Ind. Eng. Chem. Res. 1987,26, 475-483
475
Downstream Development of Three-Dimensional Viscocapillary Film Flow Nathan E. Bixler*t and L. E. Scriven Department of Chemical Engineering and Materials Science, University Minneapolis, Minnesota 55455
of
Minnesota,
Analytic solutions of the lubrication and linearized Navier-Stokes equations are constructed for downstream development of three-dimensional steady film flow on a moving web. Variation of disturbances in the cross-web direction is represented by Fourier decomposition. The analyses reduce to eigenproblems: the eigenvalues are the decay rates of the corresponding disturbance modes, i.e., eigenvectors. Solutions to the linearized Navier-Stokes equations are valid for all Reynolds numbers, capillary numbers, and cross-web wavenumbers. The results predict rates of surface leveling and, thus, are useful for the design of coating devices and for three-dimensional stability analyses of coating flows. Extensions are made to upstream spatial propagation of three-dimensional disturbances to steady film flow on a moving web and to upstream propagation and downstream development of three-dimensional disturbances t o steady channel flow. 1. Introduction Downstream of almost every device for coating a flexible substrate, or web, is a zone in which the deposited film asymptotically approaches a fully developed regime. The fully developed regime is usually solid body translation or nearly solid body motion because the force of gravity is usually small in comparison with the viscous drag exerted by the moving web, and furthermore, the role of gravity vanishes altogether when the web is horizontal. Surface leveling, the steady approach to a downstream plug-flow regime, i.e., solid body translation, is analyzed here. The analysis is a three-dimensional generalization of Higgins’ (1982) analysis of the decay of small, two-dimensional departures upstream from the fully developed regime when the deposited film is incompressible Newtonian liquid having uniform viscosity and surface tension. Gravity is regarded as negligible, and thus either capillary forces, viscous forces, or both lead to downstream decay of an upstream departure from plug flow in the film-forming zone. Viscous forces resist liquid deformation and thus play a dual role: (1)they oppose transverse, i.e., cross-web, redistribution of liquid and (2) they cause velocity variations perpendicular to the web to decay. Capillary forces resist departure from a planar meniscus and thus also play a dual role: (1) they promote transverse redistribution of liquid that tends to flatten the meniscus and (2) they counteract downstream, i.e., down-web, meniscus curvature that results from decay of velocity variations perpendicular to the web. Three possibilities are illustrated in Figures 1-3. Figure 1 shows a transversely symmetric (in the z direction) disturbance decaying to a plug profile, which is the problem that Higgins (1982) investigated. Here viscous forces are driving and capillary forces are resisting decay to uniform plug flow. Figure 2 shows the situation when a coating device is either not well aligned or not uniformly fed uniform plug flow is reached only after the liquid is redistributed across a long transverse distance, ordinarily a very slow process. Here the primary effects of viscous and capillary forces are, respectively, to resist and to drive the transverse redistribution of liquid. In Figure 3 the coating is ribbed by a fluid mechanical instability in the forming zone; here the liquid needs to +Current address: Fluid Mechanics and Heat Transfer I, Division 1511, Sandia National Laboratories, Albuquerque, NM 87185.
0888-5885f 87 f 2626-0475$01.50 f 0
redistribute only over relatively short transverse distances. In this situation, the roles of viscous and capillary forces are truly dual because fluid redistributes in the downstream and transverse directions over approximately the same length scale. The present investigation focuses on understanding the rate and m a n n e r of spatial decay of a disturbed flow to its final plug-flow state. There are two primary applications: the first application is to coating die design; the second application is to coating stability. Whereas the aim of most coating devices is to apply a uniform layer of liquid onto a moving web, in practice the liquid film is marred by irregularities due to coating die imperfections or misalignment, mechanical vibrations, pressure nonuniformity or fluctuation, and perhaps even fluid mechanical instabilities-such as the ribbing instability (Pearson, 1960; Taylor, 1963; Coyle et al., 1986). What is crucial is that these imperfections should be kept small so that the fiial film thickness is maintained uniform to within some tolerance. Surface leveling may play an important role in achieving this goal by reducing the amplitude of coating imperfections. Quantitative estimates of leveling rates can aid in the design of coating dies by making clear which disturbances level so slowly that they cannot be tolerated. The present investigation also pertains to the construction of a Robin boundary condition, i.e., one that relates variables to their derivatives, at an outflow boundary of a finite difference simulation or a finite element analysis of a coating flow or other free surface flow. The idea is that knowledge of how a flow decays to the fiial plug-flow regime can be incorporated advantageously into a boundary condition at an outflow boundary. Higgins (1980,1982) proposed a Robin boundary condition which is valid at outflow boundaries where the flow is two-dimensional. His boundary condition was tested and found to be superior to Dirichlet (fixing the dependent variables) or Neumann (fixing the derivatives of the dependent variables) boundary conditions when imposed at the outflow boundary of a slot coater (Bixler and Scriven, 1980). The analysis here extends Higgins’ Robin boundary condition for two-dimensional flows to one for three-dimensional flows. This new boundary condition is useful, not only in analyses of three-dimensional flows but also in analyses of the stability of two-dimensional flows to three-dimensional disturbances. The authors have incorporated this boundary condition advantageously into their analysis of the onset of the ribbing instability in a slot coater (Bixler, 1982; Bixler and Scriven, 1987), and 0 1987 American Chemical Society
476 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987
2. Asymptotic Equations and the Lubrication Approximation The momentum and mass conservation equations for steady viscous flows of incompressible, isothermal, Newtonian liquids are Figure 1. Redistribution of transversely symmetric film.
