.--------

-I-cose -I

2 E R

. Figure 3 Possible values for the radius of curvature R of the coating meniscus for the slot coater at small capillary numbers.

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COATING FLOWS 71

possible. Figure 3 shows that a pressure difference across the gap increasesthe utility of the coating device by enabling the coating meniscus to adopthigher curvatures and thus produce thinner coatings.

According to Figure 3, the radius of curvature of the coating meniscuscan be arbitrarily large. This is because the effect of gravity on the shape ofthe coating meniscus has been neglected. If gravity is retained in theanalysis, there is an upper limit to the radius of curvature of the coatingmeniscus at its point of tangency with the substrate, as suggested by thedashed line in Figure 3.

Tallmadge et al. (1979) experimentally studied the slide coater, in whichliquid flows down an inclined plane that ends a small distance from thesubstrate. They did not impose a pressure difference across the gap. As thespeed was increased at constant flow rate, a point was reached where itbecame impossible to preserve a two-dimensional flow. The film wouldeither narrow and resist being drawn to the proper width or, morecommonly, split into two or more parts. This "split point" marks thecrossing of the lower boundary in Figure 3. On the other hand, as coatingspeed was decreased, a point was reached where not all of the liquidsupplied was taken up by the moving substrate, and dripping was observedbelow the point of application. This "drip point" marks the crossing of theleft, dashed boundary in Figure 3.

Coatin9 Flows Metered by the Coatin9 MeniscusDIP COATING AT LOW CAPILLARY NUMBER In a self-metering coating flow,the coating thickness is a dependent variable. The best-known example of aself-metering coating flow is free withdrawal, or dip coating, in which thesubstrate is withdrawn from a large reservoir of liquid (Figure 4). This typeof coating flow has been reviewed by Tallmadge & Gutfinger (1967). Equation (3), R, the radius of curvature of the static meniscus at the coatingpoint, is uniquely determined by the hydrostatic pressure field to be[a/2pG(1 + sin ~)]t/2. Unlike premetered coating, the radius of curvature ofthe coating meniscus is not free to vary, with the result that the coatingthickness becomes the dependent variable in Equation (3). Thus the coatingthickness depends upon the fluid properties and the substrate speed, whichis usually a disadvantage. Ruschak (1974) compared the predictions Equation (3) for vertical withdrawal (~ = 0) with the low-capillary-numberdata of Moray (1940) and found good agreement only at capillary numbersbelow about 0.01. The quantitative usefulness of Equation (3) is limitedindeed.

DIP COATING AT HIGH CAPILLARY NUMBER Many attempts have been madeto improve upon the predictions of the equation of Landau & Levich for dip

COATING FLOWS 71

possible. Figure 3 shows that a pressure difference across the gap increases the util ity of the coating device by enabling the coating meniscus to adopt hig her curvatures and thus produce thinner coating s.

According to Fig ure 3, the radius of curvature of the coating meniscus can be arbitr arily larg e. This is because the effect of g ravity on the shape of the coating meniscus has been neglected. If gravity is retained in the analysis, there is an upper limit to the radius of curvature of the coating meniscus at its point of tangency with the substrate, as suggested by the dashed line in Fig ure 3.

Tallmadg e et al. (1979) experimentally studied the slide coater, in which liquid flows down an inclined plane that ends a small distance from the substrate. They did not impose a pressure difference across the gap. As the speed was increased at constant flow rate, a point was reached where it became impossible to preserve a two-dimensional flow. The film would either narrow and resist being drawn to the proper width or, more commonly, split into two or more parts. This "split point" marks the crossing of the lower boundary in Figure 3. On the other hand, as coating speed was decreased, a point was reached where not all of the liquid supplied was taken up by the moving substrate, and dripping was observed below the point of application. This "drip point" marks the crossing of the left, dashed boundary in Fig ure 3.

Coating Flows Metered by the Coating Meniscus DIP COATING AT LOW CAPILLARY NUMBER In a self-metering coating flow, the coating thickness is a dependent variable. The best-known example of a self-metering coating flow is free withdrawal, or dip coating, in which the substrate is withdrawn from a larg e reservoir of liquid (Fig ure 4). This type of coating flow has been reviewed by Tallmadge & Gutfinger (1967). In

Equation (3), R, the radius of curvature of the static meniscus at the coating point, is uniquely determined by the hydrostatic pressure field to be

[o/2pG(1 + sin O()] 1/2. Unlik premetered coating , the radius of curvature of the coating meniscus is not free to vary, with the result that the coating thickness becomes the dependent variable in Equation (3). Thus the coating thickness depends upon the fluid properties and the substrate speed, which is usually a disadvantag e. Ruschak (1974) compared the predictions of

Equation (3) for vertical withdrawal (0( = 0) with the low-capillary-number data of Moray (1940) and found g ood ag reement only at capillary numbers below about 0.01. The quantitative usefulness of Equation (3) is limited indeed.

DIP COATING AT HIGH CAPILLARY NUMBER Many attempts have been made to improve upon the predictions of the equation of Landau & Levich for dip

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72 RUSCHAK

coating at capillary numbers higher than 0.01. One useful result is that thereis an upper limit to the film thickness. Derjaguin (1945) and Van Rossum(1958) studied the fully rectilinear flow far above the liquid reservoir andfound that there is a maximum possible flow rate that can be lifted by themoving surface against gravity. This maximum flow rate is 2SD/3, wherethe film thickness D is given by (ItS/pG cos ~)1/2. This result is consistentwith g cos 0 being of order unity in Equation (1), i.e. consistent with viscousand gravitational effects being in balance. Derjaguin concludes that this willbe the film thickness approached as the capillary number becomes largewith inertial effects remaining negligible. More recent theoretical studies byHomsy & Geyling (1977) and by Tuck (1983) support this contention.Experiments, however, have generally yielded a value of d = DID that isless than unity at high capillary number. For example, Van Rossum (1958)could not produce films thicker than d = 0.68, while Groenveld (1970)reports 0.66, and the highest value obtained by Spiers et al. (1974) is about0.8. Moreover, after reaching its maximum value, d decreases slightly withfurther speed increases.

Marques et al. (1978) have performed numerical experiments on theboundary-layer equations that may account for this apparent lack ofagreement between experiment and theory. The boundary-layer equationsare in fact the equations for nearly rectilinear flow when the inertial andgravitational terms are retained but the surface-tension term is discarded.In the numerical experiments a film thickness greater than d -- 0.67 could

Fioure 4 Free withdrawal from a pool of liquid.

72 RUSCHAK

coating at capillary numbers higher than 0.01. One useful result is that there is an upper limit to the film thickness. Derjaguin (1945) and Van Rossum

(1958) studied the fully rectilinear flow far above the liquid reservoir and found that there is a maximum possible flow rate that can be lifted by the moving surface against gravity. This maximum flow rate is 2SD'j3, where the film thickness D' is given by (IlSjpG cos rx)1!2. This result is consistent with g cos e being of order unity in Equation (1) , i.e. consistent with viscous and gravitational effects being in balance. Derjaguin concludes that this will be the film thickness approached as the capillary number becomes large with inertial effects remaining negligible. More recent theoretical studies by

Homsy & Geyling (1977) and by Tuck (1983) support this contention. Experiments, however, have generally yielded a value of d = DjD' that is less than unity at high capillary number. For example, Van Rossum (1958) could not produce films thicker than d = 0.68, while Groenveld (1970) reports 0.66, and the highest value obtained by Spiers et al. (1974) is about 0.8. Moreover, after reaching its maximum value, d decreases slightly with

further speed increases. Marques et al. (1978) have performed numerical experiments on the

boundary-layer equations that may account for this apparent lack of agreement between experiment and theory. The boundary-layer equations are in fact the equations for nearly rectilinear flow when the inertial and gravitational terms are retained but the surface-tension term is discarded.

