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A Nonlinear Three-Dimensional Rupture Theory of Thin Liquid Films
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
190, 250–252 (1997)
CS974867
NOTE A Nonlinear Three-Dimensional Rupture Theory of Thin Liquid Films hT / Çr(h 01Çh) / Çr(h 3Ç2Çh) Å 0.
A process of nonlinear three-dimensional rupture of thin liquid films is numerically analyzed for the first time. With the rupture time being successfully calculated, it has been possible to develop a more complete rupture theory for thin liquid films. In contrast to the linear analysis indicating the shortest rupture time of thin liquid films to be the same for both two- and three-dimensional rupture, the nonlinear analysis reveals that the latter proceeds faster than the former. In particular, among all three-dimensional disturbance modes, the symmetric one makes the thin liquid films rupture fastest. It is concluded that the rupture process develops at a point rather than along a line on thin liquid films. q 1997 Academic Press
Key Words: nonlinear rupture theory; thin liquid films; threedimensional.
This equation governs the behavior of three-dimensional long-wave interfacial disturbances on a static film (define h Å 1 ) subject to the van der Waals attraction. In solving Eq. [1], we first apply linear stability theory to study the stability of the static film. By letting h Å 1 / H(X, Y, T ) and linearizing the equation in H, the linearized system of Eq. [1] is derived as follows: HT / HXX / HYY / HXXXX / 2HXXYY / HYYYY Å 0.
The importance research into the rupture of thin liquid films for the understanding of processes in colloid systems, such as flotation, foams and emulsions, coalescence of bubbles and droplets, and vapor and condensation on a solid surface is widely recognized. On the biological front, this phenomenon involves small scale deformations of membranes, such as the onset of microvilli in normal and neoplastic cells, attachment of cells and microorganisms, and phagocytosis. Recently this research has been applied to the basic study of aerospace technology, such as foams which attach to the surfaces of nozzles when jets blast out from propellers (1–3). ˚ , the When liquid coating films on a flat plate are as thin as 100–1000 A films appear unstable. A number of reports have discussed the stability and/ or rupture problems of these films. First, some research based on the linear theory was carried out (4, 5). Nevertheless, the use of linear theory cannot correctly predict the rupture behavior of thin films perturbed by small disturbances. This motivated research employing a nonlinear theory to analyze the rupture process (6–9). For all the above analyses, a two-dimensional disturbance model has been used to solve the rupture problem and led to the conclusion that the rupture process develops along a line on the thin liquid film. Regarding the study of three-dimensional disturbances, Williams and Davis (6) have derived a long-wave interfacial equation that can be used to govern the dynamics of the three-dimensional rupture process. However, to our knowledge, no analysis of the three-dimensional rupture process has been reported. Therefore, in the present work, we deal with the three-dimensional disturbance problem in detail by employing a supercomputing system to perform numerical analyses. On the basis of the analysis results, we develop a complete rupture theory for thin liquid films.
v Å q2 0 q4,
where
AID
JCIS 4867
/
6g25$$$521
[4] q
2
2
qÅ a /b . It is convenient for us to specify various modes of rupture that are present in all three-dimensional disturbances. Define a Å q cos u and b Å q sin u, 0 £ u £ p /2.
qM Å
1
q
2
and vM Å
1 4
[6]
and then the shortest linear rupture time TL is found to be [7]
Obviously, we can obtain from Eq. [7], analytical result of the linear theory, that the predicted shortest rupture time of thin liquid films is the same for both two- and three-dimensional disturbances. However, considering the
250
05-19-97 11:11:52
[5]
When u Å 0 or u Å p /2, it is obvious from Eq. [3] that the disturbance is a two-dimensional mode. The other values of u represent three-dimensional disturbances. The small disturbance with the maximum growth rate vM occurs at qM , which is the wavenumber of the maximum growth rate. By letting d v /dq Å 0 and using Eq. [4], it is easy to find
TL Å 04 ln(H0 ).
