Free coating of non-newtonian liquids onto a vertical surface

Chtmical EngineeringScience.1975.Vol. 30, pp. 379-395. PergamonPres.

Printed in Great Britain

FREE COATING OF NON-NEWTONIAN LIQUIDS ONTO A VERTICAL SURFACE R. P. SPIERS, C. V. SUBBARAMAN and W. L. WILKINSON Postgraduate School of Studies in Chemical Engineering, University of Bradford, Bradford, Yorks BD7 IDP (Receiued 10 December 1973;accepted 24 September 1974) Abstract-The coating of non-Newtonian liquids onto a vertical surface continuously withdrawn from the liquid bath is considered. An analytic treatment is presented for purely viscous non-Newtonial liquids using the Ellis and generalised Bingham models both of which may be reduced to a new theory for power-law fluids. The theories give a relationship between the dimensionless film thickness, T,, and the Capillary number, C,, as a function of the fluid physical properties and the parameters of the viscous model. The dimensionless groups have been generalisedto allow for non-Newtonianbehaviour. The power-lawand Ellis model predictions are compared with previous theoretical studies and shown to be consistent with known limits. Experimental data are also presented for a wide range of non-Newtonian fluids and compared with the new theories.

1. INTRODUCTION The

entrainment of a liquid film on to a belt or plate which is continuously withdrawn from a bath of liquid, illustrated in Fig. 1, is a problem of considerable importante in several industries concerned with the application of liquid coatings to solid substrates. An analysis of the problem of predicting the film thickness as a function of withdrawal speed and the physical properties of the fluid has been presented previously by the present authors[l] for Newtonian fluids. Previous experimental and theoretical work was also discussed. However, many of the fluids used in the coating industries are significantly non-Newtonian in character, typical examples being protective coatings such as polymerie materials or paints, and it is therefore necessary to extend the analysis to tater for materials of this type. Several workers have previously considered the free coating of purely viscous non-Newtonian fluids on to vertical surfaces. Two idealised fluid models have been used, namely the power-law [2-51 and Ellis [6-81 models. The Ellis model is preferable to the power-law as it describes the correct low shear behaviour of many pseudoplastic fluids, low stresses existing at the free surface in the withdrawal process. Also, in most cases, a high shear Newtonian parameter is unnecessary as only moderate shear stresses are encountered at the wal1in the film region. Thus the Ellis model may be used as an adequate description of the viscous behaviour, but it has been observed [9-111that some so called power-law fluids (particularly aqueous pH controlled Carbopol 934 solutions) exhibit a yield stress. This has not been included in any of the analyses presented until now. Following the new analysis for Newtonian fluids by the present authors[l], a new theory for Ellis model fluids is

Fig. 1. Profile of a liquid film adhering to a vertical moving surface.

presented below, while to take the yield stress phenomenum into account the theory for the generalised Bingham model is also derived. It is shown that both of these theories reduce to a new power-law theory which may be compared with previous theories. This shows the widely different results for the various theories and their inconsistencies. In most cases it was originally claimed that no agreement with experimental data was obtained. It is shown that the new power-law theory is consistent with the known theoretical limits and agrees well with previous data for claimed power-law fluids. Both of the new theories are tested, using viscometric data to which the

379

380

R. P. SPIERS,C. V. SIJBBARAMAN and W. L. WILKINSON

fluid models had been fitted, but for a very limited number of the test fluids used. Results for the ftlm thickness for all fluids are, however, presented on a power-law basis and compared with the predictions of the new power-law theory. This, in cormection with the use of the more detailed viscous models, provides a basis for comparing the theory and data but on viscous behaviour only. It is also wel1known that many of the fluids used in this investigation are viscoelastic but as yet no quantitative analysis of these effects in withdrawal has been presented. Indeed only one qualitative examination has been given in the case of cylinder withdrawal[l2] and no quantitative measurements of the fluids’ elastic properties, in this case, were made. Guttìnger[L, 101claimed his aqueous Carbopol 934 solutions to be inelastic on the basis of no observable recoil on cessation of motion. However, only the results for a very low concentration (0.16%) Carbopol solution have been found to.agree with the new power law theory. Gutfìnger also tested one viscoelastic fluid-15% aqueous CMC for which no agreement with theory has been obtained. Roy[l2] has also used Carbopo1934 solutions from which he observes no recoil while the solutions of Carbopol 941 and CMC were observed to be viscoelastic. He also found no agreement between theory [ 131 and experiment. Kapoor [ 1l] has, however, demonstrated small elastic recoil in 0.08% aqueous Carbopol. From the very limited quantitative tests performed here it may be inferred that all the fluids used in the present work are viscoelastic, previously used qualitative tests being insensitive except in the case of gross properties. The role of hydrodynamic and viscoelastic forces has, however, been considered quantitatively in blade coating[l4], where shear rates arë generally several orders of magnitude higher than those encountered in withdrawal, though only the first normal stress differente was considered.

2. DJlVELOPMEh’T OF ELLISMODELTHEORY

become

dp _d-,-+pg

(1)

dy ch

O=!i!P dy with the boundary conditions, assuming that Idh /dx 16 1, u(x, 0) =

u

(9

7&,h)=O +$=

(8 -p -&(x,

h)

(2)

(iii)

where conditions 2(ii) and (iii) represent the resolved shear and normal stresses respectively at the free surface. Their formulation has been discussed for Newtonian fluids by the present authors[l]. Taking the form of the Elhs model used by Gutfmger and Tallmadge 121 -g=

T& t blTJ-1).

(3)

The velocity protlle and flowrate in the dynamic meniscus region are given respectively by: ut,&,)($-,,)+--&

u(y)=

x

(

dp gpg

a )

M-Y)

q= uh-~($+~(~+pg)a*

.+’ -

ha+]1 (5)

The flux, q, in the constant film thickness region is given by:

From the process illustrated in Fig. 1, we may consider the fluid film in three regions as developed _ by_ Landau and Levich[lS] viz. -

(6)

Region 1: zone far removed from the surface of the liquid in the bath and where the film thickness is constant at ho; Region 2: the dynamic meniscus region in which the lìlm thickness, h, varies with height above the bath, x; Region 3: zone close to the bath surface where only surface tension forces are considered-the static meniscus region.

In comparison with the Newtonian theory [ 11it may be seen that to evaluate the viscous component of &(x, h) we must elirninate the pressure gradient, dpldx, between Eqs. (4) and (5). This may not be carried out directly for any a, but if:

Under the assumption of inertia free, one-dimensional flow, the equations in the dynamic meniscus region

Equation (5) may be expanded by the binomial theorem to give

‘5!&1* pg dx

Free coatingof non-Newtonianliquidsonto a verticalsurface

381 (11)

(7)

This, however, gives a rather unwieldy equation for the velocity prollle on substitution into Eq. (4). A simpler but very limited case may be examined by putting (Y= 2 whence Eqs. (4) and (5) may be combined diiectly to give: u(y)=

Under this set of limiting conditions Eq. (10) may be compared with Eq. (3) to give, for Eq. (11): &(x, h)

h).