N E e t * V t= V 2 t - V p v
Figure 2. Redistribution of nonuniformly fed film.
the idea has been extended to other coating flow analyses (Kistler and Scriven, 1984). To the best of our knowledge, only Fall (1978) has previously investigated the downstream decay of threedimensional disturbances. Fall used lubrication theory to analyze the problem. His results are limited to small wavenumbers and large capillary numbers. (Fall’s claim that his analysis is valid for small capillary numbers is shown to be incorrect in section 3.) Higgins (1980) was the first to treat the analogous two-dimensional problem in its full generality. The treatment, to follow in section 4, closely resembles that of Higgins, which, in turn, was inspired by that of Wilson (1969), who examined the approach to Poiseuille flow in a channel. Both Higgins and Wilson were able to use a stream-function representation for their two-dimensional flows; here the three-dimensional analogue of the stream function, a vector potential (Aris, 1962; Morse and Feshbach, 1953), is used to facilitate representation of threedimensional flows. Earlier investigations that were less general than Higgins’ include those of Bretherton (1961),Cox (19621, Coyne and Elrod (1969), Groenveld and van Dortmund (1970), and Ruschak (1974). None of these analyses account for liquid inertia and so represent no more than limiting cases. Two methods are used here to analyze the asymptotic approach to plug flow. The first relies on the lubrication assumptions and leads to an approximate solution; the second constructs a solution of the exact governing equations linearized about the final plug-flow regime. Each method incorporates normal mode analysis to account for variations in the transverse direction (z coordinate). Both methods ultimately reduce to eigenproblems: each eigenvalue is the decay rate of the mode represented by the corresponding eigenfunction. In section 2 the asymptotic equations, boundary conditions, and lubrication equation are laid out. In sections 3 and 4 the respective solutions of the lubrication and exact asymptotic equations are constructed. Section 5 contains limiting formulas and numerical solutions of the exact asymptotic equations. The results are compared with those of Fall and Higgins in section 6. Finally, in section 7 Wilson’s (1969) analysis is extended to include threedimensional disturbances.
t
=0
(2.2)
Here t is the velocity field in the liquid with web speed, U , as the velocity unit; 9 is the pressure field, with yU/h, as the pressure unit, where y is dynamic viscosity and h,, the film thickness of the downstream plug-flow regime, is the unit of length; and NRe Uh,/v is the Reynolds number, where v is kinematic viscosity. The liquid adheres to the moving web and so
t=i
Figure 3. Redistribution of ribbed film.