In the numerical experiments a film thickness greater than d = 0.67 could

Figure 4 Free withdrawal from a pool of liquid.

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COATING FLOWS 73

not be produced, and so significant inertial effects may explain theexperimental data at high capillary number.DIP COATING AT INTERMEDIATE CAPILLARY NUMBER To bridge the gapbetween the results at high and low capillary numbers, two-dimensionalflow equations must be faced. By the estimates given earlier for the limits ofapplicability of Equation (3), the effect of gravity in the film-entrainmentregion becomes important at about the same capillary number that theassumption of nearly rectilinear flow breaks down. That this is so is alsoevident from the two-dimensional finite-element calculations of Ruschak(1982), which show Equation (3) failing at a capillary number of about 0.01,even though gravity is not included in the calculations.

Nonetheless, the assumption of nearly rectilinear flow has often beenretained in analyses of dip coating. The effect of gravity in the film-entrainment region has been successfully incorporated into the analysis ofthe equations of nearly rectilinear flow by Lee & Tallmadge (1975), whosolved the equations numerically, and also by Wilson (1982), who obtainedan explicit result by the method of matched asymptotic expansions.Unfortunately, the inclusion of gravity alone does not greatly expand therange of validity of Equation (3). In analyses where nearly rectilinear flowhas not been assumed, some, but not all, of the terms estimated to be oforder 62 have been kept. The papers by Spiers et al. (1974) and by Esmail Hummel (1975) are examples.

Another difficulty in extending the analysis of Landau & Levich to highercapillary numbers lies in matching the solution to the differential equationgiving the meniscus shape of the film-entrainment region with the staticmeniscus associated with the reservoir. Whenever gravity or other higher-order terms are retained, the procedure of Landau & Levich for matchingcurvatures breaks down. That this problem can be rationally approached isevidenced by the analyses of Lee & Tallmadge and Wilson just mentioned.Unfortunately, ad hoc measures pervade the literature. In some cases thedifferential equation for the film shape has been illogically altered toproduce a new differential equation that does permit the curvaturematching of Landau & Levich; examples include the papers by White &Tallmadge (1965) and Spiers et al. (1974). In other cases the static meniscusis assumed to extend up to the stagnation point, which is always present onthe meniscus. The differential equation for the static meniscus is applied as aboundary condition at the stagnation point. This procedure does notguarantee continuous slope and curvature along the entire meniscus,contrary to the claims of its first proponents, Esmail & Hummel (1975), andthere would seem to be nothing special about the stagnation point on themeniscus that would permit one to deduce that dynamic effects are

eOA TING FLOWS 73

not be produced, and so sig nificant inertial effects may explain the experimental data at hig h capillary number.

DIP COATING AT INTERMEDIATE CAPILLARY NUMBER To bridg e the g ap between the results at hig h and low capillary numbers, two-dimensional

flow equations must be faced. By the estimates g iven earlier for the limits of applicability of Equation (3), the effect of g ravity in the film-entrainment reg ion becomes important at about the same capillary number that the assumption of nearly rectilinear flow break s down. That this is so is also evident from the two-dimensional finite- element calculations of Ruschak

(1982), which show Equation (3) failing at a capillary number of about 0.01, even thoug h g ravity is not included in the calculations.

Nonetheless, the assumption of nearly rectilinear flow has often been retained in analyses of dip coating . The effect of g ravity in the filmentrainment reg ion has been successfully incorporated into the analysis of the equations of nearly rectilinear flow by Lee & Tallmadg e (1975), who solved the equations numerically, and also by Wilson (1982), who obtained an explicit result by the method of matched asymptotic expansions.

Unfortunately, the inclusion of g ravity alone does not g reatly expand the rang e of validity of Equation (3). In analyses where nearly rectilinear flow has not been assumed, some, but not all, of the terms estimated to be of order 02 have been k ept. The papers by Spiers et al. n974) and by Esmail &

Hummel (1975) are examples. Another difficulty in extending the analysis of Landau & Levich to hig her

capillary numbers lies in matching the solution to the differential equation g iving the meniscus shape of the film-entrainment reg ion with the static meniscus associated with the reservoir. Whenever g ravity or other hig herorder terms are retained, the procedure of Landau & Levich for matching curvatures break s down. That this problem can be rationally approached is evidenced by the analyses of Lee & Tallmadg e and Wilson just mentioned.

Unfortunately, ad hoc measures pervade the literature. In some cases the differential equation for the film shape has been illog ically altered to produce a new differential equation that does permit the curvature matching of Landau & Levich; examples include the papers by White &

Tallmadg e (1965) and Spiers et al. (1974). In other cases the static meniscus is assumed to extend up to the stag nation point, which is always present on the meniscus. The differential equation for the static meniscus is applied as a boundary condition at the stag nation point. This procedure does not g uarantee continuous slope and curvature along the entire meniscus, contrary to the claims of its first proponents, Esmail & Hummel (1975), and there would seem to be nothing special about the stag nation point on the meniscus that would permit one to deduce that dynamic effects are

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negligible below this point. What is more, the differential equation for thestatic meniscus that is applied at the stagnation point has been incorrectlytaken from the book of Deryagin & Levi (1964, p. 36); a change in thepositive direction of the X axis produces a change in the sign of the firstderivative of the film thickness, which has apparently been overlooked.Although Nigam & Esmail (1980) later admitted the ad hoc nature of thisboundary condition, it continues to appear in papers (Tekic & Jovanovic1982), and in its incorrect form at that.

Although improved agreement with experimental data at moderateand/or high capillary numbers is claimed for many analyses subsequent tothat of Landau & Levich, this does not really justify ad hoc mathematicalprocedures. Perhaps the best way to solve the dip-coating problem atcapillary numbers higher than 0.01 is by a two-dimensional numericalmethod. Lee & Tallmadge (1974) used a finite-difference scheme and hadsome success for capillary numbers larger than about 1.0. Current finite-element methods can handle this problem more easily and without therestriction on the capillary number.

Coating Flows Metered Upstream of the Coating MeniscusBLADE COATING In dip coating the flow rate is metered by the coatingmeniscus, and as a result the coating thickness is sensitive to the fluidproperties and the substrate speed. There is, however, an important class ofself-metering coating flows in which the metering takes place upstream ofthe coating meniscus. The coating meniscus essentially responds to animposed flow rate, much as in premetered coating. While these coatingflows are not as accurate as a premetered coating flow, a substantial degreeof independence of the coating thickness from the fluid properties and thesubstrate speed can be achieved.

