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Three-Dimensional Nonlinear Rupture Theory of Thin Liquid Films on a Cylinder
Journal of Colloid and Interface Science 256, 480–482 (2002) doi:10.1006/jcis.2002.8493
NOTE Three-Dimensional Nonlinear Rupture Theory of Thin Liquid Films on a Cylinder The three-dimensional nonlinear rupture theory of thin liquid films on a cylinder is presented and studied in this note. The thin liquid film with the effect of intermolecular forces was modeled by a continuum theory, and the three-dimensional evolution equation of liquid films on a cylindrical surface was derived based on a long wavelength approximation. Both linear stability theory and nonlinear numerical method were adopted to solve this evolution equation. The linear stability analysis fails to distinguish the threedimensional mode from the two-dimensional one in terms of maximum disturbance growth rate and always yields a rupture time larger than the nonlinear solution. In contrast, the nonlinear numerical results clearly show that among three disturbance modes, the two-dimensional annular disturbance one yields the longest rupture time, the two-dimensional axisymmetric disturbance one yields the second longest, and the three-dimensional disturbance one does the shortest. Accordingly, it can be concluded that the rupture status of thin films on cylinder is most likely evolved in the threedimensional disturbance mode as predicted. C 2002 Elsevier Science (USA) Key Words: nonlinear; three-dimensional; rupture theory; thin liquid film; cylinder; stability theory.
INTRODUCTION The dynamic stability and the related rupture behavior of thin liquid films on a solid substrate is an important research topic in the fields of biological science (e.g., the action of healing, converging, and coalescence during the rupture process of cell membranes [1–3], mechanical engineering (e.g., the flowing phenomenon of films on a working surface during coating, laser cutting, and aggregation of condensed vapor on the interior wall of a tube [4, 5], and chemical engineering (e.g., the study of wetting and floating-sieving and the application of foams and emulsions [6, 7]). Recently, the application of film rupture theory has been extended to the fields of aerospace and medication (8, 9). The stability of an extremely thin liquid film is influenced substantially by the van der Waals potential, such that a tiny disturbance may drive it to be unstable or even lead to film rupturing (10). The earliest study of the stability and rupture theory of liquid films was initiated by Ruckenstein and Jain (11), who modeled a thin liquid film as a continuum and used a linear stability theory to examine its dynamic stability. Williams and Davis (12) adopted a long wavelength approximation to derive the corresponding nonlinear evolution equation of a film surface. Burelbach et al. (13) further studied the accuracy of numerical solution to the nonlinear evolution equation. On the other hand, Sharma and Ruckenstein (14) used a perturbation method to obtain the approximate solution to nonlinear rupture time, and this solution scheme was later modified by Hwang et al. (15). Hwang et al. (16) also used the long wavelength approximation to derive the three-dimensional nonlinear evolution equation and numerically calculated the corresponding rupture time. 0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
Because the geometry of a cylinder is relatively complex, the research on the rupture problem of liquid films on it remains quite limited (17–19). Chen and Hwang (20) examined the nonlinear rupture theory of liquid films on a cylinder, and their numerical results indicated that the lateral capillarity would accelerate film rupturing. However, the nonlinear rupture studies of such a subject performed by the above researchers are mainly limited to the two-dimensional simplified model, and the corresponding truly three-dimensional nonlinear rupture theory is scant in the literature and definitely requires a further investigation.
FORMULATION In this section, the physical model of a rupture theory on a cylinder is mathematically presented, the long wavelength approximation is used to derive the three-dimensional nonlinear rupture evolution equation associated with two simplified two-dimensional forms, and the linear stability theory is adopted to yield the corresponding linear solution. The physical model describing a thin liquid film on a cylinder (radius = a) is illustrated in Fig. 1. Here, h is the local dimensionless thickness of the film. The aforementioned long wavelength evolution equation derived by Williams and Davis (12) is given by h T + ∇ · (h −1 ∇h) + ∇ · (h 3 ∇ 2 ∇h) = 0,
[1]
where the subscript T denotes the normalized time parameter. The above equation also governs the behavior of three-dimensional long-wave interfacial disturbances on a static film (of an average dimensionless thickness h¯ = 1) subjected 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 introducing H = h − 1 and linearizing the equation in H (here H is the perturbation denoting the three-dimensional disturbance imposed on the static film), the linearized system of Eq. [1] is derivated as follows: HT + (Hx x x + Hx yy /r 2 + Hx /r 2 + Hx )x + (Hyyy /r 2 + Hx x y + Hy /r 2 + Hy ) y /r 2 = 0,
[2]
where the subscript x, subscript y, and r denote the normalized z coordinate, normalized θ coordinate, and normalized cylinder radius, respectively. If we use normal mode for the disturbance amplitude, it can be written as H = H0 exp(ωT + ikx + imy) + c.c.,
[3]
Where H0 is the initial disturbance amplitude, ω is the growth rate of the disturbance, and k and m are the wavenumbers. If we substitute Eq. [3] into Eq. [2], then the characteristic equation can be obtained as ω = (k 2 + m 2 /r 2 )[(1 + 1/r 2 ) − (k 2 + m 2 /r 2 )].