e
(2.1)
at y = O
(2.3)
At the free surface, where the liquid is in contact with gas, which is taken here to be inviscid and pressureless (i.e., the uniform pressure in the gas is chosen as the pressure datum), the viscous shear stress in the liquid vanishes and normal viscous and pressure stress in the liquid are balanced by capillarity due to interfacial curvature; hence, (bii) x ii = 0 Nc,f:iiii = 2% a t y = h (2.4) Here h is the outward-pointing unit normal to the interface; f is the stress tensor, which, under the restrictions that the liquid is incompressible and Newtonian, is
f = [ V t + ( V t ) T ] - vp
(2.5)
??is the mean curvature of the interface, and Nc, 5 yU/u is the capillary number which expresses the relative importance of viscous forces as compared with capillary forces, u being the surface tension. The interface is taken to be impermeable iieii =
o
at y = h
(2.6)
Far downstream the flow is a solid-body translation: t - j p-0 h-1 asx-m (2.7)
At the upstream end of the asymptotic zone, denoted here by x = 0, the disturbance from plug flow is represented by E u o(u,z) fi = i + cu0 (2.8) Here t is a parameter which represents the magnitude of the disturbance. The position denoted by x = 0 is arbitrary, and thus the size o f t is open. If the point, x = 0, is chosen far enough downstream from a coating device, t can be made as small as the analysis may require, namely, small enough that terms in the equations of motion, (2.1) and (2.2), and boundary conditions, (2.3)-(2.7), which are higher than first-order in t can be dropped. However, higher-order solutions could also be constructed, as discussed by Higgins (1980). The zeroth-order solution, t = 0, of eq 2.1-2.7 is the asymptotic plug-flow regime U=i P=O H = l (2.9) Capital letters are used to denote the base flow. When t is small, the solution of (2.1)-(2.7) can be approximated by t=U+tu p=P+tp h=H+th (2.10) = tVI12h Here u , p , and h are disturbances to the velocity field, pressure field, and free surface elevation, respectively, and
Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 477 satisfy the equation system obtained by neglecting terms of the order greater than E when (2.10) is substituted in (2.1)-( 2.7)
NRe au/ax = v 2 u - vp
(2.11)
v.u = 0
(2.12)
u=O
at y = O at y = 1
(t-j)X j = 0
Ncat:jj = VI,2h j - u = ah/&
(2.13)
at y = 1 at y = 1
(2.15) (2.16)
Solutions of (2.11)-(2.16) are constructed in section 2.4. Equations 2.11-2.16 can be simplified by means of the lubrication approximation (Cameron, 1966), which is valid when the flow is nearly rectilinear and convective inertial forces are small; i.e., the Reynolds number is effectively zero. When this approximation holds, (2.11)-(2.16) reduce to the set a2uII/ay2 = vI,p
(2.17)
v*u= 0
(2.18)
uII-O
h-0
p-0
uII
at y = O
(2.19) j.u = 0
-Neap = VI,2h at y = l
auII/ay = 0
-
uoII
asx-a
at x = 0
(2.20) (2.21)
(2.22)
The subscript I1 again indicates that vectors have components in only the streamwise and transverse directions, x and z. The pressure field can be found by solving (2.17) with boundary conditions 2.19 and 2.20a, integrating that result and (2.18) over y from the web to the free surface, and combining the results of these integrations to eliminate velocity. The result is VI&3V~g/6) = (aK/ax)/2
(2.23)
This differs from the well-known lubrication result by a factor of on the left side because the velocity profile here is semiparabolic rather than parabolic. Finally, because pressure in this approximation is not a function of the filmwise coordinate, y, (2.20b) can be used to eliminate pressure in favor of film thickness: vII.[K3vII(v,~h)]= 3~~~ aK/ax When
t
‘0.5 -
(2.14)
V, is the two-dimensionalgradient which has components in the streamwise and transverse directions, x and z.
u=O
2 6
(2.24)
0.1
(2.25)
(i.e., to first order in t , it reduces to this). To our knowledge, this equation has not appeared in the literature before. It is solved in the following section.
1
10
NCa/N3
Figure 4. Ratio of the slowest decay rate to wavenumber as a function of the ratio of capillary number to wavenumber cubed as predicted by the lubrication theory-eq 3.2, 3.3, and 3.4a-and by the full asymptotic analysis when the Reynolds number is zero-eq 5.1.
and N is the transverse wavenumber of the disturbance or, alternatively, one wavenumber in a Fourier representation of the disturbance. If Nca/W is finite, there are four independent roots of the characteristic equation, (3.2), and thus, there are four independent solutions of the form shown in (3.1). However, only two of these roots are positive, as required in order to satisfy the far-downstream boundary condition, (2.21). When the ratio of the capillary number to the cube of the wavenumber, Nca/W,is small, these are approximated by
a / N = 1 f (3Nca/4N3)1/2
(3.3)
When Nc,/W is large, they are approximately
a/N = W / 3 N c a
CY/N (3Nca/Nj)1/3
(3.4)