One example of such a self-metering coating device is the blade coater,commonly used to apply coatings during the manufacture of paper(Gartaganis et al. 1978). A simple model of a blade coater by t~reener Middleman (1974) illustrates the principle. The face of the blade has a smallslope 6 with respect to the rigid substrate, so that the liquid is drawn into aslowly converging region (Figure 5). The blade is presumed to be floodedupstream (an excess of liquid is supplied), and surface-tension effects arepresumed to b~ n~gligible so that th~ pressure can bc taken to be sensiblyatmospheric at each end of the blade. In terms of the dimensionless groupb = ~L/E, the ratio of the coating thickness to the gap is given by DIE =(1 b)/(2 + b). The most important feature of this result is that the coatingthickness is independent of fluid properties and substrate sp~cd.

Often the blade is not rigidly fixed with respect to the substrate. Rather,it is loaded by one means or another, and the loading controls the

74 RUSCHAK

neg lig ible below this point. What is more, the differential equation for the static meniscus that is applied at the stag nation point has been incorrectly taken from the book of Deryag in & Levi (1964, p. 36) ; a chang e in the positive direction of the X axis produces a change in the sig n of the first derivative of the film thickness, which has apparently been overlooked.

Although Nig am & Esmail (1980) later admitted the ad hoc nature of this boundary condition, it continues to appear in papers (Tekic & Jovanovic 1982) , and in its incorrect form at that.

Althoug h improved ag reement with experimental data at moderate and/or high capillary numbers is claimed for many analyses subsequent to that of Landau & Levich, this does not really justify ad hoc mathematical procedures. Perhaps the best way to solve the dip-coating problem at capillary numbers hig her than 0.01 is by a two-dimensional numerical method. Lee & Tallmadg e (1974) used a finite-difference scheme and had some success for capillary numbers larger than about 1.0. Current finite

clement methods can handle this problem more easily and without the restriction on the capillary number.

Coating Flows Metered Upstream of the Coating Meniscus BLADE COATING In dip coating the flow rate is metered by the coating meniscus, and as a result the coating thickness is sensitive to the fluid properties and the substrate speed. There is, however, an important class of self-metering coating flows in which the metering takes place upstream of the coating meniscus. The coating meniscus essentially responds to an imposed flow rate, much as in premetered coating. While these coating

flows are not as accurate as a premetered coating flow, a substantial degree of independence of the coating thickness from the fluid properties and the substrate speed can be achieved.

One example of such a self-metering coating device is the blade coater, commonly used to apply coatings during the manufacture of paper

(Gartaganis et al. 1978) . A simple model of a blade coater by Greener & Middleman (1974) illustrates the principle. The face of the blade has a small slope [) with respect to the rigid substrate, so that the liquid is drawn into a slowly converging region (Figure 5) . The blade is presumed to be flooded upstream (an excess of liquid is supplied) , and surface-tension effects are presumed to be neg ligible so that the pressure can be taken to be sensibly atmospheric at each end of the blade. In terms of the dimensionless g roup

b = [)L/E, the ratio of the coating thickness to the g ap is g iven by D/E = (1 + b)/(2 + b). The most important feature of this result is that the coating thickness is independent of fluid properties and substrate speed.

Often the blade is not rig idly fixed with respect to the substrate. Rather, it is loaded by one means or another, and the loading controls the

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COATING FLOWS 75

k/////////////////////////////~~/ ~

Fixture 5 Blade coater flooded upstream.

coating gap. If A is the force per unit width applied to the blade, thenA6z/6#S = In(1 +b)-2b/(2+b). Thus b and the gap E follow when A isspecified. For the special case b

76 RUSCHAK

between the roller surfaces where the coating meniscus forms, namely W--Ef(m) (Coyne & Elrod 1970). The fractional change in distance between theroller surfaces over this transition region is therefore expected to be on theorder of,Sm, and if this is required to be small, then the roller surfaces can beconsidered parallel over the transition region. Thus the flow field dividesnaturally into two parts: a two-dimensional portion between parallelsurfaces in the immediate vicinity of the coating meniscus, and a nearlyrectilinear portion elsewhere (Taylor 1963, Ruschak 1982).

The flow problem near the coating meniscus that arises in this way issketched in Figure 7 for the case a = 1. Taylor (1963) pointed out that thedimensionless flow rate q = Q/SW depends only upon the capillarynumber. The asymptotic pressure gradient upstream can be expressed interms of q as dp/dx = 12(1 -q)/f2. Indeed, this pressure gradient providesone of the boundary conditions at x = m that the solution to the equationsfor nearly rectilinear flow between the rollers must satisfy. Unfortunately,this boundary condition is not useful until q is determined as a function ofthe capillary number by solving the difficult flow problem of Figure 7. In

Fioure 6 Roll coater flooded upstream.

76 RUSCHAK

between the roller surfaces where the coating meniscus forms, namely W = Ef(m) (Coyne & Elrod 1970). The fractional chang e in distance between the roller surfaces over this transition reg ion is therefore expected to be on the order of bm, and if this is required to be small, then the roller surfaces can be considered parallel over the transition reg ion. Thus the flow field divides naturally into two parts: a two-dimensional portion between parallel surfaces in the immediate vicinity of the coating meniscus, and a nearly rectilinear portion elsewhere (Taylor 1963, Ruschak 1982) .

The flow problem near the coating meniscus that arises in this way is sk etched in Fig ure 7 for the case a = 1. Taylor (1963) pointed out that the dimensionless flow rate q == Q/SW depends only upon the capillary number. The asymptotic pressure g radient upstream can be expressed in terms of q as dp/dx = 12(1 _q)!f2. Indeed, this pressure g radient provides one of the boundary conditions at x = m that the solution to the equations for nearly rectilinear flow between the rollers must satisfy. Unfortunately, this boundary condition is not useful until q is determined as a function of the capillary number by solving the difficult flow problem of Fig ure 7. In

----------------x

t

Figure 6 Roll coater flooded upstream.

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COATING FLOWS 77

fact, Taylor resorted to special experiments to measure q. Later, Coyne &Elrod (1970) determined q approximately (for a closely related flowproblem) by a method that involves assuming a functional form for thevelocity field, and Ruschak (1982) solved for q without making simpli-fications by using the finite-element method. When the capillary number isless than about 0.01, Equation (3) can be used to compute q. With W/2 asthe appropriate radius of curvature for the coating meniscus, the result isq = 1.34c2/3. According to Ruschaks results, q is given approximately by0.54c1/2 for 0.01 < c < 0.1, and q approaches a constant value of about 0.41as c ~ ~. Thus the pressure gradient of the nearly rectilinear flow should be12k/f2 where the coating meniscus forms, and k decreases from 1 to about0.59 as the capillary number increases from 0 to ~.

Frequently, approximate expressions have been used for the pressuregradient where the coating meniscus forms. Dowson & Taylor (1979),Coyne & Elrod (1970), Savage (1977a), Sullivan & Middleman (1979), Bixler (1982) have critically discussed many of these. The Swift-Steiberapproximation, for instance, is to set the pressure gradient to zero where thecoating meniscus forms, but as has just been seen, this is never strictly true.According to the popular Prandtl-Hopkins condition, the interface formsat a stagnation point in the nearly rectilinear flow field (i.e. at a point whereu and ur are both zero). This leads to a pressure gradient of 12(2/3)/f2, whichis close to the correct result at capillary numbers near unity.