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
[3]
where H0 is the initial disturbance amplitude, v is the growth rate of the disturbance, and a and b are the wavenumbers in the X and Y directions, respectively. If we substitute Eq. [3] into Eq. [2], then the characteristic equation can be obtained as
II. FORMULATION Consider a thin liquid film on a horizontal plate, shown schematically in Fig. 1. Here, h is the dimensionless thickness of the local film. The available model equation derived by Williams and Davis (6) is given by
[2]
Here, H is the perturbation denoting the three-dimensional disturbance imposed on the static film. If we use a normal mode for the disturbance amplitude, it can be written as H(X, Y, T ) Å H0 exp( vT )cos( aX )cos( bY ),
I. INTRODUCTION
[1]
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251
NOTE
FIG. 1. Pictorial representation of a thin liquid film on a plate.
nonlinear effects, the estimate of TL by Eq. [7] becomes incomplete. It follows to obtain the nonlinear rupture time TN directly from the nonlinear system given by Eq. [1]. We solve Eq. [1] with the initial condition H(X, Y, 0) Å 1 / H0 cos( aX )cos( bY ) by using the finite difference numerical method. Moreover, center differences are employed for the space variable, while the midpoint rule is used for the time variable and also the Newton–Raphson iteration method is used to find a spatially periodic solution on the fixed interval 0 p /(qM cos u ) £ X £ p /(qM cos u ) and 0 £ Y £ 2p /(qM sin u ). Burelbach et al. (10) showed that if the time interval DT is fixed at 0.01, spatial effects become unimportant when the spatial-mesh number is larger than 20. In this paper, we choose DT Å 0.01, the spatial mesh 30 1 30, H0 Å 0.05–0.25, and the range of u is chosen to be 0– p /4 by considering the symmetrical condition of the disturbance modes with respect to u Å p /4. The convergent error limit of the iteration method is selected to be 10 06 . When some local regions of the interface disappear, the time integration will be terminated. Under those conditions, we can determine the nonlinear rupture time TN .
III. RESULTS AND DISCUSSION A schematic view of the film surface when rupture occurs ( T Å TN ) is shown in Fig. 2. It should be noted that the nonlinear rupture state profiles a point at the places near the rupture regions.
FIG. 3. Rupture time of thin liquid film versus initial disturbance amplitude for different modes of disturbance.
Figure 3 presents the variation of nonlinear rupture time TN as a function of initial disturbance amplitude H0 for various modes of disturbance. It is obvious that the rupture time decreases when the u increases in the region of 0 £ u £ p /4. The minimum rupture time occurs at u Å p /4, which represents a symmetric mode of the three-dimensional disturbance. Under such circumstances, it can be concluded that the rupture of the thin film develops at a point, not along a line, as indicated by the two-dimensional rupture theory. Also, the rupture time decreases when the initial disturbance amplitude H0 increases. Furthermore, the relative difference in TN between the two- and three-dimensional disturbances discussed in the previous section does not obviously change as H0 varies. To quantitatively compare the results between the three-dimensional and the two-dimensional rupture processes, we take a specific H0 Å 0.05 for both processes and obtain the result that the rupture time TN Å 6.123 for the three-dimensional mode ( u Å p /4) and TN Å 6.8864 for the two-dimensional mode ( u Å 0), where the nonnegligible difference is about 11%. This is because only a unidirectional force effect is involved in the two-dimensional rupture process. However, in the case of three-dimensional disturbances, bidirectional force effects should be considered to enhance the rupture behavior and consequently shorten the rupture time.
IV. CONCLUSIONS In the present work, we concentrate on the analysis of the three-dimensional nonlinear rupture of thin liquid films. Analyses by the linear theory indicate that the predicted shortest rupture time of thin liquid films is the same for both two- and three-dimensional disturbances. On the other hand, use of a supercomputing system has made it possible to perform high speed numerical calculations for finding the three-dimensional nonlinear rupture time TN . The numerical analysis results reveal that the rupture time for the three-dimensional process is shorter than that for the two-dimensional process. In conclusion, three-dimensional nonlinear analysis of the rupture of thin liquid films indicates that the rupture develops at a point in the solid-liquid interface.