The first of the roots in (3.4) matches Fall’s (1978). The solutions of (3.2) at intermediate values of N C a / Pare easily found numerically by Newton’s method. The smallest positive root, a / N , which corresponds to the most slowly decaying mode of disturbance in the lubrication approximation, is shown in Figure 4 for the parameter, Nca/W, ranging from 0.01 to 10. 4. Exact Asymptotic Solution
Any solenoidal vector field, i.e., one that satisfies (2.2), can be represented by a vector potential (Aris, 1962): u=VXA
is small, (2.24) is well approximated by vI+h = 3~~~ ah/ax
0 0.0 1
V*A=O
(4.1)
The vector potential, A , can be made solenoidal without loss of generality, a property known as gauge invariance (Morse and Feshbach, 1953). Taking the vector to be solenoidal simplifies the curl of the velocity field, V X u = V X V X A = -V2A
+ V(V*A)= -V2A
(4.2)
3. Lubrication Solution Solutions of (2.25), which rest on the lubrication approximation, can be constructed by separating variables and have the form h = e-ox[dlcos Nz + d2 sin Nz] (3.1)
a result exploited below. The differential equation which A must satisfy can be found by taking the curl of the asymptotic approximation to the momentum equation (2.11), and using (4.2) to replace velocity
where CY is an exponential decay rate that satisfies the characteristic equation
Solutions, A , for which variables are separable, have the form
[(a/N2- 112- 3 ( N c a / N j ) ( ~ / N )= 0
(3.2)
V4A - NRev2(aA/aX) = 0
A = e-ax[ciSi cos Nz
+ ci’Sisin Nz]
(4.3)
(4.4)
478 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987
Here (Y is again an exponential decay rate; ci and ci'are sets of unknown vector coefficients, and i = 1-4. The functions Sican be assembled in a matrix vector
[Si] = [COS(PY), sin (PY), COS (YY),sin (yy)IT
(4.5)
that conveniently represents the dependence of the vector potential on the filmwise coordinate, y; P and y are
P
(01'
-
W)"'
y
(a2- N2
+ aNRE)'/'
(4.6)
Summation over a repeated index is to be understood in (4.4) and below. Equation 4.4 contains altogether 24 scalar coefficients, three in each of the four ciand three in each of the four c / . However, eight of them are constrained by the requirement in (4.1) that the vector be solenoidal and four more disappear in taking the curl to get the velocity field (discussed more fully by Bixler (1982)). Thus, the velocity field as constructed in (4.1) contains 12 unknown scalar coefficients. It is advantageous to construct the velocity and pressure fields from the vector potential solution in (4.4) because the boundary conditions in (2.12)-(2.15) are then easier to apply. The velocity field can be computed from (4.1) and has the form u = [D.u..C + D / u ,.'CIS.e-ax 1
v
V
E;] p::z E] E
E
D,'
k cos Nz
u l [ sin P -
(4.15)
a2 ; sin y + u2[cos y - cos P I + hl- = 0 O l P
(4.16)
P2 + Y2
u2[ '(2 Y
COS
P ) - COS y
1
+ u4[cos y] = 0
(4.17)
(4.7)
J
Here Diand D/ are
u3 = -u1
[-:'::E]
(4.18)
(4.8)
k sin Nz
uijc and uijlCare the respective entries in the ith row and j t h column of the matrices
[":
UJj
:; U'?
: U'j
:I
w4
p ul'
UZi
U,,,
u;,
u3,
w2
u:,
:q
(4.9)
wql
Because the coefficients uijcand ui,lCare related by (4.1) to vector coefficients ciand ci', only 12 of them are independent. The pressure field can be found by substituting velocity as given by (4.7) into the asymptotic momentum equation, (2.11), and integrating. It has the form p = [cos (Nz)pi+ sin ( N ~ ) p / l S ~ e - (4.10) ~~
Here p i and p / , i = 1-4, are scalar coefficients and are related by the asymptotic momentum equation to the velocity coefficients. Finally, the free surface elevation must have the following form in order to satisfy the kinematic boundary condition 2.13 h = [cos (Nzlh, + sin (Nz)h,']e-""
Equations 4.16-4.19 are four homogeneous equations in four unknowns. A solution exists only if the determinant of the coefficients in these equations is zero. This occurs when
w[ NC*
sin
P Y
cos y - - cos
sin y
(5P4 + 2P2y2+ y4) COS p y4) sin
COS
-+
P + -(P4 + 6P2y2+ Y
P sin y
- 4p2(p2
+ r2)= 0
(4.20)
which is the characteristic equation for decay rate, Eigenvectors (unnormalized) satisfy
(4.11)
The coefficients in (4.7), (4.10), and (4.11) must be chosen so that the governing equations and boundary conditions, (2.11)-(2.16), are satisfied. However, it is convenient to choose first a particular phase of the disturbance in the transverse coordinate, z. A convenient choice is to let a free surface crest be located at z = 0, i.e., ah/& = 0 and d 2 h / d 2 2 < 0 there. With this choice the coefficients that have primes in (4.7), (4.10), and (4.11) drop out. (Were the choice to be to retain only the coefficients with primes, the following results would be identical except that the signs of some terms containing wavenumbers would change; the characteristic equation for decay rate, a, would be identical.) Equations 2.11-2.16 impose the following relationships upon the remaining unknown coefficients
y
1
u, = 2 cos p -
Y2 +
P2
cos 7
(4.21)
Y2 + P2 u2 = 2 sin P - ___ sin y
(4.22)
~
P2 rP
Y u4 = -2- sin P
P
hl =
CY.