[4]
When ω = 0, the cutoff wave-number kc is determined to be: kc = [(1 + 1/r 2 ) − m 2 /r 2 ]1/2 . 480
[5]
NOTE
FIG. 1.
Illustrative view of three-dimensional surface profile of a liquid film on a cylinder.
Moreover, setting dω/dk = 0 yields the maximum growth rate of disturbance ωm and the corresponding wave-number of maximum growth rate km as given below: km = [(1 + 1/r 2 )/2 − m 2 /r 2 ]1/2
481
and
ωm = (1 + 1/r 2 )1/2 /4.
[6]
From these, the linear solution of shortest rupture time TRL is obtained as follows: TRL = −ln(H0 )/ωm .
[7]
If the annular effect around the cylindrical perimeter is neglected, then the purely three-dimensional disturbance mode becomes the two-dimensional axisymmetric disturbance mode (i.e., m = 0). On the other hand, if the axial effect along the cylindrical long axis is neglected (i.e., liquid film evolves regardless of the axial capillarity), then the three-dimensional mode becomes the twodimensional annular disturbance mode (i.e., k = 0).
set to be −π/k ≤ x ≤ π/k and 0 ≤ y ≤ 2π/m. The Newton–Raphson iterative method is introduced to solve Eq. [2], and the convergence tolerance is set to be 10−6 . Whenever the local thickness of the film becomes zero, the calculation is terminated. And such a termination time is regarded as the nonlinear rupture time TRN . For all nonlinear numerical results presented in this short note, H0 = 0.025, S = 1.0, and A = 0.001 are assigned. Table 1 summarizes the relationship of the maximum disturbance growth rate ωm , linear rupture time TRL , and nonlinear rupture time TRN for the twodimensional annular mode at cylinder radius a = 80–150 and annular wavenumber m = 1–4. From this table, it is found that (1) ωm increases with increasing a, so it does with m at small m; (2) both TRL and TRN increase with increasing a; (3) TRN decreases with increasing m; and (4) TRN is always shorter than TRL . The computed values of TRN for the three-dimensional disturbance mode a = 150, m = 1, 2, 3 are 1.33, 1.29, and 1.26, respectively. By comparing these values with the last row of Table 1, it is observed that (1) like the two-dimensional
RESULTS AND DISCUSSION The nonlinear three-dimensional evolution equation of free surface of a liquid film on a cylinder, Eq. [2] is a strongly nonlinear partial differential equation, which is only solvable through a numerical scheme. The initial condition is h(X, Y, T ) = 1 + H0 exp(ωT ) cos(kx) cos(my). The finite difference method is adopted in this note, where the central finite difference scheme is employed for the spatial coordinates and the Crank–Nicholson implicit method is used for the time domain. Periodical boundaries along the x and y coordinates are
TABLE 1 Relationship of ωm , TRL , and TRN for the Two-Dimensional Annular Mode at a = 80 to 150 and m = 1 to 4 A
m
ωm
TRL
TRN
80
1 2 1 2 3 1 2 3 4
0.1563 0.3320 0.1000 0.2800 0.1800 0.0444 0.1541 0.278 0.2370
8.8722 4.1752 13.8629 4.9511 7.7016 31.1916 8.9976 5.3779 5.8443
3.1753 1.3120 4.7200 1.5300 1.2750 10.064 2.8741 1.5927 1.2960
100
150
FIG. 2. Variation of TRN with k for the three-dimensional disturbance mode at m = 0–3, a = 150.
482
NOTE
annular disturbance mode, TRN of the three-dimensional disturbance mode also decreases with increasing m; (2) under the same condition (a, m, k), TRN of the three-dimensional disturbance mode is much smaller than that of the twodimensional annular disturbance mode. Additional analysis results (which are not listed herein) also indicate that TRN for the three-dimensional disturbance mode decreases with decreasing a because a smaller a implies a higher lateral capillarity as well as a more unstable film system. Figure 2 shows the relationship between TR and k at a = 150 for the twodimensional axisymmetric disturbance mode (m = 0) and the three-dimensional disturbance mode (m = 1–3). From this figure, it is observed that for both modes TRN initially decreases with increasing k, but it starts to increase when it passes its minimum. TRN for the three-dimensional disturbance mode is also smaller than that for the two-dimensional axisymmetric disturbance mode. In summary, under the same condition, TRN for the three-dimensional disturbance mode is the smallest among three modes. This can be well explained by the fact that the rupture of liquid film for the three-dimensional disturbance mode initiates at a point within a trough region but not along a line for the two-dimensional case (16).