Besides the pressure gradient, the pressure where the coating meniscusforms is also required as a boundary condition and may be estimated(Taylor 1963) or obtained by the solution of the flow field of Figure (Ruschak 1982). When the capillary number is large, this pressure will be the order ofltS/W, which is small compared with the characteristic pressureofl~S/fiE and may be taken as zero. When the capillary number is small, onthe other hand, the pressure will be of the order of 2~r/W, and the boundarycondition becomes p = - 26/fc, which can be of some consequence if c issmaller than 6. For simplicity, this latter expression is used throughoutwhat follows.

Figure 7 Coating meniscus between parallel surfaces.

COATING FLOWS 77

fact, Taylor resorted to special experiments to measure q. Later, Coyne & Elrod (1970) determined q approximately (for a closely related flow problem) by a method that involves assuming a functional form for the velocity field, and Ruschak (1982) solved for q without making simpli

fications by using the finite-element method. When the capillary number is . less than about 0.01, Equation (3) can be used to compute q. With W /2 as

the appropriate radius of curvature for the coating meniscus, the result is q = 1.34c2/3 According to Ruschak's results, q is g iven approximately by 0.54c1/2 for 0.01 < c < 0.1, and q approaches a constant value of about 0,41 as c -+ 00. Thus the pressure g radient of the nearly rectilinear flow should be

12k/f2 where the coating meniscus forms, and k decreases from 1 to about 0.59 as the capillary number increases from 0 to 00.

Frequently, approximate expressions have been used for the pressure g radient where the coating meniscus forms. Dowson & Taylor (1979),

Coyne & Elrod (1970), Savag e (1977a), Sullivan & Middleman (1979), and Bixler (1982) have critically discussed many of these. The Swift-Steiber approximation, for instance, is to set the pressure g radient to zero where the coating meniscus forms, but as has just been seen, this is never strictly true.

According to the popular Prandtl-Hopk ins condition, the interface forms at a stag nation point in the nearly rectilinear flow field (i.e. at a point where

u and uy are both zero). This leads to a pressure g radient of 12(2/3)1f2, which is close to the correct result at capillary numbers near unity.

Besides the pressure g radient, the pressure where the coating meniscus forms is also required as a boundary condition and may be estimated

(Taylor 1963) or obtained by the solution of the flow field of Fig ure 7 (Ruschak 1982). When the capillary number is larg e, this pressure will be on the order of .uS/W , which is small compared with the characteristic pressure of .uS/bE and may be taken as zero. When the capillary number is small, on the other hand, the pressure will be of the order of 2a/W , and the boundary condition becomes p = - 2{)lfc, which can be of some consequence if c is smaller than (). For simplicity, this latter expression is used throug hout what follows.

s

w

S Figure 7 Coating meniscus between parallel surfaces.

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78 RUSCHAK

The value of 2 varies little with 6 and c and is close to 4/3, as when therollers are flooded both upstream and downstream. Benkreira et al. (1981)made 1500 experimental measurements for forward roll coating (a > 0) capillary numbers between 0.03 and 15, and for b between about 0.04 and0.3..They found 2 = 1.31 with a standard deviation of 0.49/o. Pitts & Greiller(1961) measured flow rates for 6 between about 0.06 and 0.3 and for a = They found 2 to lie-between 1.26 and 1.38, and they noted that 2 decreasesslightly with increasing speed (capillary number). According to the theo-retical results of Ruschak (1982), for capillary numbers at even multiplesof 10 between 0.01 and 10, the predicted values of 2 are, respectively,1.36, 1.33, 1.30, and 1.29 for 3 = 0.06. Thus the flow rates predicted byGatcombe (1945) for flooded rollers are expected to be correct within a fewpercent for typical values of c and 6, and as a result, the rollers meter theliquid in much the same way as the blade coater.

The position of the coating meniscus is simply related to 2 and q.From the definition of q, the distance between the roller surfaces where thecoating meniscus forms is W = Q/Sq = E2/q, so that f= 2/q andm = [2(2/q- 1)] 1/2. When the capillary number is so small that q is muchsmaller than 1, the coating meniscus forms too far from the nip (m >> 1)for the assumption of nearly rectilinear flow to be valid up to the vicinityof the coating meniscus, and a fully two-dimensional analysis is in order. Asthe capillary number increases, so too does q increase, and the meniscus isdrawn toward the nip.

It is necessary to know in forward roll coating (a > 0) how the flowdivides between the two rollers, or in other words what is the coatingthickness on each roller? Pitts & Greiller (1961) observed that two regionsof recirculation are usually (but not always) present upstream of the coatingmeniscus. Savage (1982) noted that the equations for nearly rectilinear flowpredict the stagnation point marking the onset of these recirculations. Onthe assumption that the streamline passing through this point ultimatelybecomes the free-surface streamline (an assumption consistent with theobservations of Pitts & Greiller), he was able to predict quantitatively howthe flow divides. Actually, C.-S. ih pursued this idea some yearsearlier (seeHintermaier & White 1965) and obtained a result that, although correct, islimited to the special case where a is nearly 1. With 21 and 2~ the flow ratesultimately associated with rollers 1 and 2 so that 21 + 22 = 2, Savages resultmay be expressed as 22/21 = a1"5. This form of the result was pointed out byD. J. Coyle (personal communication, 1983) and follows immediately fromSavages work. Savage presented experimental data that support his result,and Benkreira et al. (1981) used a least-squares analysis of their 1500 datapoints to obtain 22/21 = 0.87aTM (the asymmetry for a = 1 was attributed

78 RUSCHAK

The value of A varies little with (j and c and is close to 4/3, as when the rollers are flooded both upstream and downstream. Benk reira et a1. (1981) made 1500 experimental measurements for forward roll coating (a> 0) at capillary numbers between 0.03 and 15, and for (j between about 0.04 and 0.3. They found ..1= 1.31 with a standard deviation of 0.4%. Pitts & Greiller

(1961) measured flow rates for (j between about 0.06 and 0.3 and for a = l. They found A to lie -between 1.2 6 and 1.38, and they noted that A decreases slig htly with increasing speed (capillary number) . According to the theoretical results of Ruschak (1982) , for capillary numbers at even multiples of 10 between 0.01 and 10, the predicted values of A. are, respectively, 1.36, 1.33, 1.30, and 1.29 for b = 0.06. Thus the flow rates predicted by

Gatcombe (1945) for flooded rollers are expected to be correct within a few percent for typical values of c and b, and as a result, the rollers meter the liquid in much the same way as the blade coater.

The position of the coating meniscus is simply related to A and q. From the definition of q, the distance between the roller surfaces where the coating meniscus forms is W = QISq = EA.lq, so that f = Ajq and

m = [2(Ajq_1)]1/2. When the capillary number is so small that q is much smaller than 1, the coating meniscus forms too far from the nip (m 1) for the assumption of nearly rectilinear flow to be valid up to the vicinity of the coating meniscus, and a fully two-dimensional analysis is in order. As the capillary number increases, so too does q increase, and the meniscus is drawn toward the nip.