-[
NRe
1
Y2 + P2 + ___ sin y
,-j sin P cos y
(4.23)
P2
1 - - cos p sin y
1
(4.24j
and the remaining coefficients are defined in (4.12)-(4.15). There is an infinite number of eigenvalues, i.e., decay rates, that satisfy the characteristic equation, (4.20); they
Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 479 the disturbance because it persists farthest downstream. In this case the eigenfunction expansion is trivial.
Table I. Dependence of the Two Slowest Decay Rates, aI and aII,on Reynolds and Capillary Numbers when the Wavenumber Is 0.1 NQ. 0.1 1 10 100 NC. 0.01 “I 0.0032 0.0033 0.0033 0.0038 0.3135 0.3026 0.1988 0.0217 “I1 0.1 “I 0.0003 0.0003 0.0003 0.0003 0.5356 0.4971 0.2242 0.0248 “I1
5. Limiting Cases and Numerical Results Two methods are employed here to solve the characteristic equation, (4.20). The first is to construct approximate solutions valid for limiting parameter values. The second is to employ secant iteration on a digital computer to track roots as parameters are varied; results of doing so are in Tables 1-111. A useful limit occurs when inertia is negligible compared with viscous forces, i.e., when the Reynolds number is approximately zero. In this limit the characteristic equation, (4.20),reduces to (Bixler, 1982)
~~
1
“I “I1
10
cy1
100
“I1 “I “I1
0 0.6959 0 0.7313 0 0.7353
0 0.6217 0 0.6467 0 0.6495
0 0.2287 0 0.2291 0 0.2292
0 0.0250 0 0.0251 0 0.0251
are generally complex numbers. Only eigenvalues with positive real parts correspond to modes that decay with downstream distance and obey the far-downstream boundary condition, (2.14). A unique solution for a given upstream boundary condition, (2.15), could be constructed from the above analysis as follows: (1) the velocity profile at the upstream boundary is expanded in the Fourier representation in the transverse coordinate, z; (2) for each wavenumber in the Fourier expansion, the set of decay rates that obey the characteristic equation, (4.17),and have positive real parts is found, and the corresponding eigenvectors as defined in (4.21)-(4.24), (4.12)-(4.15), (4.7), (4.10), and (4.11) are computed; and (3) each Fourier component of the upstream velocity profile is then expanded in the infinite set of eigenfunctions. The final result is a double expansion: one is a Fourier expansion; the other is an expansion in the eigenfunctions defined above. It is noteworthy that the differential equation, (4.3), is not self-adjoint and thus eigenfunctions defined in (4.21)-(4.24), (4.12)-(4.15), (4.7), (4.10), and (4.11) are not an orthogonal set on the interval, y = [0,1]. Th’is means that a biorthogonal expansion would be needed as explained elsewhere (Bixler, 1982). On the other hand, if the upstream boundary denoted by x = 0 is far enough downstream of the coating device, the perturbation to the flow field is well approximated by the dominant mode of
2a[cos2 p - 6’1
+ [ p cos p sin p - P 2 ] / N c a= 0
When wavenumber is zero, this is identical with Higgins’ (1982) eq 3.4. Equation 5.1 is easily solved for limiting cases of large and small capillary numbers. When capillary forces dominate viscous forces so that the capillary number is small, the dominant decay rate is = N - (3Nca/N)”’ for N,, > 1, N = NRe = 0 (5.6) Another useful limit occurs when inertia overwhelms viscous forces, i.e., when the Reynolds number is very large. In this limit the characteristic equation, (4.20), reduces to sin p cos y - (P/r)cos p sin y = 0 for NRe >> 1 (5.7) When the wavenumber is large, this further simplifies to for N >> 1, NRe >> 1 (5.8) (tan y ) / y = 1 / N for which the slowest decay rate is approximately =
(P+ a2)/NR,
for N
>> 1, NRe >> 1 (5.9)
When the wavenumber is small, the slowest decay rate is given by (5.3), provided that N4NR,/Nca ’ 4
:a: x-VELOCITY, U
y-VELOCITY, v
I-VELOCITY,
w
Figure 9. Scaled x - , y-, and z-velocity profiles of first mode for NRe = 100 and N,, = 1: (--) N = 0.501;(--) N = 1; (--) N = 2; ( - - -) N = 3.16;(-.-) N = 10.
0 0.1
1 .o
10.0
0
WAVE NUMBER, N
Figure 7. Behavior of the first five spatial decay rates, a1-a5, 88 the wavenumber varies when NRe = 100 and N,, = 1: (-) real part; (-- -) imaginary part. ORTHOGRAPHIC PROJECTION OF STREAKLINES I NR.= 100
0.333
I
YLLZJLJ
0.0 0.000 0 . 3 3 3 0 . 6 6 7 1.000 TRANSVERSE LENGTH, N z / T
0.0
I
I
1.o DOWNSTREAM LENGTH, ax
U lli
t
K
>
a
4
Ya
J
2.0
Figure 8. Portrait of six streaklines of the most slowly decaying disturbance in a translating liquid film.