ACKNOWLEDGMENTS FIG. 2. Three-dimensional film profiles as simultaneously viewed from the X and Y directions during rupture. H0 Å 0.05, u Å p /4, and T Å TN .
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JCIS 4867
/
6g25$$$521
05-19-97 11:11:52
The authors acknowledge with appreciation the financial support provided by the National Science Council (Grant NSC 86-2212-E-033-011) and by
coida
252
NOTE
the National Center for High-performance Computing of the Republic of China (Grant NCHC 86-01-005).
REFERENCES 1. Derjaguin, B. V., ‘‘Theory of Stability of Colloids and Thin Films’’ (R. K. Johnston, Transl.) Consultants Bureau, New York, 1989. 2. Hunter, R. J., ‘‘Foundations of Colloids Science.’’ Oxford Univ. Press, Oxford, 1986. 3. Edwards, D. A., Bernner, H., and Wasan, D. T., ‘‘Interfacial Transport Processes and Rheology.’’ Butterworth–Heinemann, Stoneham, 1991. 4. Sheludko, A., Adv. Colloid Interface Sci. 1, 391 (1967). 5. Ruckenstein, E., and Jain, R. K., Chem. Sci. Faraday Trans. II 70, 132 (1974). 6. Williams, M. B., and Davis, S. H., J. Colloid Interface Sci. 90, 220 (1982). 7. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 113, 456 (1986).
AID
JCIS 4867
/
6g25$$$521
05-19-97 11:11:52
8. Chen, J. L., and Hwang, C. C., J. Collloid Interface Sci. 167, 214 (1994). 9. Hwang, C. C., Chen, J. L., and Shen, L. F., Phys. Rev. E. 54, 3013 (1996). 10. Burelbach, J. P., Bankoff, S. G., and Davis, S. H., J. Fluid Mech. 195, 463 (1988). Chi-Chuan Hwang* ,1 Chaur-Kie Lin* Wu-Yih Uen† *Department of Mechanical Engineering †Department of Electronic Engineering Chang Yuan University Chung Li, Taiwan 32023, Republic of China Received November 8, 1996; accepted March 13, 1997
1
To whom correspondence should be addressed.
coida
190, 250–252 (1997)
CS974867
NOTE A Nonlinear Three-Dimensional Rupture Theory of Thin Liquid Films hT / Çr(h 01Çh) / Çr(h 3Ç2Çh) Å 0.
A process of nonlinear three-dimensional rupture of thin liquid films is numerically analyzed for the first time. With the rupture time being successfully calculated, it has been possible to develop a more complete rupture theory for thin liquid films. In contrast to the linear analysis indicating the shortest rupture time of thin liquid films to be the same for both two- and three-dimensional rupture, the nonlinear analysis reveals that the latter proceeds faster than the former. In particular, among all three-dimensional disturbance modes, the symmetric one makes the thin liquid films rupture fastest. It is concluded that the rupture process develops at a point rather than along a line on thin liquid films. q 1997 Academic Press
Key Words: nonlinear rupture theory; thin liquid films; threedimensional.
This equation governs the behavior of three-dimensional long-wave interfacial disturbances on a static film (define h Å 1 ) subject to the van der Waals attraction. In solving Eq. [1], we first apply linear stability theory to study the stability of the static film. By letting h Å 1 / H(X, Y, T ) and linearizing the equation in H, the linearized system of Eq. [1] is derived as follows: HT / HXX / HYY / HXXXX / 2HXXYY / HYYYY Å 0.