=a$(X,

(W

Then from Eqs. (5) and (6), the pressure gradient is satisfied, through the normal stress condition (2)(iii), by

U+a(F”2-~)(~-hy) d’h ‘b(-(x,h))] udx’-ä~dx ax

-

+f(F”‘-&)‘((h

-y)3-h31

(8)

=

U(h _ ho) + 3+amho

b(pg)‘ho’

where 4 F=bh’(Uh

-- ah’ 3

4a2 -q)+$-

%$-c&f&

h)),]‘”

&c,

(13)

where Noting that in the dynamic meniscus region we have &($(x,h))=

($&[$(~

-q)+&r

Equation (8) may be differentiated with respect to x, from which, via a suitable form of the Ellis model, we may obtain the viscous contribution to &(x, h), i.e.

$(x,h)=-;F-"'[$+&]g 4q dh 1 az dh -3hZdx+SJz

(9)

Now, from Eq. (3), it is easily seen that the form of the Ellis model chosen is inadequate to calculate the viscous stress &.(x, h) from [c?u/c?x](x,h). We chose, therefore, a more general form which, though not directly comparable with Eq. (3), is of use when the rate of strain is known, i.e.

This is thus an approximate theory for the Ellis model at cr=2only. If we now introduce the genera1 dimensionless groups, which may be simplified for a = 2:

RE; 0

(10) where h is the second invariant of the rate of strain tensor $ and Si/ is the Kronecker delta given as 1 if i = j, 0 otherwise. In evaluating the second invariant we shall ignore the terms (aulax)’ in comparison with (au la y )*,although this may not be strictly valid at the free surface, y = h [see also boundary conditions-Eq. (2)]. We shall therefore make the approximation that, through the continuity equation, the stress, &, at the surface is given by:

T=h 0

!!@”

0 (

T,

=

ho

u

co=p

h+ (

(15)

)

(“=+‘) )

and putting F(L)=$[~($x,h))]

C,

=

(

y;;‘)“”

R. P. SPIER& C. V. SUBBARAMAN and W. L. WILKINSON

382

where F(L) is dimensionless, Eq. (13) becomes:

fx

L’C,

1 d3L

cdR’+y(L) c

Co

11 .

(16)

This may be compared with the theory derived by Tallmadge[6] by putting a = 2 and omitting the stram rate term given by F(L). In his theory, for al1 values of a, Tallmadge invokes a thin meniscus approximation to expand the left hand side of Eq. (16) before linearizing. This may be stated as (for a = 2):

d’L c, dR)

T,,2>’ 1

It is, however, easily shown from Eq. (16) that this necessary condition is invalid for the full range of integration of Eq. (16), i.e. m< R ~0 where R = 0 is located at the bath surface. The thin meniscus approximation may be avoided by adding [-Tla+‘] to both sides of Eq. (16) (putting a = 2) it being noted that, in general, for a being an integer, this is equivalent to completion of the square, tube, etc., of the left hand side. The resulting equation is easily linearized by the introduction of L = 1 + P where E 6 1. Expanding and ignoring terms of O(E’) and above, the modified Eq. (16) may be rearranged to give:

This failure of convergente is probably due to the approximations introduced into the evaluation of &(x, h) from Eq. (10). Removal of these approximations, and having the parameters to describe both forms of the Ellis model, Eqs. (3) and (lO), might solve the convergente problem of F(L). Thus, omitting the non-convergent terms of F(L) we may arrive at a form of Eq. (17) which is compatible with boundary condition (18) giving: F(L) = -2&-)f-&

(+r%).

(19)

A comparison with the power law theory, which may be derived from Eqs. (5) and (6), evaluating [au/&](x, h) from Eq. (4), shows that Eq. (19) is equivalent to evaluating the stram rate from the power-law velocity profile and using the low shear parameter of the Ellis model to calculate the extra stress, &(x, h). Hence, we may obtain in general:

(20) Thus in the case of a = 2, we have from Eq. (17):

3

‘5++2 C, dR

2 F(L) - (g-

(21)

+ T,3n >

+ $ (L - 1)(1- T: - T?) t (2

0

t T13”)2]1’2= 0. (17)

This is the new Ellis model theory for a = 2 only and is equivalent to the new Newtonian theory[l]. It is found, though, that, despite linearization Eq. (17) in incompatible with the Landau and Levich boundary condition: d3L $ndR’=O

(18)

required for the curvature matching procedure. Examination of Eq. (17) reveals the problem to lie in the strain rate term F(L), and it is found that the only term compatible with Eq. (18) is:

or, in the general case, by combining Eq. (20) with the dimensionless forms of Eqs. (5) and (6) and linearizing, using the method given above, we have: ’ d3L 2coti c;dR’-c1at1

(l-

T0~/3_~)$($~)

_ T1(n+l)ja+ T,“+‘+a+2 L x(1-T:-T,“+‘)t3Co

(L _ 1)

L’C,

‘~-%“(1_T,‘n-~

CldR x$

($g)]]]“’

Cl at1

) =o.

(22)

Equation (22) is found to be compatible with the Landau and Levich boundary conditions, Eq. (18), and on

Free coating of non-Newtonian liquids onto a vertical surface integration using the initial conditions

dL d2L lim L = l,z=z=O R-

(23

383

Application of the curvature matching procedure gives a theory similar to Eq. (26) but with the curvature coefficient given by Eq. (29). Further simplification of Eq. (28) omitting the low shear stress parameter, and introducing the transformation:

we obtain: .$= [(a t 2)““C,]“‘R d2L ;m Jjj.5 = ae,(a, Co, c,, T,, T,)

(24)

and putting a = l/n power-law theory:

where aEl is termed the curvature coefficient. A theory for predicting the final film thickness may be obtained under the curvature matching precedure of Landau and Levich[lS] by combining Eq. (24) with the limiting curvature at the top of the meniscus derived from capillary statics (Region (3) of Fig. l), vis. lim % L-dR

= D0fi

(25)

’

(30)

and b”” = l/K, yields a new

n

I

Tl(n+l)/”

2ntl

1n

EO

where T, = ho(pg/IW”)“(“+”and C, = (KU”/uh~“-‘). Integration with the initial conditions (23) gives, for the curvature coefficients:

The new theory for predicting the final film thickness is then given by: T

aEt(a,Co,Cl, To,TI) =‘(a+‘)

=

1

2c1

[

1

(26)

(32) whence, via curvature matching, the power law theory is given by

noting the relationship: Do’=CoTo*=C,T,

(o+Wa

.

-P-c’&?

