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Asymptotics of a Pinhole
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
191, 514–516 (1997)
CS974982
NOTE Asymptotics of a Pinhole z r 0 as r r ` ,
It is well known that, in a thin liquid film, corresponding to any particular thickness there is precisely one axisymmetric equilibrium solution to the Laplace–Young capillary equation; i.e., corresponding to each film thickness there is a unique pinhole radius which has been shown to be unstable (1973, Taylor and Michael, J. Fluid Mech. 58, 625–639). We deduce an asymptotic relationship between film thickness and pinhole height in the limit of very thin films. q 1997 Academic Press Key Words: asymptotics; surface tension; contact angle; pinhole.
The (unstable) pinhole shape in an otherwise uniform horizontal liquid film of infinite extent is modeled by the Laplace–Young capillary equation. We consider the axisymmetric situation where the film thickness H, the liquid density r, the liquid/air surface tension s, and acceleration due to gravity g are assumed to be known. In (5), a different formulation is used where the pinhole size is assumed to be known; the present formulation is more natural as film thickness can more readily be experimentally determined. The alternative formulation is more convenient if one wishes to extend to higher orders. This work may be considered an extension of that of ( 1, 3, 4, and 6 ) who considered asymptotic expressions for the shape of the meniscus near a small cylinder. Their results may be interpreted in the context of a pinhole where the contact angle u £ p / 2. We extend here to the more general case where 0 õ u £ p and the meniscus turns back on itself ( see Fig. 1 ) . This situation is not covered by the work of the aforementioned authors. Mathematically the extension of the solution is traceable to the sign ( zr ) in [1] , which changes sign when the orientation of the profile passes through the vertical. We use cylindrical polars r , z with z measured downward from the top surface ( Fig. 1 ) . Nondimensionalize lengths with the thickness of the layer, so that the lower solid surface is z Å 1. Balancing the capillary pressure jump across the curved surface with the hydrostatic pressure in the liquid, the equation for the free surface is
S
zrr zr / ( 1 / z 2r ) 3 / 2 r ( 1 / z 2r ) 1 / 2
D
/ e 2z Å 0.
/
6g2e$$$$$1
[2]
2. SOLUTIONS
2.1. Inner Zero-Curvature Region
z Å z0 | r0 ln
07-11-97 07:49:52
S
q
r / r 2 0 r 20 r0
D
,
[3]
with 0 on the upper branch and / on the lower branch. Applying the contact angle boundary conditions gives
z0 Å 1 / r0 ln cot( u /2).
[4]
Note that this solution becomes invalid if u approaches p. In this region a new scaling is required (5). This is of little practical significance and a brief outline of the solution in this case is included in Section 4 for completeness. In addition, note that the radius of the pinhole measured on the substrate is ru Å r0 csc u.
2.2. Outer Capillary-Gravity Region In this region the slopes are small. Making the appropriate stretching of the radial coordinate R Å er, the governing equation becomes a Bessel equation
0zRR 0
1 zR / z Å 0. R
[5]
The solution which tends to zero at large distances is
z Å A( e )K0 (R).
514
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
JCIS 4982
for some r Å ru ,
where u is the liquid/air contact angle (see Fig. 1). We are interested in finding how the minimum radius of the pinhole r0 varies with e.
[1]
Here e 2 Å rgH 2 / s is the inverse of the Bond number and may be regarded as being the square of the undisturbed film thickness made dimensionless with the capillary length. For a 40-mm layer of water-like coating, e É 0.07, which justifies an asymptotic analysis. The boundary conditions are
AID
zr Å 0tan u at z Å 1,
Near the pinhole, one can ignore to a first approximation the final gravity term in the governing equation, the surface thus having zero net curvature in this approximation. The axisymmetric solution is easily found to be
1. INTRODUCTION AND FORMULATION
sign ( zr )
and
coidas
[6]
515
NOTE
4. BOTTOM CAPILLARY-GRAVITY REGION In the case where the contact angle approaches p, the mensicus again approaches the horizontal and the inner scaling is no longer appropriate. In this region a new small slope approximation is needed with the stretching R Å er giving rise to a new Bessel equation: zRR /
1 zR / z Å 0. R
[12]
This has general solution z Å B( e )J0 (R) / C( e )(Y0 )(R). FIG. 1. Schematic of a pinhole.
[13]
In this instance the slope condition [ 2 ] must be applied to the bottom solution rather than the inner solution. There is a boundary layer of O ( e ) in u ( 7 ) , so the slope boundary condition can be written as
2.3. Matching As r r ` the inner solution is
ezR Å 0tan u Å 0tan( p 0 eU ) Ç eU
z Ç 1 / r0 ln cot( u /2) 0 r0 ln(2r/r0 ).