The velocity components are scaled so that their maximum values are of unit magnitude. A t small values of the wavenumber, below the value at which the first two eigenvalues merge, the velocity component profiles of the dominant mode are nearly parabolic in the direction perpendicular to the web. However, the profiles become increasingly complicated as the wavenumber rises. When the wavenumber is 10, the motion is confined to a thin layer near the top of the liquid film, the layer having a thickness of about one-half the transverse wavelength, or about x/lO. This spatially oscillatory disturbance is unable to reach the web because viscous diffusion is relatively slight. Lamb’s (1945) analysis of two-dimensional standing waves on an inviscid moving stream is analogous. His result shows that the amplitude of the motion varies as e-N(l-Y)when the wavelength is short compared with the stream depth. Thus the magnitude of the velocity at a depth x / N below the free surface is only about 4% of that at the free surface. A t an intermediate Reynolds number and a small capillary number, the first two decay rates again merge and form a complex pair, as shown in Figure 10. However, in this case the complex part of the slowest decay rate is much smaller than the real part, and therefore the dominant mode decays over a much shorter length than one down-
0.1
1 .o
10.0
WAVE NUMBER, N
Figure 10. Behavior of the first six spatial decay rates, al-a6, as the wavenumber varies when NRe = 1 and Nca = 0.01: (--) real part, (- - -) imaginary part.
stream wavelength. Standing waves would likely be easily detected by the naked eye only when the imaginary part of the slowest decay rate is at least of the same order as the real part. With this criterion, the results indicate that standing waves should be “observable” only when the Reynolds number is much greater than unity and the capillary number is of the order of unity or less. Comparison of Analyses. The slowest decay rates predicted by the lubrication theory in section 3 and the exact asymptotic analysis in section 4 are compared in Figure 4. The two predictions are indistinguishable when the Reynolds number is zero (as it is taken to be in the lubrication theory) and when the wavenumber is 0.1 or less. At a larger wavenumber, the results differ markedly, the full theory predicting much slower decay rates than those of the lubrication theory. Fall’s (1978) solution, shown in Figure 4, is in good agreement with the general lubrication result of (3.2) only when Nc,/N3 is large, and therefore Fall’s assertion that his result is valid at small capillary numbers is false. Rate of Spatial Decay. The results in Tables 1-111 can be used readily to estimate the downstream distance needed to achieve a certain degree of leveling of a small disturbance. Table IV shows an example of the downstream distances needed for disturbances of wavenumbers 0, 0.1, and 1 to decay by a factor of 10 when a coating device is operating at Reynolds and capillary numbers of
482 Ind. Eng. Chem. Res., Vol. 26, No. 3, 1987 Table IV. Downstream Distances for Disturbances To Decay by a Factor of 10 when the Reynolds Number Is 1 and CaDillarv Number Is 10 wavenumber, wavelength, decay dist, Na 2xlNb (In 10)/cuab 0 (2-D) 3 0.1 62.8 700 000 1 6.3 190 a
Inverse film thickness. *Film thickness.