To compare the predictions of these three new theories it was decided to evaluate the power-law theory for n = 0.5 while the two Ellis model theories were evaluated for the following fluid parameters: a=2

b = 10.0 m’N_* sec-’

(33a)

o=0+06Nm-’ p= lOOOkgm-‘.

(.+‘)” + Tl”” T1

x(1-T:-T,“+‘)+~~R

(33)

a = 1.0 m4N-’ sec-’

d’h

the theory becomes: 1 d’L c;W-

I1 .

*“E a,L(T,, n) c ,,6 “(“+‘) V2

(27)

The predictions of this theory are given below and compared with the predictions of other simplified theories outlined below. Omission of the strain rate term from Eq. (22) reduces the theory to one similar to that given by Tallmadge[6] using the Landau and Levich approximation:

(31)

L3 d3L

(28)

This result was not, however, derived by Tallmadge who used the thin meniscus approximation noted above. Integration of Eq. (28) with boundary conditions (23) gives a curvature coefficient of the form dZL mi dRZ = aEz(a, Co, CI, Ta, TI).

(29)

Choosing a = 2 allows Eqs. (22) and (28) to be rearranged to eliminate the third derivative term in the last term of both equations. The curvature coefficients aEl, aE2and apL were evaluated from Eqs. (22), (28) and (31) using a 4th order Runge-Kutta method[lól with the start@ value procedure of Landau and Levich[lS]. Details have been given previously by the present authors[l]. The three theories were then evaluated from Eq. (26)using the coefficients aEl and aE2for the two Ellis model theories and Eq. (33) for the power-law theory. The results of these theories are tabulated in Table 1 and

384

R. P.

SPIERS,C.

V. SUBBARAMAN and W. L.

WILKINSON

Table 1. Comparisonof Ellis model theories using the parameters given by Eq. (33a) and the new power-Iaw theory, Eq. (33),for n = f CI oMI o+IOO3 ON)06 0.001 0*002 0.003 0.004 0.005 0*006 0.008 0.01 0.03 0.06 0.1 0.2 0.3 0.6 1.0 2.0 3.0 6.0 10.0

T,*

T,t

0.1489 0*1809 0.2036 0.2215

0.1418 0.1740 0.1975 0.2162

0.2636

0.2597

0.2925

0.2893

0.3149 0.3659 0.3999 0.4256 0.4612 0.4823 0.5187 0.5457 0.5820 06031 0.6385 0.6638

0.3122 0.3641 0.3984 0.4244

TA

0.1428 0.1781 0.2067 0.2323 0.2771 0.3112 0.3399 0.3649 0.3872 0.4258 0.4589

0.4816 0.5181 0.5452 Fig. 3. Curvature coefficients, opL.(T,, n )u . rt.

0.6026 0.6380 06634 09

*Power law theory, Eq. (33). tEllis model theory based on uEZ,Eq. (29). $-Ellismodel theory based on aE ,, Eq. (24)

0.9

0.7 06 T

compared graphically in Fig. 2. The power law theory was also evaluated for other values of the index, n, the curvature coefficients, CX~~(T,,n), being given in Fig. 3 and the film thickness predictions in Fig. 4. The values of (IP~ at T, = 0 were found to agree closely with those of Gutfinger[2] for the zero gravity theory. Solutions of Eqs. (22) and (28) were also sought for al1 values of a ((Y> 1) so that comparisons could be made 0.5

I

l

1

I IO+

10-1

0,4-

0.1 10-4

I 10-3 Cl

Fig. 2. Comparison of Ellis model (a = 2) and power-law (n = 3) theories.

0-5 0.4 0.3 0.2 0.1

Fig. 4. Power law theory for various n.

with experimental results for fluids whose viscous properties may be described by the Ellis model. Direct solution of these equations using the Runge-KuttaMerson method was found to be impossible even if iteration was included to evaluate the term d’L/dR3 in the fìnal term. Use of the modified Euler method, giving only second order accuracy, was also precluded when it was again found that the solution failed to converge on a second derivative even at the first step in the integration. This problem was avoided by applying the binomial expansion to Eq. (28) (The result for Eq. (22) is similar in form):

Free coating of non-Newtonian liqulds onto a vertical snrface

the expansion being valid if:

A numerical solution of Eq. (28) was then formulated using Eq. (34) whenever condition (35) was met, this being that the left hand side of Eq. (35) was less than 0.01. Under all other conditions Eq. (28) was used. Although being only second order, the modiied Euler method was used for both equations and comparison was made with the direct solution of Eq. (28) using the Runge-Kutta method with the parameters of Eq. (33a). This showed this proposed method to be valid and accurate for a = 2 with a convergente limit of 0.01 per cent at each step. However, for other values of (Y,it was found that convergente was very slow for Eq. (28) and it was found necessary to mise the convergente limit to 1 per cent at each step. This is particulary true in the theoretical predictions shown in Fig. 18 for the fluid shown in Fig. 9 and because of the crude solution convergente used, this method can only be regarded as approximate. Application of this method to Eq. (22) stil1failed to produce a solution. Other numerical methods are being investigated for more accurate solutions.

385

It is immediately clear from Fig. 5 that the various theories which have been proposed ditIer greatly in their predictions and some serieus inconsistencies are worth noting. Fiist it may be shown from the Newtonian theories of White and Tallmadge[l7] and Spiers et al. [l] that a gravity corrected theory must predict a thinner 8hn than the zero gravity theory of Landau and Levich[lS]. This must also be true for any purely viscous fluid whether it be Newtonian or not, and all the theories refer to fluids of this type. All the curves should, therfore, lie below the zero gravity theories, Curves 1 and V, whereas it is seen that the gravity corrected theory of Gutfinger (Curve 111)lies above, at least for values of the power-law Capillary number, Cr, less then 0.1. The Tallmadge theories (Curves 111and IV) also lie above Curves 1 and V at low values of Cl. The theories represented by Curves 11-IV also failed to agree with data for so-called power-law fluids[2, 3, 6, 71. The so-called empirical ‘pat&’ theories of Tallmadge (Curves VI and VII) correctly lie below Curve 1, i.e. they predict a thinner film than the zero gravity theory for all values of Cr, but they predict that the dimensionless

“OT 0.9 -

3. COMPARJSONOF POWER-LAW THEORIES Because of its simplicity, containing only two parameters, the power-law model is useful for comparison with the previous theories to check the validity to the new theory. It may be noted that for all values of Tr the power-law theory, Eq. (31), has the same rate of convergente to a constant second derivative as L + m, as the zero gravity theory of Gutfmger and Tallmadge[2]. i.e. as

Fig. 5. Comparison of theories for n = 0.5.

hence as Table 2. Theories compared in Fig. 5

d’L L*m,;rlJ+Oforalln. This rate of convergente is not found for any of the Tallmadge gravity corrected power-law theories [2-4]. A comparison of the power-law theories should also provide checks on the validity of the Ellis model theories. The predictions of the new power law theory have, therefore, been compared, in Fig. 5, with the results of the previous theories listed in Table 2, some of which were derived from Ellis model theories, for a flow behaviour index, n, of 0.5.