It is necessary to k now in forward roll coating (a> 0) how the flow divides between the two rollers, or in other words what is the coating thick ness on each roller? Pitts & Greiller (1961) observed that two regions of recirculation are usually (but not always) present upstream of the coating meniscus. Savage (1982) noted that the equations for nearly rectilinear flow predict the stag nation point mark ing the onset of these recirculations. On the assumption that the streamline passing through this point ultimately becomes the free-surface streamline (an assumption consistent with the observations of Pitts & Greiller) , he was able to predict quantitatively how the flow divides. Actually, c.-s. Yih pursued this idea some years earlier (see

Hintermaier & White 1965) and obtained a result that, althoug h correct, is limited to the special case where a is nearly 1. With Al and ..12 the flow rates ultimately associated with rollers 1 and 2 so that ..11 + ..12 = A., Savag e's result may be expressed as ..12/..11 = a1.s. This form of the result was pointed out by

D. J. Coyle (personal communication, 1983) and follows immediately from Savage's work. Savage presented experimental data that support his result, and Benk reira et al. (1981) used a least-squares analysis of their 1500 data points to obtain A2/A1 = 0.87a1.6S (the asymmetry for a = 1 was attributed

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COATING FLOWS 79to gravitational effects). Equation (3) predicts an exponent of 1.67 for verysmall capillary numbers.

Instability of Coating Flows: Ribbing LinesThe calculation of the steady, two-dimensional flow field, and sometimesthe determination of the range of conditions over which such a solution tothe governing equations exists, is often just a prelude to a determination ofthe stability of the flow. The flow instabili~ty called ribbing lines has receiveda great deal of attention in the literature. This instability may be observed inroll coating and slide coating, for example. The flow field for ribbing lines issteady but three dimensional. More specifically, the flow field becomesperiodic in the third dimension, and the coating itself becomes ribbed anduneven. Such an event is usually intolerable in a coating process, and so thestudy of this and other instabilities is of considerable importance.

The theoretical calculations and experimental observations of Saffman &Taylor (1958) provided the first insights into the ribbing instability. Saffman& Taylor used the equations for nearly rectilinear flow, expanded to includethe third dimension, to evaluate the stability of a coating meniscus betweenparallel surfaces (Figure 7) to small disturbances. They found that the flowis unstable if the pressure gradient corresponding to the rectilinear flowupstream of the meniscus, namely dP/dX = 12#S(1--q)/W2, is positive.Because q cannot exceed 0.41 (Ruschak 1982), this is always the case.

Arguments by Pitts & Greiller (1961) are useful in trying to understandthis result. They consider the evolution of the disturbance to be quasi-steady-state, and they suppose that at some location in the third dimensionthe coating meniscus moves slightly upstream because of a disturbance. Thelinear pressure field of positive slope upstream of the displaced meniscusremains the same as that behind the undisturbed meniscus, but it is shiftedalong the X axis by the distance the displaced meniscus has moved. Itfollows that at any position X, the pressure behind the disturbed meniscusis more positive than that behind the undisturbed meniscus. Therefore, flowwill occur from the region behind the disturbed meniscus toward the regionbehind the undisturbed meniscus, and the disturbance will be reinforced.

In forward roll coating with equal roller speeds, the position of thecoating meniscus is near the nip where the roller surfaces are nearly parallelif the capillary number is sufficiently large, as was seen in the precedingsection. In light of the findings of Saffman & Taylor, it is not surprising thatforward roll coating is subject to the ribbing-line instability. According tothe experimental observations of Pitts & Greiller.(1961), Mill & South(1967), and Greener et al. (1980), forward roll coating is stable at sufficientlylow capillary numbers but eventually becomes unstable as the capillary

COATING FLOWS 79

to gravitational effects). Equation (3) predicts an exponent of 1.67 for very small capillary numbers.

Instability of Coating Flows: Ribbing Lines The calculation of the steady, two-dimensional flow field, and sometimes the determination of the range of conditions over which such a solution to the governing equations exists, is often just a prelude to a determination of the stability of the flow. The flow instabili.ty called ribbing lines has received a great deal of attention in the literature. This instability may be observed in roll coating and slide coating, for example. The flow field for ribbing lines is steady but three dimensional. More specifically, the flow field becomes periodic in the third dimension, and the coating itself becomes ribbed and uneven. Such an event is usually intolerable in a coating process, and so the study of this and other instabilities is of considerable importance.

The theoretical calculations and experimental observations ofSaffman & Taylor (1958) provided the first insights into the ribbing instability. Saffman & Taylor used the equations for nearly rectilinear flow, expanded to include the third dimension, to evaluate the stability of a coating meniscus between parallel surfaces (Figure 7) to small disturbances. They found that the flow is unstable if the pressure gradient corresponding to the rectilinear flow upstream of the meniscus, namely dP/dX = 12J.lS(1-q)/W2, is positive.

Because q cannot exceed 0.41 (Ruschak 1982), this is always the case. Arguments by Pitts & Greiller (1961) are useful in trying to understand

this result. They consider the evolution of the disturbance to be quasisteady-state, and they suppose that at some location in the third dimension the coating meniscus moves slightly upstream because of a disturbance. The linear pressure field of positive slope upstream of the displaced meniscus remains the same as that behind the undisturbed meniscus, but it is shifted along the X axis by the distance the displaced meniscus has moved. It follows that at any position X, the pressure behind the disturbed meniscus is more positive than that behind the undisturbed meniscus. Therefore, flow will occur from the region behind the disturbed meniscus toward the region behind the undisturbed meniscus, and the disturbance will be reinforced.

In forward roll coating with equal roller speeds, the position of the coating meniscus is near the nip where the roller surfaces are nearly parallel if the capillary number is sufficiently large, as was seen in the preceding section. In light of the findings ofSaffman & Taylor, it is not surprising that forward roll coating is subject to the ribbing-line instability. According to the experimental observations of Pitts & Greiller(1961), Mill & South

(1967), and Greener et al. (1980), forward roll coating is stable at sufficiently low capillary numbers but eventually becomes unstable as the capillary

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80 RUSCHAK

number is increased and the coating meniscus is drawn toward the nip.Thus the fact that the roller surfaces diverge to some extent evidently has astabilizing effect on the flow field.

The mechanisms by which diverging surfaces stabilize the flow field wereelucidated by Pearson (1960) in his analysis of a spreader and by Pitts Greiller (1961) in their analysis of the forward roll coater for the case a = Following Pitts & Greiller, suppose that at some location in the thirddimension the meniscus moves slightly upstream because of a disturbance.Because the roller surfaces diverge slightly, the radius of curvature of thecoating meniscus must decrease as it moves inward, and thus the pressure atthe meniscus will become more negative by the amount e(2tr/W2)dF/dX,where e is the distance the meniscus has moved and W is the distancebetween the roller surfaces at the position of the coating meniscus in thetwo-dimensional flow field. The pressure behind the undisturbed meniscusat the value of X of the displaced meniscus is more negative than thepressure at the undisturbed meniscus by the amount e dP/dX, where thepressure gradient is that for the nearly rectilinear flow just upstream ofthe coating meniscus in the two-dimensional flow field. If, at this X, thepressure is higher behind the displaced meniscus than behind the un-displaced meniscus, then liquid will flow from behind the displacedmeniscus toward the undisplaced meniscus and increase the disturbance.Thus, comparing the above two terms, the disturbance can grow wheneverdP/dX > (2tr/W2)dF/dX. For instability the pressure gradient must notonly be positive but also sufficiently large. Savage (1977a,b) proved that thisinequality is a necessary but not sufficient condition for instability, andGokhale (1983) derived a stronger necessary condition.