[7]
to O ( e ) where U is a known O ( 1 ) constant defined by u Å p 0 eU. This condition applies when z Å 1 and R Å RH for some ( unknown ) value of RH . Applying this to [13 ] we obtain the following two conditions:
[8]
B( e )J0 (RH ) / C( e )Y0 (RH ) Å 1;
As R r 0 the outer solution is z Ç A(ln(2/r e ) 0 g ),
[14]
U Å 0 B( e )J1 (RH ) 0 C( e )Y1 (RH ).
where g is Euler’s constant. Balancing the ln r terms gives A Å r0
[9]
and balancing the constant terms gives
[15]
On matching [13] to the lower branch of the inner solution [3], two more conditions are obtained, r0 Å 2C( e )/ p;
1 / r0 ln cot( u /2) 0 r0 ln(2/r0 ) Å A(ln(2/ e ) 0 g ).
z0 / r0 ln(2/r0 ) Å B( e ) / 2C( e )/ p(ln( e /2) / g ),
[10]
[16]
while matching the top capillarity-gravity solution to the upper inner solution yields two more conditions:
3. RESULTS Rearranging [10] we have
A( e ) Å r0 ; r0 (ln(2/ e ) 0 g ) Å z0 0 r0 ln(2/r0 ). 1 r0 ( e ) Å . ln(1/ e ) / ln(1/r0 ) / 2 ln(2) 0 ln cot( u /2) 0 g
[11]
Note that up to this stage the errors are O( e 2 ). Now one could solve the last equation iteratively for r0 ( e ), but that introduces an awkward secondary expansion in ln ln(1/ e )/ln( e ). Hence it is best to solve [11] numerically; i.e., given a particular undisturbed film thickness H and contact angle u, calculate the inverse Bond number e 2 Å rgH 2 / s and then solve [11] numerically for r0 . For the particular case where the contact angle u Å 3p /4, here are some typical results and a comparison with the corresponding numerical results of (2).
e
r 0 ( e) (asymptotic)
r 0 ( e) (numerical)
0.001727 0.01006 0.04737 0.10119 0.2071
0.096217 0.118757 0.15078 0.17463 0.20652
0.096218 0.118756 0.15079 0.17472 0.20710
Thus for a particular dimensionless film thickness e, if the pinhole radius is larger than r0 ( e ), the pinhole will open out. Otherwise it will close over.
AID
JCIS 4982
/
6g2e$$$$$2
07-11-97 07:49:52
[17]
Equations [ 15 – 17 ] form a system of six equations for six unknown quantities A ( e ) , r0 ( e ) , z0 ( e ) , B ( e ) , C ( e ) , and RH . A little algebra allows the elimination of A ( e ) , B ( e ) , C ( e ) , and z0 ( e ) and reduction to just two unknowns r0 and RH to be determined from Eq. [15 ] , where z0 Å r0 (ln(4/ er0 ) 0 g ); A( e ) Å r0;
C( e ) Å pr0 /2;
B( e ) Å 2r0 (ln(4/ er0 ) 0 g ).
[18]
For a given dimensionless film thickness e, [15] subject to [18] can easily be solved numerically whence the pinhole shape is given by [6], [3], and [13]. As an illustration we compare asymptotic computations of r0 and RH with the corresponding numerical estimates of (2):
e
r0 , RH (asymptotic)
r0 , R H (numerical)
2.88182.1003 0.104151 0.226608
0.05765, 0.34206 0.10818, 0.47169 0.13595, 0.53076
0.05765, 0.34206 0.10820, 0.47183 0.13605, 0.53167
coidas
516
NOTE
The asymptotic results are clearly in good agreement with the Hartland and Hartley results (2).
ACKNOWLEDGMENTS Thanks to John Hinch for suggesting this particular formulation and Claire Jordan for useful discussions.
REFERENCES 1. Derjaguin, B., Dokl. Akad. Nauk SSSR 58, 517–521 (1946). 2. Hartland, S., and Hartley, R. W., ‘‘Axisymmetric Fluid–Liquid Interfaces,’’ Elsevier, Amsterdam, 1976.
AID
JCIS 4982
/
6g2e$$$$$3
07-11-97 07:49:52
3. 4. 5. 6. 7. 8.