Curve 1 11 111 Iv

V VI VII VIII

Tbeory Guffinger’s zero gravity theory Guffinger’s gravity corrected theory Talhnadge’s modiied theory (special case of Ellis flnid) Talhnadge’s variable coefficient theory Groeveld’s zero gravity theory Talhnadge’s ‘pat&’ theory (a) Talhnadge’s ‘pat&’ theory (b) Present power-law theory

Reference 2 2 67 3 4 5 5

386

R. P. SPIERS, C. V. SUBBAA MAN and W. L. WILKINSON

thickness, Tl, approaches infinity as the Capillary number becomes large, a condition which cannot be realised theoretically or experimentally. The problem probably arises as a result of using the Newtonian value for the curvature coefficient (i.e. 0643 at T, = 0) for Curve VI, or those from theory IV[3] for Curve VII, for which there is no justifiable basis. It was, however, found that Curve VI agreed with data for a power-law fluid while no agreement was found with Curve VII[S]. The power-law theory developed here, Eq. (33), Curve VIII, is seen to be consistent with the zero gravity theory, i.e. it always predicts a thinner film and is not at variante with the known high speed, inertialess limit. Comparison with experimental data is made below. It is thus found that the Ellis model theories are consistent with the zero gravity and high speed limits by implication. However, it is also possible to demonstrate the consistency of the simplified Ellis model theory, Eq. (28), with the power-law theory, Curve VIII, (see also Fig. 2). The Tallmadge power-law and Ellis model theories[ó] were compared with each other and experimental data by Hildebrand and Tallmadge[7]. In this comparison[7] the power-law theory used was a special case of the Ellis model theory[ó] which predicts a thinner film at high speeds than the power-law theory. It may be argued, however, that one or both of these theories is incorrect from the above comparison. Invoking the following evidente we may also demonstrate the Ellis model predictions to be incorrect. Under high withdrawal speeds it might be expected that high shear stresses exist within the film and hence the low shear parameter of the Ellis model should exert little intluence. Thus the theory should approximate to the corresponding power-law theory. At low speeds, however, the low shear parameter should exert a strong iníluence to give theoretical predictions parallel to, but smaller than either the Newtonian[l7] or corresponding power-law theories. The theories presented by Hildebrand and Tallmadge[7], however, show the opposite effect, while the new theory, Eq. (28), for a = 2, shown in Fig. 2, gives the expected behaviour. The full Ellis model theory for (Y= 2 shows very strong deviation from the simplified theory and may not be valid at high Capillary numbers (the film thickness may become infinite) due to the approximations used in determining the surface strain rate term. Before comparing the new theories with experimental data for a wide variety on non-Newtonian fluids, it has been noted [9-1 l] that pH controlled aqueous Carbopol solutions exhibit a yield stress. Thus to complete the comparison of the experimental data with viscous non-Newtonian fluid theory predictions a theory is required which accounts for the existente of a yield stress.

4. DEVEU>PMENTOFGENERALISEDBINGHM1MODELTHEORY

In this section we develop a withdrawal theory for a fluid exhibiting a yield stress and whose constitutive equation is given by

whence, from Eq. (1) with boundary conditions (2)(i) and (ii), the velocity profile and flowrate are given respectively by: u(y)=

ut

-$[g% tgy-‘{[g%+$

_$)‘”

_ [;

(pp +!$

q=uh-

-2&gpg

+&&

_ZJ”“‘]

tgy’”

-Y) (37)

hylol+l)ln

[+ (pg +~)]““yP””

(38)

where for 7XY 5 Ti, h 2 y t yl in the dynamic meniscus region, yl being given by:

70=

(

pgtg

>

(h-Yd.

(39)

The flux, q, in the constant thickness region is given by:

and 70=pg(ho-Yd

(41)

i.e. for

To determine the pressure gradient in the Jynamic meniscus region we must consider the normal stress boundary condition (2)(iii). However, because of the presence of a yield stress it is impossible to determine the viscosity function to relate the stress, TL, to the stram rate, [au/ax](x, h), at the surface where the shear rate and stress are taken to be zero. We shall, therefore, invoke the Landau and Levich approximation to the pressure, whence combining Eqs. (38) and (40) we obtain:

=

U(h

-

ho)

+_$

Ih

O[ $

hoyO(“+‘)‘”

-&

1

y0(2n+*)‘” (42)

387

Free coatingof non-Newtonianliquidsonto a verticalsurface and 70=(pB-~)(h-y,)=pg(h~-yo).

(43)

Substituting into Eq. (39) we obtain a theory for the generalised Bingham model which satisfies the Landau and Levich boundary condition and may therefore be used in a curvature matching procedure:

Introducing the following dimensionless groups: [ L = h/ho

1 T;“+‘)‘“(,-A)““) =i(L-l)(l; n~~+lv” [cm+l)ln 2n; 1~en+lvn

T,“+’- d’L “” _ T tn+o,n dZ1’

ZI = ; (CP

““+’ 50= y,/ho

’

ntl

C,=KU” uho"

CI= y,/ho.

K

_ (1 _ A)<“+‘)l” -& (

_A)(“+‘)/ M _+L(L

Equations (42) and (43) becoming respectively:

x (1 _ A)(z”+I)h

-&&(L =L

_

&

, +

TF+‘)ln

1

[0(n+‘),n - -

2n: 1 l0(Dl+‘)‘” (45)

[

and 1 d’L l-pd51’ (L-51)=

1-50.

=

c

f

II 7,

T,(“+‘)l”

(1 t

Integration of Eq. (48), noting the problem of evahrating A as outlined above for the Ellis model theory, using the initial conditions (23), yields:

while from capillary statics we have: lim ~ = C,-“‘Doti L-, d{,

2n: 1 l”

giving, from the curvature matching procedure:

l)(1 t e - A)‘“+“‘” T,

=

aGBh

Tl,

[O)

c

d2 -

&

(1 t cz- A)““‘“‘“)]

(47)

where A

=(l-lo,cl-T;l”rs).

Expanding Eq. (47) and ignoring terms of O(c’) and higher we obtain: H = l(l _ T,(“+l)l”(1 _ A)lln) + _&

(49)

(Zn+l)h

&(-+wn__ [

-

1

116 Z’W’)

’

-A)‘2”+1”“)].