1and 10, respectively. As expected, long-wavelength (small wavenumber) disturbances decay most slowly, and twodimensional disturbances (zero wavenumber) decay most rapidly. Disturbances with a wavenumber near unity, such as those generated by a ribbing instability, decay by a factor of 10 after traveling a few hundred film thicknesses downstream and therefore may be tolerable if there is a long enough run of web downstream of the coating station. Construction of Robin Boundary Conditions. Higgins (1982) has shown how to construct a Robin condition, i.e., a boundary condition of the third kind, a t the outflow boundary to employ in numerical simulation of a coating flow when that flow is two-dimensional. Authors have tested a Robin condition based on Higgins’ analysis in solving a viscous-free-surface problem by the Galerkin/ finite element method (Bixler and Scriven, 1980; Bixler, 1982) and found that it improves accuracy as compared with Neumann or Dirichlet conditions, which had typically been used in earlier numerical treatments of related flows. The analysis of section 4 and results of section 5 permit construction of a Robin boundary condition which is useful for numerical solution of the equations that describe three-dimensional free surface flows. Use of a Robin outflow condition is particularly appealing in performing a three-dimensional, linear stability analysis of a two-dimensional flow because small disturbances to a base flow can be represented by a Fourier normal mode decomposition in the transverse coordinate (Bixler, 1982); thus, individual Fourier components can be singled out for study. This simplifies construction of the Robin outflow condition because only one wavenumber needs to be considered a t a time. In this case, a boundary condition of the same form as Higgins’ eq 4.5 applies: & V u = -K.(u - 13 i.VIIh = -k(h - 1)
The simplest choices for the proportionality constants, K and k, are K = aII and k = cyI. In practice it may be necessary to construct a Robin outflow condition which relates the normal and shear stresses to the velocity components, as described elsewhere (Bixler, 1982). 7. Extensions of the Exact Asymptotic Analysis The methods used in this paper to construct asymptotic solutions in a downstream free-surface zone apply equally well to an upstream free-surface zone or to upstream or downstream channel zones. The analysis of an upstream free-surface zone is identical with that culminating in (4.9)-(4.18), except that the relevant roots of the characteristic equation, (4.17), are ones with a negative real part. The analysis of the spatial decay of three-dimensional disturbances in a channel zone with upstream or downstream distance differs from that in a free-surfacezone only by the boundary conditions at the upper surface. Therefore, the methods of section 4 can be applied to extend Wilson’s (1969) analysis of the spatial decay of flow disturbances in a channel to include three-dimensional disturbances. The extension is straightforward when inertia is negligible (NRe= 0). Wilson’s eq 3.3 applies, but with 2ao replaced by /3 (a2- W)lI2. (Here the solid boundaries
are at y = 0 and 1instead of y = -1 and 1, as Wilson used, which accounts for the factor of 2.) The characteristic equation is then sin p = Itp
(7.1)
The first root of 7.1 is =N
(7.2) and larger roots can be found from Wilson’s Table I by using the relationship “1
+
= [(2aoi)2 W ] 1 / 2 i = 2, 3, etc.
(7.3)
where cyoi is Wilson’s ith eigenvalue, cyo. Solutions a t a finite Reynolds number can be constructed as well. The asymptotic governing equations for this case are [e .
ub) au ax + v i d udY b’]
= -vp
+ v2u
(7.4)
v-u = 0 (7.5) and the no-slip and no-penetration boundary conditions in combination are
at y = O , 1 (7.6) In this case, the methods of section 4 lead to a fourth-order ordinary differential equation with nonconstant coefficients for which solutions are best constructed by numerical means. u = O
Acknowledgment We thank B. G. Higgins for the inspiration to undertake this work and also for helpful comments and criticisms. This research was supported by a National Science Foundation Grant for research in capillary hydrodynamics and coating flows.
Nomenclature A = vector potential h = dimensionless perturbation to local film thickness bm = final film thickness h_ = dimensionless local film thickness % = dimensionless local mean curvature of the interface i, j , k = unit vectors in x , y , and z directions, respectively fi = outward-pointing unit normal to the interface N = transverse wavenumber Nca = capillary number, F U / u NRe= Reynolds number, Uh,/v p = dimensionless pressure perturbation p = dimensionless pressure u = dimensionless velocity perturbation ij = dimensionless velocity U = web speed x , y, z = dimensionless coordinates Greek Symbols cy
= complex decay rate
parameter defined by (4.6) y = parameter defined by (4.6) (3 = t
= dimensionless amplitude of perturbation
= dynamic viscosity, Pa s v = kinematic viscosity, m/sz u = surface tension, N/m
Literature Cited Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics; Prentice-Hall: Englewood Cliffs, NJ, 1962; p 70. Bixler, N. E. Ph.D. Thesis, University of Minnesota, Minneapolis, 1982. Bixler, N. E.; Scriven, L. E. Bull. Am. Phys. Sac. 1980,25(9), 1079. Bixler, N. E., Scriven, L. E., submitted for publication in J . Comp. Phys. 1987.
Ind. Eng. Chem. Res. 1987,26, 483-488 Bretherton, F. P. J. Fluid Mech. 1961,10(2),166. Cameron, A. Principles of Lubrication, Longmans: London, 1966; p 49. Cox, G. B.J . Fluid Mech. 1962,14(1),81. Coyle, D. J.; Macosko, C. W.; Scriven, L. E. J. Fluid Mech. 1986,in press. Coyne, J. C.; Elrod, H. G. J. Lubr. Technol. 1969,91(4),651. Fall, C. J. Lubr. Technol. 1978,100(4), 462. Grovenveld, P.; van Dortmund, R. A. Chem. Eng. Sci. 1970,25(10), 1571. Higgins, B. G. Ph.D. Thesis, University of Minnesota, Minneapolis, 1980. Higgins, B. G. Znd. Eng. Chem. Fundum. 1982,21(2),168.