The second stabilizing mechanism follows from the result of Taylor(1963) on the geometric similarity of the flow field near the coatingmeniscus. Taylors result is again that the flow rate Q at the coatingmeniscus is given by S Wq, where W is the distance between the surfaces andwhere q depends only upon the capillary number. Thus, if a disturbancecauses a section of the coating meniscus to move slightly upstream, the flowrate past it is reduced because the surfaces are closer together. A simplemass-conservation argument at the meniscus shows that this response has adampening effect on the disturbance.

Figure 8 shows a flow geometry similar to the spreader considered byPearson (1960), which I have analyzed for this review. The dimensionlessflow rate 2 is taken to be 4/3 to approximate that for the forward roll coater.In Figure 9 the capillary number above which the flow field is unstable tosmall disturbances is plotted against ~, the relative slope between thelinearly diverging surfaces. The curve has purposely been extended tovalues of ~ that are not small. In light of the stabilizing mechanisms

80 RUSCHAK

number is increased and the coating meniscus is drawn toward the nip. Thus the fact that the roller surfaces diverge to some extent evidently has a stabilizing effect on the flow field.

The mechanisms by which diverging surfaces stabilize the flow field were elucidated by Pearson (1960) in his analysis of a spreader and by Pitts &

Greiller (1961) in their analysis of the forward roll coater for the case a = 1. Following Pitts & Greiller, suppose that at some location in the third dimension the meniscus moves slightly upstream because of a disturbance.

Because the roller surfaces diverge slightly, the radius of curvature of the coating meniscus must decrease as it moves inward, and thus the pressure at the meniscus will become more negative by the amount c(2(J/W2)dF/dX, where c is the distance the meniscus has moved and W is the distance between the roller surfaces at the position of the coating meniscus in the two-dimensional flow field. The pressure behind the undisturbed meniscus at the value of X of the displaced meniscus is more negative than the pressure at the undisturbed meniscus by the amount cdP/dX, where the pressure gradient is that for the nearly rectilinear flow just upstream of the coating meniscus in the two-dimensional flow field. If, at this X, the pressure is higher behind the displaced meniscus than behind the undisplaced meniscus, then liquid will flow from behind the displaced meniscus toward the undisplaced meniscus and increase the disturbance.

Thus, comparing the above two terms, the disturbance can grow whenever dP/dX > (2(J/W2)dF/dX. For instability the pressure gradient must not only be positive but also sufficiently large. Savage (l977a,b) proved that this inequality is a necessary but not sufficient condition for instability, and

Gokhale (1983) derived a stronger necessary condition. The second stabilizing mechanism follows from the result of Taylor

(1963) on the geometric similarity of the flow field near the coating meniscus. Taylor's result is again that the flow rate Q at the coating meniscus is given by S W q, where W is the distance between the surfaces and where q depends only upon the capillary number. Thus, if a disturbance causes a section of the coating meniscus to move slightly upstream, the flow rate past it is reduced because the surfaces are closer together. A simple mass-conservation argument at the meniscus shows that this response has a dampening effect on the disturbance.

Figure 8 shows a flow geometry similar to the spreader considered by Pearson (1960), which I have analyzed for this review. The dimensionless flow rate Il is taken to be 4/3 to approximate that for the forward roll coater. In Figure 9 the capillary number above which the flow field is unstable to small disturbances is plotted against ex, the relative slope between the linearly diverging surfaces. The curve has purposely been extended to values of ex that are not small. In light of the stabilizing mechanisms

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COATING FLOWS

Figure 8 Coating meniscus between linearly diverging surfaces.

81

discussed, it is not surprising that the capillary number at which the flowbecomes unstable to small disturbances increases with the slope betweenthe surfaces. The rather substantial divergence of 10 (~ = 0.176) corre-sponds to the small capillary number of 0.054. The stabilizing mechanismsare weak indeed.

1,0

0.1

0,0t

0.0010,001

I I I I I I I1[ I I I I I I IIII I I I I IlL

UNSTABLE

//

//

I STABLE

Figure 9stability analysis following Pearson (1960);following Pitts & Greiller (1961) and Savage (1977a,b).

I I I I I IIII I I I ~ ~lll I I I I I I liO,Ol o,1 I,O

Stability of a coating meniscus between linearly diverging surfaces: linear-necessary condition for instability

Figure 8

COATING FLOWS 81

Coating meniscus between linearly diverging surfaces.

discussed, it is not surprising that the capillary number at which the flow becomes unstable to small disturbances increases with the slope between the surfaces. The rather substantial divergence of 10 (a = 0.176) corresponds to the small capillary number of 0.054. The stabilizing mechanisms are weak indeed.

0.1

c UNSTABLE

/ /

/

0.01

STABLE

/ /

/

/ /

/ /

/ /

O.OOI __ __ ____ L- ____ __ LW 0.001 0.01 0.1 1.0

Figure 9 Stability of a coating meniscus between linearly diverging surfaces: linearstability analysis following Pearson (1960); ---- necessary condition for instability following Pitts & Greiller (1961) and Savage (1977a,b).

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82 RUSCHAK

Also plotted in Figure 9 is the necessary condition for instabilitydetermined by Pitts & Greiller (1961). This condition appears to approachthe exact result asymptotically as the capillary number decreases. There isother evidence that the necessary condition for instability is useful only forsmall capillary numbers. By comparing predictions with experimental data,Savage (1977a) found that the necessary condition for instability gives reasonable estimate of the region of stability for a roller and flat-plategeometry over the capillary number range of 0.06-0.11. On the other hand,when Greener et al. (1980) compared their stability data for forward rollcoating for capillary numbers in the range 0.1-10, they found that thenecessary condition for instability substantially underestimates the regionof stability. However, under the experimental conditions of Greener et al.,the coating meniscus forms so far from the nip that a fully two-dimensionaltreatment is probably in order anyway.

Although not strictly correct, it is instructive to use the stability resultsfor flow in a linearly diverging region (Figure 9) to predict the stability of theforward roll coater. To do this it is necessary only to estimate the slopebetween the roller surfaces where the coating meniscus forms. Since themeniscus forms where f = 2/q, and since f = 1 + x2/2, it follows that thisslope is given by fi[2(2/q- 1)] 1/2. Moreover, 2 is close to 4/3, and the value.of q as a function of c is known (Ruschak 1982). Thus for a given capillarynumber, q can be looked up and Figure 9 can be used to estimate the slopebetween the roller surfaces at which instability occurs. The value of fi thatgives this slope then follows immediately. The results are shown in Figure10, along with the experimental data of Mill & South (1967) and theexperimental correlations of Pitts & Greiller (1961) and Greener et al.(1980). The agreement between the predictions and the data is good. therefore seems that in forward roll coating the stability of the flow dependsprimarily upon the divergence between the roller surfaces where the coatingmeniscus locates : the greater the divergence, the more likely the flow is to bestable.