’

(50)

Again no analytic solution is possible even though Eq. (48) reduces to the new power-law theory, Eq. (31), on putting l. = 1 (Ti = 0). However, for a Bingham plastic material (n = 1) we may obtain an approximate theory which may be compared with the Newtonian theory of White and Tallmadge 1171. Introducing the approximation l. - 1, we may use as the linearizing functions: L=l+e

Tl(“+‘)/” c@‘+‘)l* K

(1 = lo(1-k e). Equation (45) becoming:

-&l

(48)

(46)

Rearranging Eq. (46) to evaluate 5, and Eq. (45) as above to give a linearized form, we equate the right hand side of the modified Eq. (45) to H and introduce L = 1 + E where Je1& 1. H

-A)‘2”+“‘“].

d’L v-T1

n+1+ Tl(“+‘)l”+ (L - 1) 1

R. P. SPIERS,C. V. SUBBARAMAN and W. L. WILKINSON

n -- 2n + 1 T,0l+lYn 60(n+lSl n+l 2n + 1 x 5J2”+‘)‘” )1

+ _$ [

Ll1(.+r)ln

given in Table 3 and plotted in Fig. 6, for &,= 0*7,0.8 and 0.9. Using the same values of fo, the generalised Bingham model, Eq. (50), is compared with the new power-law theory for n = 0.5, the results being given in Table 4 and in Fig. 7. It is seen that in al1 cases the presence of a yield stress leads to an increase in the film thickness predicted. From the results for the Bingham plastic material (Fig. 6), it may be seen that the approximate theory agrees closely with the full theory at low Ca numbers while at high values a considerable increase in film thickness over the full theory is predicted. If a specific fluid were chosen, however, it

n=o. (51) 11

(*“+l),n --(n Y 1)
Under the approximation & = 1, we find J,= L, whence putting n = 1 and introducing the transformation: X =5’[l_3T~(~_6)]“33’/3.

Equation (50) reduces to the Landau and Levich equation which has a solution of the form:

Table 3. Comparisonof Binghamplastic modeltheories,Eq. (50) [theory (i)] and (52) [theory ($1 for n = 1, with the White and TallmadgeNewtoniantlleory[l7](& = 1) ca

lim 7 L-dx

= 0643.

Applying the curvatnre matching procedure, we obtain an approximate theory for a Bingham plastic material:

where T.

= ho

(aPU>“,

Ca = @

u’

Equation (52) reduces to the White and Tallmadge Newtonian gravity corrected theoryIl71 when & = 1, while putting &,=O, implying plug flow in the film, the Landau and Levich zero gravity theory is obtained. As noted in the evaluation of the Ellis model theory it is possible to calculate a set of genera1 curves (given the ratio TJTI in the case of the Ellis model), or the film thickness prediction for a specific lluid given the viscous model parameters and its physical properties. This applies equally to the generalised Bingham model theories given above. The parameter &(Os S0I 1) may be chosen or evaluated as a function of the thickness TI, given the yield stress and the fluid’s physical properties, although the restrictions on the value of l0 must be observed.

09001 om03 0.0006 om1 om3 0.006 0.01 0.02 0.03 0.06 0.1 0.2 0.3 0.6 1.0 2.0 3.0 4.0 6.0 8.0 10.0

.. (.Y=O.9 &&!*) 1 (ii)

50=0.7 (ii) 6)

0.1984

0.20010.1999 0.23830.2382 0.26560.2655 0.28290.2851 0.28510.28740.2874 0.2898 0.28% 0.33220.3357 0.33580.33910.3395 0.3430 0.3431 0.36630.3707 0.37100.37530.3760 0.3802 0.3809 0.39260.3979 0.39850.40350.4046 0.4095 0.4107 0.4367 0.437704438 044570.4515 0.4539 0.45250.4602 0.46150.46840.4708 0.4773 0.4804 0.49200*5015 0.50340.51170.5155 0.5230 0.5280 0.52150.5325 0.53500.54450.5495 0.5577 0.5647 0.56170.5751 0.57850.58970.5967 060590.6161 0.58520600006041 0.61620.6247 0.6343 06468 0.62460.6420 06474 0.66130.6726 068300.7002 0.65300.6724 0.67880.69410.7078 0.7187 0.7397 0.69000.7124 0.72020.73750.7545 0.7659 0.7930 0.71070.7349 0.74360.76190.7812 0.7926 0.8238 0.7249 0.74430.7715 0.78180.80150.8251 0.8361 0.8750 0.7575 0.76740.7965 0.80830.82880.8560 0.8666 0.9114

b0,

I

I

l

1

I

09

oe 0.7 0.6

-

5. COMF’ARBON WTl’H ZERO YIELDSTRKSSTHEORIE3 To determine the influence of a yield stress on the final film thickness two comparisons have been made. Firstly, for the Bingham plastic material (n = l), the full theory, Eq. (50), and the simplilied theory, Eq. (52), are compared with the White and Tallmadge theory [ 171.The results are

&=0.8 . (ii) 6)

O,I0 10-3

Eq.

(50) -Full

---. Eq. (52)

@21 i0+

1 IO-'

theory

-Simplified l I

theory

-

I 10

102

Cl

Fig. 6. Comparison of Newtonian[l7] and Bingham plastic theories.

Free coating of non-Newtonian liquids onto a vertical surface

389

Table 4. Comparison of generalised Bingham model theory, Eq. (50)with the power-law theory, Eq. (33),for n = 0.5 (,=

06-

TI CI oGOO1 oMI3 o@M6 O+IOl 0033 0+)06 0.01 0.02 0.03 0.06 0.1 0.2 0.3 0.6 1.0 2.0 3.0 6.0 lO,O

Jo= l 0.1489 0.1809 0.2036 0.2215 0,2636 0.2925 0.3149 0.3467 0.3659 0.3999 0.4256 0.4612 0.4823 85187 0.5457 0.5820 06031 0.6385 06638

0.7

-S,.O~El O.?-

&=0.9

l$=O.8

0.2244 0.2678 0.2978 0.3212 0.3545 0.3748 0.4106 0.4379 0.4758 0.4984 0.5375 0.5664 06057 0.6285 06669 0.6947

0.1512 0.1845 0.2083 0.2272 0.2722 0.3035 0.3279 0.3629 0.3842 0.4222 0.4512 0.4918 0.5168 05581 0.5895 0.6323 0.6572 0.6995 0.7301

l0 = 07

Q,= 09

04-

&=l-0

T 057 0.4 0.3 -

0.2304 0.2770 0.3095 0.3351 0.3719 0.3946 0.4349 0.4659 0~5095 0.5357 0.5814 0.6157 0.6626 0.6901 0.7370 0.7711

should be found that, at high speeds (large Ca), the yield stress theories approximate closely to their corresponding viscous model theories. This is because, at high speeds, high shear stresses will exist in the film, and, hence, unless the yield stress is very high, it will exert little influence. Correspondingly at low speeds the influence of the yield stress should be more marked. 6. EXPERIMENTAL PRCKXLWRR