483
Kistler, S. F.; Scriven, L. E. Znt. J. Num. Meth. Fluids 1984,4(3), 207. Lamb, H. Hydrodynamics, 6th ed.; Dover: New York, 1945;p 464. Morse, P. M.; Feshbach, H. Methods of Theoretical Physics; McGraw-Hill: New York, 1953;Part I, p 210. Pearson, J. R. A. J. Fluid Mech. 1960,7(4), 481. Ruschak, K.J. Ph.D. Thesis, University of Minnesota, Minneapolis, 1974. Taylor, G. I. J. Fluid Mech. 1963,16(4), 595. Wilson, S.J . Fluid Mech. 1969,38(4),793. Received for review June 14,1985 Accepted June 24, 1986
Mass Transfer in Wetted-Wall Column with Cocurrent Laminar Liquid-Liquid Flow Satoru Asai,* Jun’ichi Hatanaka, Toshiya Kimura, and Hidekazu Yoshizawa Department of Chemical Engineering, University of Osaka Prefecture, Sakai, Osaka 591, J a p a n
Film- and continuous-phase mass transfers in liquid-liquid systems were studied using a cocurrent laminar wetted-wall column of a modified form. The film-phase mass-transfer coefficients, obtained by dissolution of MIBK, 1-butanol, and cyclohexanol in water, were in good agreement with the Beek-Bakker model, demonstrating the advantage of this column because of its negligible end effect. The continuous-phase mass-transfer coefficients were measured by the transfer of I2 dissolved in continuous phases of ethylhexyl alcohol, toluene, and n-hexane into the interface, where I2 disappeared by an instantaneous irreversible reaction with Na2S203in the aqueous film phase. These coefficients were in satisfactory agreement with the penetration theory when the driving force of solute I2 was evaluated by the method proposed in this paper. In all fundamental studies for liquid-liquid mass transfer, it is desirable to use an experimental apparatus in which the interfacial area and hydrodynamics are well defined. However, few such apparatus have been available. As one kind of suitable apparatus, cocurrent laminar wetted-wall columns have been used by Maroudas and Sawistowski (1964) and Bakker et al. (1967) for mass transfer in binary and multicomponent liquid-liquid systems. Their data were compared with the penetration theory. However, these types of wetted-wall columns are subject to an end effect caused by accumulation of surface-active impurities near the lower end of the wetted wall, and appropriate corrections for this effect must be made. Otherwise, intentional spilling of the film phase over the edge of the run-off tube into the continuous phase, which is not generally practiced, must be used to prevent accumulation of impurities. Unfortunately, however, a reasonable correction to account for the end effect is unlikely to be possible when the mass transfer is accompanied by chemical reaction, since the equivalent length of the end effect varies in an unpredictable manner with the ratio of the reaction rate to the diffusion rate of the transferring solute (Hikita et al., 1967). Furthermore, in previous studies (Maroudas and Sawistowski, 1964; Bakker et al., 1967), the flow rates of both phases were adjusted so that negligible velocity gradients were produced across the interface in order to demonstrate the application of the penetration theory. Hence, the effects of velocity gradients due to viscous drag created by the adjacent phase have not been established. In addition, the mass-transfer coefficients in the continuous phase have not been directly measured. This work presents individual measurements of filmand continuous-phase mass-transfer coefficients in the 0888-5885/87/2626-0483$01.50/0
presence of any velocity gradient at the liquid-liquid interface, using a cocurrent laminar wetted-wall column similar to the one used for gas-liquid systems by Hikita et al. (1967,1976). It has been shown to have a negligible end effect. The observed mass-transfer coefficients are discussed from the aspect of comparison with the model of Beek and Bakker (1961).
Experimental Section The wetted-wall column used in this work is depicted in Figure la. The falling film was formed on the surface of a vertical glass rod 13 or 14 mm in diameter. The rod had a hemispherical end. The internal diameter of the column was 46 mm. The annular space between the rod and column wall was filled with the continuous phase, presaturated with water. The film-phase liquid entered the column through an annular space between a nozzle of 15.6-mm i.d. and a wetted-wall rod, flowed onto the surface of the wetted-wall rod in a laminar liquid film without rippling, and ran down a slender rod 3 mm in diameter into the run-off tube. The liquid level in the run-off tube was adjusted so that the stagnant film of surface-active impurities, building up from the liquid level, just covered the falling film on the slender rod, as shown in Figure lb. Thus, the end effect due to the presence of the stagnant film could be expected to be negligible. The hemispherical end of the wetted-wall rod was taken to be equivalent to a cylindrical vertical rod with length equal to 0.46 times the diameter of the rod. This has been confirmed theoretically and experimentallyfor gas-liquid systems (Hikita et al., 1967). Physical properties of the systems used are shown in Table I. In the experiments on film-phase mass transfer, the dissolution rates of three organic solvents were measured 0 1987 American Chemical Society