The importance research into the rupture of thin liquid films for the understanding of processes in colloid systems, such as flotation, foams and emulsions, coalescence of bubbles and droplets, and vapor and condensation on a solid surface is widely recognized. On the biological front, this phenomenon involves small scale deformations of membranes, such as the onset of microvilli in normal and neoplastic cells, attachment of cells and microorganisms, and phagocytosis. Recently this research has been applied to the basic study of aerospace technology, such as foams which attach to the surfaces of nozzles when jets blast out from propellers (1–3). ˚ , the When liquid coating films on a flat plate are as thin as 100–1000 A films appear unstable. A number of reports have discussed the stability and/ or rupture problems of these films. First, some research based on the linear theory was carried out (4, 5). Nevertheless, the use of linear theory cannot correctly predict the rupture behavior of thin films perturbed by small disturbances. This motivated research employing a nonlinear theory to analyze the rupture process (6–9). For all the above analyses, a two-dimensional disturbance model has been used to solve the rupture problem and led to the conclusion that the rupture process develops along a line on the thin liquid film. Regarding the study of three-dimensional disturbances, Williams and Davis (6) have derived a long-wave interfacial equation that can be used to govern the dynamics of the three-dimensional rupture process. However, to our knowledge, no analysis of the three-dimensional rupture process has been reported. Therefore, in the present work, we deal with the three-dimensional disturbance problem in detail by employing a supercomputing system to perform numerical analyses. On the basis of the analysis results, we develop a complete rupture theory for thin liquid films.
v Å q2 0 q4,
where
AID
JCIS 4867
/
6g25$$$521
[4] q
2
2
qÅ a /b . It is convenient for us to specify various modes of rupture that are present in all three-dimensional disturbances. Define a Å q cos u and b Å q sin u, 0 £ u £ p /2.
qM Å
1
q
2
and vM Å
1 4
[6]
and then the shortest linear rupture time TL is found to be [7]
Obviously, we can obtain from Eq. [7], analytical result of the linear theory, that the predicted shortest rupture time of thin liquid films is the same for both two- and three-dimensional disturbances. However, considering the
250
05-19-97 11:11:52
[5]
When u Å 0 or u Å p /2, it is obvious from Eq. [3] that the disturbance is a two-dimensional mode. The other values of u represent three-dimensional disturbances. The small disturbance with the maximum growth rate vM occurs at qM , which is the wavenumber of the maximum growth rate. By letting d v /dq Å 0 and using Eq. [4], it is easy to find
TL Å 04 ln(H0 ).
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
[3]
where H0 is the initial disturbance amplitude, v is the growth rate of the disturbance, and a and b are the wavenumbers in the X and Y directions, respectively. If we substitute Eq. [3] into Eq. [2], then the characteristic equation can be obtained as
II. FORMULATION Consider a thin liquid film on a horizontal plate, shown schematically in Fig. 1. Here, h is the dimensionless thickness of the local film. The available model equation derived by Williams and Davis (6) is given by
[2]
Here, H is the perturbation denoting the three-dimensional disturbance imposed on the static film. If we use a normal mode for the disturbance amplitude, it can be written as H(X, Y, T ) Å H0 exp( vT )cos( aX )cos( bY ),
I. INTRODUCTION
[1]
coida
251
NOTE
FIG. 1. Pictorial representation of a thin liquid film on a plate.
nonlinear effects, the estimate of TL by Eq. [7] becomes incomplete. It follows to obtain the nonlinear rupture time TN directly from the nonlinear system given by Eq. [1]. We solve Eq. [1] with the initial condition H(X, Y, 0) Å 1 / H0 cos( aX )cos( bY ) by using the finite difference numerical method. Moreover, center differences are employed for the space variable, while the midpoint rule is used for the time variable and also the Newton–Raphson iteration method is used to find a spatially periodic solution on the fixed interval 0 p /(qM cos u ) £ X £ p /(qM cos u ) and 0 £ Y £ 2p /(qM sin u ). Burelbach et al. (10) showed that if the time interval DT is fixed at 0.01, spatial effects become unimportant when the spatial-mesh number is larger than 20. In this paper, we choose DT Å 0.01, the spatial mesh 30 1 30, H0 Å 0.05–0.25, and the range of u is chosen to be 0– p /4 by considering the symmetrical condition of the disturbance modes with respect to u Å p /4. The convergent error limit of the iteration method is selected to be 10 06 . When some local regions of the interface disappear, the time integration will be terminated. Under those conditions, we can determine the nonlinear rupture time TN .
III. RESULTS AND DISCUSSION A schematic view of the film surface when rupture occurs ( T Å TN ) is shown in Fig. 2. It should be noted that the nonlinear rupture state profiles a point at the places near the rupture regions.