Experimental results for film thickness have been obtained for a wide variety of non-Newtonian fluids by the verticai withdrawal of a continuous flat helt from a bath of the fluid. Details of the fluids used are given in Table 5 and these may be considered in four groups: Carbopol934 in both water and glycerol based solutions without pH control, Separan AP30 in glycerol and water based solutions (considered separately) and aqueous Carbopo1934 with pH control. These fluids allow a wide range of non-Newtonian viscous properties to be covered. Tbe apparatus, which has been previously described by the present authors[l], in studies on Newtonian fluids, consisted essentially of two 20 cm diameter pulleys over which a continuous 5 cm brass helt was stretched. The belt was drawn through the fluid by rotating the lower pulley by means of a variable speed motor. The thickness of the fluid film adhering to the belt was measured by means of a non contact capacitance probe, a technique again described by the authors[l]. The necessary physical properties of the experimental fluids are given in Table 5, the density being determined by hydrometer and the surface tension by ring tensiome-

0.2 ~

OZ’ C3

1

IO-'

IO-

I

10

102

Cl Fig. 7. Comparison of power-law and generahsed Bingham theories (n = f). ter, where possible. The use of this method is ruled out in the case of many of the polymer solutions used here due to their elastic properties, although the method of Roy [12] might be employed. Other methods of determining the surface tension are also ruled out because of the properties of the experimental fluids. A value of Oofj Nm-’ was thus chosen as being representative of the solvents used and should be accurate to within 10 per

cent. The viscous properties were obtained using a rotational concentric cylinder viscometer-a Haake RVl, the temperature being controlled by means of a thermostatically governed water bath. Samples were tested before and after use in the coating rig. The geometry and construction of this instrument, however, limits the shear rate range obtainable to approximately 10 sec-‘-1000 sec-’ without the use of large gaps between the cylinders for which shear rate corrections would be necessary. The viscometric data was fitted to the power-law model, the parameters being given in Table 5 and used in calculating the dimensionless film thickness and speed. As noted above in deriving the various theories, the low shear behaviour of the fluids is important and hence should be determined, the information not being available from measurements in the concentric cylinder viscometer. Five of the fluids used were therefore selected as being representative of those given in Table 5 and tested in a tone-and-plate system on a Weissenberg Model R18 Rheogoniometer. The opportunity was also taken to examine the elastic nature of these fluids. The fluids selected were nos. 4, 8, 16, 18 and 19 of Table 5, these covering the four groups of experimental fluids. Both pH controlled aqueous Carbopo1934 solutions were tested, as these materials have been widely used in withdrawal studies[2, 10,121. The flow curves obtained are given in Figs. 8-12, and in al1 cases the total normal force on the lower platen was measured. The results show a normal pseudo-plastic behaviour for fluids 4,8 and 16 (Figs. 8-10) although the flow curve for 0.6% aqueous Separan AP?”

R. P.

390

SPIER& c.

V.

thJBBARAMAN

and W. L.

WILIUNSON

Table5. Experimentalíluids(non-Newtonian)

Solution

n

1.0% Carbopol-Water 1.9% Carbopol-Water 1.5%Carbopol in 80% Glycerol solution 1.0% Carbopol in 80% Glycerol solution 0.5% Carbopol in 80% Glycerol solution 0.5% Carbopol in Glycerol 1.05% Polyacrylamide in 80% Glycerol solution 0.625% Polyacrylamide in 80% Glycerol solution 0~500%Polyacylamide in 85% Glycerol solution 0.400% Polyacrylamide in 75% Glycerol solution 0.300% Polyacrylamide in 30% Glycerol solution 0900% Polyacrylamide-Water 0.818% Polyacrylamide-Water 0.750% Polyacrylamide-Water 0665% Polyacrylamide-Water O@O%Polyacrylamide-Water 0.475% Polyacrylamide-Water 0.150% Carbopol-Water (Neutralised) 0.258% Carbopol-Water (Neutralised)

I

Iö’ 10-2

IO-'

I

I

I

10

Shear rate,

I 102

I

103

104

sec-’

Fig. 8. Flow curve 1%Carbopo1934in 80/20glycerol/water.

exhibits a pronounced sigmoidal shape. Of the other two fluids, fluid 4 was found to have thickened considerably between use in the coating rig and testing in the rheogoniometer. The viscometric data for fluid 8 (0625% Separan AP30 in 80% glycerol solution) was thus the only data that could be fitted to the Ellis model and allow comparisons of predicted and actual film thickness. For

NTm ksrn

1 2

19.5 19.5

0.330 1.100

0,610 OMI 0.550 o@xl

1000 1000

3

17.2

5.300

0.545 OMI

1240

4

19.5

1.250

0.575 0*060 1150

5 6

17.0 17.5

0.090 2600

0900 0.060 1000 0.950 0.060 1260

7

20.0

IlTJO

0.350 o@xl

8

19.0

0.450

0.775 0.060 1230

9

22.0

0.430

0.750 o+l60 1230

10

20.0

0.150

0.825 O+-hjO 1210

11 12 13 14 IS 16 17 18 19

20.0 20.0 22.8 25.0 26.0 27.5 27.0 17.2 18.0

0.016 5.400 3.500 2,800 1.800 1.320 0.365 0900 ll.000

1.000 0.314 0.367 0.374 0.393 0.400 0.515 0,520 0.342

oTI 0.060 0.060 0.060 0.060 o#jO OMI OQ OTkíO

1300

1000 1000 IOOO 1000 1000 1000 IOOO 1000 1100

these three fluids a measurable normal force was obtained, though only generally at shear rates greater than 10 sec-‘, thus giving in Figs. 8-10, the first normal stress differente uncorrected for inertia effects. Of the two pH controlled aqueous Carbopol solutions (fluids 18and 19, Figs. 11,12), neither of which exhibited a measurable normal force in the shear rate range used, the lower concentration solution was found to approximate to the power law model over a wide shear rate range (Fig. 11). A yield stress may be present in this material, a non zero torque being observed long after the cessation of shear in a similar manner to Gutfinger [ 101,but extrapolation of the rather scattered data at low shear rates (due to evaporation at the surface and the use of a very sensitive torsion bar) proved this to be very small (less than 1 dyne cm-‘). The remaining aqueous solution was found to possess a highly non-linear flow curve with a yield stress of approximately 11.5 dyne cm-*, this being of significance when compared to the expected maximum shear stresses in the film [O(lOOdynecm-‘)]. The low shear (cl sec-‘) behaviour could not be predicted from the high shear rate data (> 10 sec-‘), the low shear behaviour being found to approximate to the generalised Bingham model with n eO.7. As no normal force could be detected in these two

391

Free coating of non-Newtonian liquids onto a vertical surface

‘04r---l-l

;;;_

103-

102 -

Y

.

10’-

E z lo-z10-3

I

l

I

I

I

I

10-2

lö’

I

10

102

103

S hear rate, .

Shearstress. Tl2

10’

sec-’

Fig. ll. 0.15%Carbopol-water (pH6).