Computer Methods for Flow SimulationThe flow near a coating meniscus is nearly always two dimensional. This istrue even when, as in roll coating, the major portion of the flow field hasnearly parallel streamlines. Moreover, viscous, inertial, surface tension, andgravitational effects may all be important ; the flow geometry may be highlyirregular and complicated (for example, some fluid boundaries may becompliant) ; and sometimes, as in the photographic industry, several liquidlayers may be applied simultaneously. Often non-Newtonian effects areimportant. Moreover, an analysis of the two-dimensional, steady-state flow

82 RUSCHAK

Also plotted in Figure 9 is the necessary condition for instability determined by Pitts & Greiller (1961). This condition appears to approach the exact result asymptotically as the capillary number decreases. There is other evidence that the necessary condition for instability is useful only for small capillary numbers. By comparing predictions with experimental data, Savage (1977a) found that the necessary conditio:n for instability gives a reasonable estimate of the region of stability for a roller and flat-plate geometry over the capillary number range of 0.06--0.11. On the other hand, when Greener et al. (1980) compared their stability data for forward roll coating for capillary numbers in the range 0.1-10, they found that the necessary condition for instability substantially underestimates the region of stability. However, under the experimental conditions of Greener et al., the coating meniscus forms so far from the nip that a fully two-dimensional treatment is probably in order anyway.

Although not strictly correct, it is instructive to use the stability results for flow in a linearly diverging region (Figure 9) to predict the stability of the forward roll coater. To do this it is necessary only to estimate the slope between the roller surfaces where the coating meniscus forms. Since the meniscus forms where f = A/q, and since f = 1 + x2/2, it follows that this slope is given by J[2(A/q-l)]l/2. Moreover, A is close to 4/3, and the value of q as a function of c is k nown (Ruschak 1982). Thus for a given capillary number, q can be look ed up and Figure 9 can be used to estimate the slope between the roller surfaces at which instability occurs. The value of J that gives this slope then follows immediately. The results are shown in Figure

10, along with the experimental data of Mill & South (1967) and the experimental correlations of Pitts & Greiller (1961) and Greener et al.

(1980). The agreement between the predictions and the data is good. It therefore seems that in forward roll coating the stability of the flow depends primarily upon the divergence between the roller surfaces where the coating meniscus locates: the greater the divergence, the more lik ely the flow is to be stable.

Computer Methods for Flow Simulation The flow near a coating meniscus is nearly always two dimensional. This is true even when, as in roll coating, the major portion of the flow field has nearly parallel streamlines. Moreover, viscous, inertial, surface tension, and gravitational effects may all be important; the flow geometry may be highly irregular and complicated (for example, some fluid boundaries may be compliant); and sometimes, as in the photographic industry, several liquid layers may be applied simultaneously. Often non-Newtonian effects are important. Moreover, an analysis of the two-dimensional, steady-state flow

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0.01 0,1

E/RFi#ure lO Stability of forward roll coating: data by Mill & South (1967);experimental correlation of Pitts & Greiller (1961); -- experimental correlation Greener et al. (1980); theoretical prediction based on the stability of a coating meniscusbetween linearly diverging surfaces (Figures 8 and 9).

is often of little value if the stability of the flow cannot be assessed. Theserequirements are largely beyond the means of classical analysis.

In recent years computer methods for flow simulation have advanced tothe point where most steady-state, two-dimensional flow fields can beefficiently calculated. Kistler & Scriven (1983) recently reviewed thesemethods. They included as examples calculated flow fields for roll,extrusion, slide, and curtain coating. Kobayashi ct al. (1982) did parameter study on the slide-coater flow field and found good agreementbetween a computed and an experimentally obtained flow profile. Kistler

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COATING FLOWS 83

STABLE

0.01---------L---L-L-L------------ 0.001 0.01 0.1

E/R

Figure 10 Stability of forward roll coating : data by Mill & South (1967); - experimental correlation of Pitts & Greiller (1961); experimental correlation of Greener et al. (1980); _ theoretical prediction based on the stability of a coating meniscus between linearly diverging surfaces (Figures 8 and 9).

is often of little value if th e stability of th e flow cannot be assessed. Th ese requirements are larg ely beyond th e means of classical analysis.

In recent years computer meth ods for flow simulation h ave advanced to th e point wh ere most steady-state, two-dimensional flow fields can be efficiently calculated. Kistler & Scriven (1983) recently reviewed th ese meth ods. Th ey included as examples calculated flow fields for roll, extrusion, slide, and curtain coating . Kobayash i et a1. (1982) did a parameter study on th e slide-coater flow field and found g ood ag reement between a computed and an experimentally obtained flow profile. Kistler

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Also plotted in Figure 9 is the necessary condition for instabilitydetermined by Pitts & Greiller (1961). This condition appears to approachthe exact result asymptotically as the capillary number decreases. There isother evidence that the necessary condition for instability is useful only forsmall capillary numbers. By comparing predictions with experimental data,Savage (1977a) found that the necessary condition for instability gives reasonable estimate of the region of stability for a roller and flat-plategeometry over the capillary number range of 0.06-0.11. On the other hand,when Greener et al. (1980) compared their stability data for forward rollcoating for capillary numbers in the range 0.1-10, they found that thenecessary condition for instability substantially underestimates the regionof stability. However, under the experimental conditions of Greener et al.,the coating meniscus forms so far from the nip that a fully two-dimensionaltreatment is probably in order anyway.

Although not strictly correct, it is instructive to use the stability resultsfor flow in a linearly diverging region (Figure 9) to predict the stability of theforward roll coater. To do this it is necessary only to estimate the slopebetween the roller surfaces where the coating meniscus forms. Since themeniscus forms where f = 2/q, and since f = 1 + x2/2, it follows that thisslope is given by fi[2(2/q- 1)] 1/2. Moreover, 2 is close to 4/3, and the value.of q as a function of c is known (Ruschak 1982). Thus for a given capillarynumber, q can be looked up and Figure 9 can be used to estimate the slopebetween the roller surfaces at which instability occurs. The value of fi thatgives this slope then follows immediately. The results are shown in Figure10, along with the experimental data of Mill & South (1967) and theexperimental correlations of Pitts & Greiller (1961) and Greener et al.(1980). The agreement between the predictions and the data is good. therefore seems that in forward roll coating the stability of the flow dependsprimarily upon the divergence between the roller surfaces where the coatingmeniscus locates : the greater the divergence, the more likely the flow is to bestable.