Kralchevsky, P. A., et al., J. Colloid Interface Sci. 112, 108 (1986). Lo, L. L., J. Fluid Mech. 132, 65 (1983). O’Brien, S. B. G., Quart. Appl. Math., in press. O’Brien, S. B. G., J. Colloid Interface Sci. 183, 51–56 (1996). O’Brien, S. B. G., J. Fluid Mech. 233, 519 (1991). Taylor, G. I., and Michael, D. H., J. Fluid Mech. 58, 625–639 (1973). S. B. G. O’Brien
Department of Mathematics University of Limerick Limerick, Ireland Received December 31, 1996; accepted May 9, 1997
coidas
191, 514–516 (1997)
CS974982
NOTE Asymptotics of a Pinhole z r 0 as r r ` ,
It is well known that, in a thin liquid film, corresponding to any particular thickness there is precisely one axisymmetric equilibrium solution to the Laplace–Young capillary equation; i.e., corresponding to each film thickness there is a unique pinhole radius which has been shown to be unstable (1973, Taylor and Michael, J. Fluid Mech. 58, 625–639). We deduce an asymptotic relationship between film thickness and pinhole height in the limit of very thin films. q 1997 Academic Press Key Words: asymptotics; surface tension; contact angle; pinhole.
The (unstable) pinhole shape in an otherwise uniform horizontal liquid film of infinite extent is modeled by the Laplace–Young capillary equation. We consider the axisymmetric situation where the film thickness H, the liquid density r, the liquid/air surface tension s, and acceleration due to gravity g are assumed to be known. In (5), a different formulation is used where the pinhole size is assumed to be known; the present formulation is more natural as film thickness can more readily be experimentally determined. The alternative formulation is more convenient if one wishes to extend to higher orders. This work may be considered an extension of that of ( 1, 3, 4, and 6 ) who considered asymptotic expressions for the shape of the meniscus near a small cylinder. Their results may be interpreted in the context of a pinhole where the contact angle u £ p / 2. We extend here to the more general case where 0 õ u £ p and the meniscus turns back on itself ( see Fig. 1 ) . This situation is not covered by the work of the aforementioned authors. Mathematically the extension of the solution is traceable to the sign ( zr ) in [1] , which changes sign when the orientation of the profile passes through the vertical. We use cylindrical polars r , z with z measured downward from the top surface ( Fig. 1 ) . Nondimensionalize lengths with the thickness of the layer, so that the lower solid surface is z Å 1. Balancing the capillary pressure jump across the curved surface with the hydrostatic pressure in the liquid, the equation for the free surface is
S
zrr zr / ( 1 / z 2r ) 3 / 2 r ( 1 / z 2r ) 1 / 2
D
/ e 2z Å 0.
/
6g2e$$$$$1
[2]
2. SOLUTIONS
2.1. Inner Zero-Curvature Region
z Å z0 | r0 ln
07-11-97 07:49:52
S
q
r / r 2 0 r 20 r0
D
,
[3]
with 0 on the upper branch and / on the lower branch. Applying the contact angle boundary conditions gives
z0 Å 1 / r0 ln cot( u /2).
[4]
Note that this solution becomes invalid if u approaches p. In this region a new scaling is required (5). This is of little practical significance and a brief outline of the solution in this case is included in Section 4 for completeness. In addition, note that the radius of the pinhole measured on the substrate is ru Å r0 csc u.
2.2. Outer Capillary-Gravity Region In this region the slopes are small. Making the appropriate stretching of the radial coordinate R Å er, the governing equation becomes a Bessel equation
0zRR 0
1 zR / z Å 0. R
[5]
The solution which tends to zero at large distances is
z Å A( e )K0 (R).
514
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
JCIS 4982
for some r Å ru ,
where u is the liquid/air contact angle (see Fig. 1). We are interested in finding how the minimum radius of the pinhole r0 varies with e.
[1]
Here e 2 Å rgH 2 / s is the inverse of the Bond number and may be regarded as being the square of the undisturbed film thickness made dimensionless with the capillary length. For a 40-mm layer of water-like coating, e É 0.07, which justifies an asymptotic analysis. The boundary conditions are
AID
zr Å 0tan u at z Å 1,
Near the pinhole, one can ignore to a first approximation the final gravity term in the governing equation, the surface thus having zero net curvature in this approximation. The axisymmetric solution is easily found to be
1. INTRODUCTION AND FORMULATION
sign ( zr )
and
coidas
[6]
515
NOTE
4. BOTTOM CAPILLARY-GRAVITY REGION In the case where the contact angle approaches p, the mensicus again approaches the horizontal and the inner scaling is no longer appropriate. In this region a new small slope approximation is needed with the stretching R Å er giving rise to a new Bessel equation: zRR /
1 zR / z Å 0. R
[12]
This has general solution z Å B( e )J0 (R) / C( e )(Y0 )(R). FIG. 1. Schematic of a pinhole.
[13]
In this instance the slope condition [ 2 ] must be applied to the bottom solution rather than the inner solution. There is a boundary layer of O ( e ) in u ( 7 ) , so the slope boundary condition can be written as
2.3. Matching As r r ` the inner solution is
ezR Å 0tan u Å 0tan( p 0 eU ) Ç eU
z Ç 1 / r0 ln cot( u /2) 0 r0 ln(2r/r0 ).