+ Flrst normal stress differente. T,, - ~~~

0.26%

N

106 10-z

I IO-'

i

I

I

I

I

10

102

10"

Shear

rate,

‘6 z

Carbapol-Water

(pH

5)

102

sec-’

Fig. 9. Flow curve 0.625%Separan AP30 in 80/20glycerol/water.

5,o-ll lö3 lö2

IC

10

Iö’ Shear

rate,

102 Id

104

sec-1

10

Fig. 12. 0.26%Carbopol-water (pH5). IC 7. COMPARL$ON

10

.

Sheor stress. TIZ + First normal stress differente. T,, - ~~~

Iö

IOlo-

2

I

I

I

I

I

lo-’

I

10

10P

10”

Sheor

rate.

1,

sec-’

Fig. 10. Flow curve 0.6% Separan AP30 in water.

fluids, it was decided to test them tmder oscillatory shear conditions, measuring the amplitude and phase shift of the output torque compared with the input sine wave to the plate. The comparison was made through a digital transfer function analyser connected to the rheogoniometer, the phase shifts corresponding to a viscoelastic fluid response in both cases[ll]. Harmonics were also present in the output. Thus these five íluids are seen to be viscoelastic and by inference the remainder are taken to be viscoelastic also. The viscometric data taken from the two instruments were also found to compare well despite the limitations of the concentric cylinder geometry.

WITH

EXPFXWENT

To compare experimental results and theoretical predictions for the film thickness the dimensionless groups are based on the power-law parameters evaluted from the viscometric data obtained in the Haake viscometer and given in Tabfe 5. In comparisons with other theories the dimensionless groups are evaluated from the model parameters obtained by fitting the necessary model to the viscometric data. To test the new power-law theory comparisons are made with Gutfinger’s data[2], details of the fluids being given in Table áand the present data. As noted above, Fluid 18most closely approximates to the power-law model and is similar in concentration to those used by Gutfinger. A comparison of the power-law theory and Gutfinger’s data is given in Fig. 13 and with the data for the aqueous Carbopol solutions (Fluids 18 and 19) in Fig. 14. It is seen that the results for Fhrids 18 and Gl agree wel1 with the new power law theory and these are compared separately again in Fig. 15 with the power-law theory for n = 0.5. The Tallmadge patch theoryD1 (Curve VI of Fig. 5) for n = 0.5 is also plotted as Tallmadge claimed good agreement of this theory with the data for Fluid Gl. However, as stated above the theory appears invalid theoretically and hence should only be regarded as a cor-

392

R. P.

SPIER~,

C. V. SUBBARAMAN and W. L.

Fig. 13. Comparison of Gutfinger’sdata withpower-law theory.

0.5-

Fluid 18 19 Symbol l A

0.2 -

1 I

I IO-’

IO+

10

Cl

Fig. 14. Comparison of Carbopol 934 (neutralised) data with power-law theory.

WILKINSON

observed for Fluid 19 (Fig. 14) while the theory for the generalised Bingham model, fitted to the low shear viscometric data of this fluid, is not plotted as it will show a considerable increase in the fllm thickness prediction over that using the power-law based on the high shear data. The observed film thicknesses lie considerably below this curve. Taking the remaining fluids in three groups covering the non pH controlled Carbopo1934 solutions, glycerol-water solutions of Separan AP30 and aqueous Separan solutions, we first consider Fluids 1-6 from Table 5. From Fig. 16 it is seen that Fluids 5 and 6 are almost Newtonian and agree reasonably wel1with the new Newtonian theory [l], although, for Fluid 6, a considerable drop in film thickness is found at high speeds. This may be due to significant inertia forces, a breakdown in the static meniscus approximation, or an inertia-elastic forces interaction. The remaining Fluids, 1-4, all give f%n thicknesses considerably below those predicted by the new power-law theory, while, as Fluid 4 has been found to exhibit normal stresses in steady shear, it is reasonable to assume Fluids 1-6 to be viscoelastic. In the case of the Separan AP30glyceroLwater solutions @luids 7-11, Fig. 17), good agreement is obtained with the power-law theory except for Fluid 7. It may also be noted that for Fluids g-10 close agreement is only generally obtained at low speeds with the power-law theory, but, from Fig. 3 it is possible that the overall speed range may be better approximated by the Ellis model

relation. Gutfinger[Z] claimed that other of his Carbopol

solutions (G2 and G3) were power-law in nature but the data do not appear to agree with the new power-law theory, Eq. (33) (Fig. 13). The remaining results for Gutfinger’s íluids also do not agree with the new theory although the data and theory for the viscoelastic solution, GS (1.5% aqueous CMC) approach each other closely at low speeds. However, it is seen that in all cases, except Fluid Gl, Gutfìnger’s results indicate a reduction in film thickness compared with the power law predictions. This is also

Table 6. Free coating of non-Newtonian Auids; Key to the fluids used by Gutfinger[ 101

Sohttion

Key to the Auid

K Ns” m-’

n

Gl G2 G3 G4 G5 G6

0601 2.260 3.440

0*560 0.412 0.374

0.16% Carbopol-Water 0.18% Carbopol-Water O,l% Carbopol-Water 0.20% Carbouol-Water 1.5% C.M.C.lWater 0.40% Carbopol-Water

7.760

0.108

3.140 24.740

0.631 0.244

1.0 0.9

@4-

‘..

06

n-

0.7

7;

0.5 0.6

0.6 0.5 0.4

0.8 TJ

0.3

.’

04

10-3

A’

Fluid Symbol

02 n.,

18 GI l A

“,

I

I

lo-’

10-2 Cl

Fig. 15. Comparison of data with theories for n = f.

I 100

olo-3

I

10+

i

I

10-1

c,

100

1

10'

Fig. 16. Comparison of Carbopol/glycerol/water solutions with power-law theories.

Free coating of non-Newtonian liquids onto a vertical surface

393

theory, Eq. (26), which is approximately equivalent to the authors’ Newtonian theory[l]. To test the theory it was decided to attempt to fit the viscometric data for Fluid 8, obtained from the rheogoniometer (Fig. 9) to the Ellis model. An accurate least squares fit could not be obtained and indeed sensible parameters could only be obtained by fitting the low shear rate data Ilrst and then applying least squares techniques to the remaining power-law part. The following parameters were obtained:

high concentration solution (Fluid 7). Thus, even though Fluid 8 has been shown to exhibit considerable normal stresses, it is possible that, in the speed range tested in the coating rig, the results of Fluids 8-11 are dominated by viscous forces, while the higher concentration of Separan in Fluid 7 may greatly increase in elastic forces in comparison with the viscous forces. This tentative conclusion is partly supported by the results for the aqueous Separan solutions (Fluids 12-17). Closest agreement between experimental results and a = 1.544 the power-law theory predictions is obtained in Fig. 19for Fluids 16 and 17 only, these being the lowest concentraa = 2.263 m’N_’ sec-’ tion solutions. Again the viscous behaviour is probably b = 0.534 rn2- N-” sec-’ better described by the Ellis model, and hence better agreement of theory and experiment might be obtained. although this produced errors of the order of 10 per cent However, the shape of the flow curve obtained for Fluid at parts of the fit. They were used to re-evaluate the 16 (Fig. lO), should also be noted, although a greater range dimensionless film thicknesses and Capillary numbers of of viscometric data might have allowed the fitting of a Fluid 8. They were further used to evaluate the Ellis viscous model and hence film thickness predictions. The model theory [Eq. (28)] using the method of combination remaining results for Fluids 12-15, al1 of high concentrawith Eq. (34) to calculate the curvature coefficients, (YEZ. tion, all lie below the power-law theory. It may also be The result of this theory with the experimental data and concluded that, from the results of the tests on Fluid 16, the power-law theory for n = 0.7 are shown in Fig. 18. see Fig. 10, al1 of these solutions are viscoelastic, and The experimental results are seen to lie below the possibly that in the coating experiments with Fluids 12-17 theoretical predictions as observed with other fluids when presented on a power-law basis, but, despite the errors in the Ellis model parameters for this fluid and the crude numerical solution of the theory, reasonable agreement is obtained. By comparison with the theory for (Y= 2 it may be found that solution of Eq. (22), the complete Ellis model theory, predicts the general upward trend of the experimental data provided that a solution can be found. Thus despite the crudity of the Ellis model in approximating the viscous behaviour of the fluid, the fluid’s viscoelastic properties, and the numerical solution of the theory, limited agreement is obtained for this particular fluid. Use of improved numerical techniques and a more general viscous model would help to resolve the influence of elastic forces. In contrast, a considerable reduction in lìlm thickness in comparison with the power-law theory is obtained for the Fig. 18. Comparison of thickness data and theoretical prediction for Fluid No. 8.

0.8 0.7 6

06 n

0.6

E5

05 04 m3

0.7 04 @9 1.0

02 0-1

Fig. 17. Comparison of Separan AP30/glycerol/water results with power-law theory.

CES Vol 30. NO 44

0.7 0.6

T; 0.5 0.4 0.3 0.2 0.1

Symbol

0 A 0 0 0

Fig. 19. Comparison of Separan AP30/water results with power law theory.

394

R. P.

SPIER%C.

V. SUBBARAMAN and W. L.

the elastic forces are relatively higher compared to the viscous forces than in the experiments with Fluids 8-11 which contain large amounts of glycerol. 8. CONCLUSIONS The new purely viscous non-Newtonian theories derived here are not entirely satisfactory for predicting the film thickness on flat plates withdrawn vertically. For the power law theory it is shown that the new theory is not at variante with the zero gravity theory nor with the high speed limit. Unfortunately, only one fluid (0.15% aqueous Carbopol, pH 6) has been found which closely approximates to the power-law model over a wide shear rate range, results for the film thickness agreeing in genera1 with predictions. However, it is shown that this fluid is similar to the remaining eighteen used in this investigation in being viscoelastic. The new Ellis model theories, particularly Eq. (26), may be used provided that the viscous properties of the fluid are adequately measured. It is again found to give agreement with experimental measurements of film thickness when it may be supposed that viscous forces are dominant. Thus, from the experimental results presented here, it is found that closer agreement between theory and experiment is found for low polymer concentrations in high glycerol content solvents. However, the grade and type of polymer used in preparing these fluids are likely to be of equal importante and hence there is a great need to determine both the viscous and elastic properties of any fluid completely before use in coating experiments. Unfortunately, the few results taken for the selected fluids in the rheogoniometer tests are inadequate to provide correlations of the thickness data taking viscoelasticity into account. It is found, in general, that viscoelasticity tends to reduce the film thickness as compared to that predicted by purely viscous theory. This seems to coníïrm the observations of Roy [ 121in cylinder withdrawal. NOTATION

A a b CO Cl

parameter in generalised Bingham model theory, Eq. (47) zero shear parameter of Ellis model, m2N-’ sec-’ Eq. (3) high shear parameter of Ellis model, rn*=N-” sec-’ Eq. (3) Capillary number for Ellis model, U/aa Capillary number for Ellis model, ( Uhg*-‘/bu’)“u and power-law model KU”/uho”-‘, also generalised Bingham model KU”/uho”-’ Capillary number for Bingham plastic, +/u dimensionless fìlm thickness, h&g/u)l’*

Ca DO (1) er1 rate of strain tensor, sec-’

F F(L) lfi h ho I2

K K’ L n n’ P R” TO

T,

ff x Y Yo Yl

WILKINSON

parameter of Ellis model theory, Eq. (8) surface stress function defined by Eq. (15) gravitational acceleration, m sec-* linearized function in generalised Bingham model theory, Eq. (47) film thickness, m final constant tim thickness, m second invariant of the rate of strain tensor, sec-’ power-law and generalised Bingham parameter, NS” m-* Ellis model parameter, NS” m-* [Eq. (lO)] dimensionless film thickness, hlho power law and generalised Bingham model index Ellis model index, Eq. (10) pressure, N m-* volumetric flowratelunit belt width, m* sec-’ dimensionless x-co-ordinate, x/ho dimensionless film thickness for Ellis model, ho(apg/U)“* and for Bingham plastic, ho(pg/@)‘” dimensionless film thickness for Ellis model, ho(b (Pg)” / u)“(a+l) dso power-law and generalised Bingham model, ho(Pg/Ku”)““+” velocity in x-direction, m sec-’ belt velocity, m sec-’ co-ordinate direction parallel to belt, m co-ordinate direction perpendicular to belt, m parameter in constant thickness region relating to yield stress, Eq. (41). parameter in dynamic meniscus region relating to yield stress, Eq. (39).

Greek svmbols

Ellis model index, Eq. (3) limiting constant curvature for Ellis model theory, Eq. (24) limiting constant curvature for Ellis model theory, Eq. (29) limiting constant curvature for generalised Bingham theory Eq. (49) limiting constant curvature for power-law theory, Eq. (32) Kronecker delta zero shear parameter of Ellis model, Eq. (10) plastic viscosity of Bingham plastic, N s m-2 density, kg rn-’ surface tension, N m-’ yield stress, Nm-’ stress components, N m-* dimensionless x-co-ordinate for power-law theory [(a + 2)““Cl]‘“R

Free coating of non-Newtonian liquids onto a vertical surface 51 dimensionless x-co-ordinate for generalised Bingharn model theory, Cl”‘x/ho x transformed 5, co-ordinate, I3[I 3 Tc,%:/2 - 5:/6)1}” & dimensionless generalised Bingham model parameter, yo/ho 51 dimensiottless generalised Bingbam model

parameter, yJho

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