Computer Methods for Flow SimulationThe flow near a coating meniscus is nearly always two dimensional. This istrue even when, as in roll coating, the major portion of the flow field hasnearly parallel streamlines. Moreover, viscous, inertial, surface tension, andgravitational effects may all be important ; the flow geometry may be highlyirregular and complicated (for example, some fluid boundaries may becompliant) ; and sometimes, as in the photographic industry, several liquidlayers may be applied simultaneously. Often non-Newtonian effects areimportant. Moreover, an analysis of the two-dimensional, steady-state flow

82 RUSCHAK

Also plotted in Figure 9 is the necessary condition for instability determined by Pitts & Greiller (1961). This condition appears to approach the exact result asymptotically as the capillary number decreases. There is other evidence that the necessary condition for instability is useful only for small capillary numbers. By comparing predictions with experimental data, Savage (1977a) found that the necessary conditio:n for instability gives a reasonable estimate of the region of stability for a roller and flat-plate geometry over the capillary number range of 0.06--0.11. On the other hand, when Greener et al. (1980) compared their stability data for forward roll coating for capillary numbers in the range 0.1-10, they found that the necessary condition for instability substantially underestimates the region of stability. However, under the experimental conditions of Greener et al., the coating meniscus forms so far from the nip that a fully two-dimensional treatment is probably in order anyway.

Although not strictly correct, it is instructive to use the stability results for flow in a linearly diverging region (Figure 9) to predict the stability of the forward roll coater. To do this it is necessary only to estimate the slope between the roller surfaces where the coating meniscus forms. Since the meniscus forms where f = A/q, and since f = 1 + x2/2, it follows that this slope is given by J[2(A/q-l)]l/2. Moreover, A is close to 4/3, and the value of q as a function of c is k nown (Ruschak 1982). Thus for a given capillary number, q can be look ed up and Figure 9 can be used to estimate the slope between the roller surfaces at which instability occurs. The value of J that gives this slope then follows immediately. The results are shown in Figure

10, along with the experimental data of Mill & South (1967) and the experimental correlations of Pitts & Greiller (1961) and Greener et al.

(1980). The agreement between the predictions and the data is good. It therefore seems that in forward roll coating the stability of the flow depends primarily upon the divergence between the roller surfaces where the coating meniscus locates: the greater the divergence, the more lik ely the flow is to be stable.

Computer Methods for Flow Simulation The flow near a coating meniscus is nearly always two dimensional. This is true even when, as in roll coating, the major portion of the flow field has nearly parallel streamlines. Moreover, viscous, inertial, surface tension, and gravitational effects may all be important; the flow geometry may be highly irregular and complicated (for example, some fluid boundaries may be compliant); and sometimes, as in the photographic industry, several liquid layers may be applied simultaneously. Often non-Newtonian effects are important. Moreover, an analysis of the two-dimensional, steady-state flow

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the details of the flow field of which it is a part, and so it is not generallypossible to measure the apparent dynamic contact angle in one flowgeometry and then apply this angle as a boundary condition in another flowgeometry. The apparent dynamic contact angles reported by Ngan &Dussan V. (1982) for liquid displacing air between glass slides at lowcapillary number are the most direct supporting evidence for this conclu-sion. Ngan & Dussan V. found that the spacing between the slides affectsthe value of the apparent dynamic contact angle. Kistler & Scriven (1982)provided further support by calculating the variation of the apparentdynamic contact angle with changes in flow parameters for curtain coatingat high capillary numbers. Less direct support comes from the experimentalfinding that the speed of onset of air entrainment can be changed by alteringthe flow field (Perry 1967, Levi & Akulov 1964, Levi 1966). It is, in fact, common observation in the coating art that the speed of air entrainmentdepends upon the coating method.

Modeling of the flow near the three-phase line is complicated by thebreakdown of the classical hydrodynamic equations and/or boundaryconditions in this region. Huh & Scriven (1971) showed that the equationsand boundary conditions of classical hydrodynamics make the unac-ceptable prediction that the flow of liquid in the neighborhood of a contactline exerts an unbounded force on the substrate. They suggest that theclassical no-slip boundary condition may not be valid near the wetting line.Although subsequent investigators have shown that allowing slip in theimmediate vicinity of the three-phase line removes this force singularity(Dussan V. 1979), the problems remain of finding a realistic model for slipand specifying a contact angle at the three-phase line.

Hansen & Toong (1971) realized that hydrodynamic forces can cause theslope of the meniscus to change rapidly very near the three-phase line.Indeed, they predicted that significant changes in slope occur over distancestoo small to be resolved with an optical microscope. They conclude that, ingeneral, the apparent dynamic contact angle is not equal to the actualcontact angle at the three-phase line, and that much if not all of thevariation of the apparent dynamic contact angle with speed can beattributed to hydrodynamic bending of the meniscus very near thesubstrate, rather than to changes in the actual contact angle. Some laterinvestigators (e.g. Hocking & Rivers 1982) have assumed that the actualdynamic contact angle does not vary at all with speed, so that changes inapparent dynamic contact angle are entirely attributable to hydrodynamicbending of the meniscus over a region very close to the substrate.

By incorporating the disjoining pressure of Deryagin, Teletzke et al.(1984) modified the classical hydrodynamic equations to include the effectsof fluid microstructure over distances from the substrate so small (less than

COATING FLOWS 85

the details of the flow field of which it is a part, and so it is not generally possible to measure the apparent dynamic contact angle in one flow geometry and then apply this angle as a boundary condition in another flow geometry. The apparent dynamic contact angles reported by Ngan &

Dussan V. (1982) for liquid displacing air between glass slides at low capillary number are the most direct supporting evidence for this conclusion. Ngan & Dussan V. found that the spacing between the slides affects the value of the apparent dynamic contact angle. Kistler & Scriven (1982) provided further support by calculating the variation of the apparent dynamic contact an gle with chan ges in flow parameters for curtain coating at high capillary numbers. Less direct support comes from the experimental finding that the speed of onset of air entrainment can be changed by altering the flow field (Perry 1967, Levi & Akulov 1964, Levi 1966) . It is, in fact, a common observation in the coating art that the speed of air entrainment depends upon the coating method.

Modeling of the flow near the three-phase line is complicated by the breakdown of the classical hydrodynamic equations and/or boundary conditions in this region. Huh & Scriven (1971) showed that the equations and boundary conditions of classical hydrodynamics make the unacceptable prediction that the flow of liquid in the neighborhood of a contact line exerts an unbounded force on the substrate. They suggest that the classical no-slip boundary condition may not be valid near the wetting line.

Although subsequent investigators have shown that allowing slip in the immediate vic init y of the three-phase line removes this force singularity (Dussan V. 1979), the problems remain of finding a realistic model for slip and specifying a contact angle at the three-phase line.

Hansen & Toong (1971) realized that hydrodynamic forces can cause the slope of the meniscus t o change rapidly very near the three-phase line.

Indeed, they predicted that significant changes in slope occur over distances too small to be resolved with an optical microscope. They conclude that, in general, the apparent dynamic contact angle is not equal to the actual contact angle at the three-phase line, and that much if not all of the variation of the apparent dynamic contact angle with speed can be attributed to hydrodynamic bending of the meniscus very near the substrate, rather than to changes in the actual contact angle. Some later investigators (e.g. Hocking & Rivers 1982) have assumed that the actual dynamic contact angle does not vary at all with speed, so that changes in apparent dynamic contact angle are entirely attributable to hydrodynamic bending of the meniscus over a region very close to the substrate.

By incorporating the disjoining pressure of Deryagin, Teletzke et al. (1984) modified the classical hydrodynamic equations to include the effects of fluid microstructure over distances from the substrate so small (less than

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1/~m) that the liquid is no longer homogeneous, They predict that below certain speed no air is entrained, but that above this speed an air film isentrained that is at first