FIG. 3. Rupture time of thin liquid film versus initial disturbance amplitude for different modes of disturbance.
Figure 3 presents the variation of nonlinear rupture time TN as a function of initial disturbance amplitude H0 for various modes of disturbance. It is obvious that the rupture time decreases when the u increases in the region of 0 £ u £ p /4. The minimum rupture time occurs at u Å p /4, which represents a symmetric mode of the three-dimensional disturbance. Under such circumstances, it can be concluded that the rupture of the thin film develops at a point, not along a line, as indicated by the two-dimensional rupture theory. Also, the rupture time decreases when the initial disturbance amplitude H0 increases. Furthermore, the relative difference in TN between the two- and three-dimensional disturbances discussed in the previous section does not obviously change as H0 varies. To quantitatively compare the results between the three-dimensional and the two-dimensional rupture processes, we take a specific H0 Å 0.05 for both processes and obtain the result that the rupture time TN Å 6.123 for the three-dimensional mode ( u Å p /4) and TN Å 6.8864 for the two-dimensional mode ( u Å 0), where the nonnegligible difference is about 11%. This is because only a unidirectional force effect is involved in the two-dimensional rupture process. However, in the case of three-dimensional disturbances, bidirectional force effects should be considered to enhance the rupture behavior and consequently shorten the rupture time.
IV. CONCLUSIONS In the present work, we concentrate on the analysis of the three-dimensional nonlinear rupture of thin liquid films. Analyses by the linear theory indicate that the predicted shortest rupture time of thin liquid films is the same for both two- and three-dimensional disturbances. On the other hand, use of a supercomputing system has made it possible to perform high speed numerical calculations for finding the three-dimensional nonlinear rupture time TN . The numerical analysis results reveal that the rupture time for the three-dimensional process is shorter than that for the two-dimensional process. In conclusion, three-dimensional nonlinear analysis of the rupture of thin liquid films indicates that the rupture develops at a point in the solid-liquid interface.
ACKNOWLEDGMENTS FIG. 2. Three-dimensional film profiles as simultaneously viewed from the X and Y directions during rupture. H0 Å 0.05, u Å p /4, and T Å TN .
AID
JCIS 4867
/
6g25$$$521
05-19-97 11:11:52
The authors acknowledge with appreciation the financial support provided by the National Science Council (Grant NSC 86-2212-E-033-011) and by
coida
252
NOTE
the National Center for High-performance Computing of the Republic of China (Grant NCHC 86-01-005).
REFERENCES 1. Derjaguin, B. V., ‘‘Theory of Stability of Colloids and Thin Films’’ (R. K. Johnston, Transl.) Consultants Bureau, New York, 1989. 2. Hunter, R. J., ‘‘Foundations of Colloids Science.’’ Oxford Univ. Press, Oxford, 1986. 3. Edwards, D. A., Bernner, H., and Wasan, D. T., ‘‘Interfacial Transport Processes and Rheology.’’ Butterworth–Heinemann, Stoneham, 1991. 4. Sheludko, A., Adv. Colloid Interface Sci. 1, 391 (1967). 5. Ruckenstein, E., and Jain, R. K., Chem. Sci. Faraday Trans. II 70, 132 (1974). 6. Williams, M. B., and Davis, S. H., J. Colloid Interface Sci. 90, 220 (1982). 7. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 113, 456 (1986).
AID
JCIS 4867
/
6g25$$$521
05-19-97 11:11:52
8. Chen, J. L., and Hwang, C. C., J. Collloid Interface Sci. 167, 214 (1994). 9. Hwang, C. C., Chen, J. L., and Shen, L. F., Phys. Rev. E. 54, 3013 (1996). 10. Burelbach, J. P., Bankoff, S. G., and Davis, S. H., J. Fluid Mech. 195, 463 (1988). Chi-Chuan Hwang* ,1 Chaur-Kie Lin* Wu-Yih Uen† *Department of Mechanical Engineering †Department of Electronic Engineering Chang Yuan University Chung Li, Taiwan 32023, Republic of China Received November 8, 1996; accepted March 13, 1997
1
To whom correspondence should be addressed.
coida