[7]
to O ( e ) where U is a known O ( 1 ) constant defined by u Å p 0 eU. This condition applies when z Å 1 and R Å RH for some ( unknown ) value of RH . Applying this to [13 ] we obtain the following two conditions:
[8]
B( e )J0 (RH ) / C( e )Y0 (RH ) Å 1;
As R r 0 the outer solution is z Ç A(ln(2/r e ) 0 g ),
[14]
U Å 0 B( e )J1 (RH ) 0 C( e )Y1 (RH ).
where g is Euler’s constant. Balancing the ln r terms gives A Å r0
[9]
and balancing the constant terms gives
[15]
On matching [13] to the lower branch of the inner solution [3], two more conditions are obtained, r0 Å 2C( e )/ p;
1 / r0 ln cot( u /2) 0 r0 ln(2/r0 ) Å A(ln(2/ e ) 0 g ).
z0 / r0 ln(2/r0 ) Å B( e ) / 2C( e )/ p(ln( e /2) / g ),
[10]
[16]
while matching the top capillarity-gravity solution to the upper inner solution yields two more conditions:
3. RESULTS Rearranging [10] we have
A( e ) Å r0 ; r0 (ln(2/ e ) 0 g ) Å z0 0 r0 ln(2/r0 ). 1 r0 ( e ) Å . ln(1/ e ) / ln(1/r0 ) / 2 ln(2) 0 ln cot( u /2) 0 g
[11]
Note that up to this stage the errors are O( e 2 ). Now one could solve the last equation iteratively for r0 ( e ), but that introduces an awkward secondary expansion in ln ln(1/ e )/ln( e ). Hence it is best to solve [11] numerically; i.e., given a particular undisturbed film thickness H and contact angle u, calculate the inverse Bond number e 2 Å rgH 2 / s and then solve [11] numerically for r0 . For the particular case where the contact angle u Å 3p /4, here are some typical results and a comparison with the corresponding numerical results of (2).
e
r 0 ( e) (asymptotic)
r 0 ( e) (numerical)
0.001727 0.01006 0.04737 0.10119 0.2071
0.096217 0.118757 0.15078 0.17463 0.20652
0.096218 0.118756 0.15079 0.17472 0.20710
Thus for a particular dimensionless film thickness e, if the pinhole radius is larger than r0 ( e ), the pinhole will open out. Otherwise it will close over.
AID
JCIS 4982
/
6g2e$$$$$2
07-11-97 07:49:52
[17]
Equations [ 15 – 17 ] form a system of six equations for six unknown quantities A ( e ) , r0 ( e ) , z0 ( e ) , B ( e ) , C ( e ) , and RH . A little algebra allows the elimination of A ( e ) , B ( e ) , C ( e ) , and z0 ( e ) and reduction to just two unknowns r0 and RH to be determined from Eq. [15 ] , where z0 Å r0 (ln(4/ er0 ) 0 g ); A( e ) Å r0;
C( e ) Å pr0 /2;
B( e ) Å 2r0 (ln(4/ er0 ) 0 g ).
[18]
For a given dimensionless film thickness e, [15] subject to [18] can easily be solved numerically whence the pinhole shape is given by [6], [3], and [13]. As an illustration we compare asymptotic computations of r0 and RH with the corresponding numerical estimates of (2):
e
r0 , RH (asymptotic)
r0 , R H (numerical)
2.88182.1003 0.104151 0.226608
0.05765, 0.34206 0.10818, 0.47169 0.13595, 0.53076
0.05765, 0.34206 0.10820, 0.47183 0.13605, 0.53167
coidas
516
NOTE
The asymptotic results are clearly in good agreement with the Hartland and Hartley results (2).
ACKNOWLEDGMENTS Thanks to John Hinch for suggesting this particular formulation and Claire Jordan for useful discussions.
REFERENCES 1. Derjaguin, B., Dokl. Akad. Nauk SSSR 58, 517–521 (1946). 2. Hartland, S., and Hartley, R. W., ‘‘Axisymmetric Fluid–Liquid Interfaces,’’ Elsevier, Amsterdam, 1976.
AID
JCIS 4982
/
6g2e$$$$$3
07-11-97 07:49:52
3. 4. 5. 6. 7. 8.
Kralchevsky, P. A., et al., J. Colloid Interface Sci. 112, 108 (1986). Lo, L. L., J. Fluid Mech. 132, 65 (1983). O’Brien, S. B. G., Quart. Appl. Math., in press. O’Brien, S. B. G., J. Colloid Interface Sci. 183, 51–56 (1996). O’Brien, S. B. G., J. Fluid Mech. 233, 519 (1991). Taylor, G. I., and Michael, D. H., J. Fluid Mech. 58, 625–639 (1973). S. B. G. O’Brien
Department of Mathematics University of Limerick Limerick, Ireland Received December 31, 1996; accepted May 9, 1997
coidas