CONCLUSIONS The objective of this short note is to theoretically study the nonlinear threedimensional rupture behavior of liquid film on a cylinder. From the analysis results, the following conclusions can be drawn. First, from the solutions of the linear stability analysis, it can be seen that the maximum disturbance growth rate ωm predicted by all of the disturbance modes are the same (referring to Eqs. [4] and [6]). This clearly shows that the linear theory can neither distinguish the three-dimensional effect from the two-dimensional one nor point out which disturbance mode is the most critical. Besides, the linear rupture time is always larger than the nonlinear solution under the same condition. Second, from the results of nonlinear analysis, it is found that the rupture time decreases with increasing annular wave-number m. Among three disturbance modes, the two-dimensional annular disturbance one yields the longest rupture time, the two-dimensional axisymmetric disturbance one does the second longest, and the three-dimensional disturbance one does the shortest. Therefore, the rupture status of liquid film on a cylinder is more likely evolved in the three-dimensional disturbance mode as predicted.
ACKNOWLEDGMENT The authors acknowledge the financial support (Grant No. NSC 89-2212-E231-005) provided by the National Science Council of the Republic of China.
3. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 106, 12 (1985). 4. Hunter, R. J., “Foundations of Colloids Science,” Oxford Univ. Press, Oxford, UK, 1986. 5. Derjaguin, B. V., “Theory of Stability of Colloids and Thin Films,” Consultants Bureau, New York, 1989. 6. Vrij, A., and Overbeck, J. Th. G., J. Am. Chem. Soc. 90, 3074 (1968). 7. Edwards, D. A., Brenner, H., and Wasan, D. T., “Interfacial Transport Processes and Rheology,” Butterworth–Heinemann, London, 1991. 8. Jain, R. K., and Ruckenstein, E., J. Colloid Interface Sci. 54, 108 (1976). 9. Scarpelli, E. M., “Surfactants and the Liquid Lining of the Lung,” Johns Hopkins Univ. Press, Baltimore, MD, 1988. 10. Sheludko, A., Adv. Colloid Interface Sci. 1, 391 (1967). 11. Ruckenstein, E., and Jain, R. K., J. Chem. Soc. Faraday Trans. II 70, 132 (1974). 12. Williams, M. B., and Davis, S. H., J. Colloid Interface Sci. 90, 220 (1982). 13. Burelbach, J. P., Bankoff, S. G., and Davis, S. H., J. Fluid Mech. 195, 463 (1988). 14. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 113, 456 (1986). 15. Hwang, C. C., Chang, S. H., and Chen, J. L., J. Colloid Interface Sci. 159, 184 (1993). 16. Hwang, C. C., Lin, C. K., and Uen, W. Y., J. Colloid Interface Sci. 190, 250 (1997). 17. Hammond, P. S., J. Fluid Mech. 137, 363 (1983). 18. Reisfeld, B., and Bankoff, S. G., J. Fluid Mech. 236, 167 (1992). 19. Herdt, G. C., J. Colloid Interface Sci. 160, 72 (1993). 20. Chen, J. L., and Hwang, C. C., J. Colloid Interface Sci. 182, 564 (1996). Chaur-Kie Lin∗ Chi-Chuan Hwang†,1 Te-Chih Ke‡ ∗ Department of Mechanical Engineering Ching Yun Institute of Technology Chung-Li, Taiwan 32020 Republic of China †Department of Engineering Science National Cheng Kung University Tainan, Taiwan 70101 Republic of China ‡Department of Construction Engineering National Yunlin University of Science and Technology Touliu, Yunlin, Taiwan 640 Republic of China Received September 5, 2001; accepted May 20, 2002
REFERENCES 1. Poste, G., and Nicolson, G. L., “Membrane Fusion,” North-Holland, Amsterdam, 1978. 2. Gallez, D., and Coakely, W. T., Progr. Biophys. Mol. Biol. 48, 155 (1986).
1 To whom correspondence should be addressed. Fax: 886-6-2766549. E-mail: [email protected].
NOTE Three-Dimensional Nonlinear Rupture Theory of Thin Liquid Films on a Cylinder The three-dimensional nonlinear rupture theory of thin liquid films on a cylinder is presented and studied in this note. The thin liquid film with the effect of intermolecular forces was modeled by a continuum theory, and the three-dimensional evolution equation of liquid films on a cylindrical surface was derived based on a long wavelength approximation. Both linear stability theory and nonlinear numerical method were adopted to solve this evolution equation. The linear stability analysis fails to distinguish the threedimensional mode from the two-dimensional one in terms of maximum disturbance growth rate and always yields a rupture time larger than the nonlinear solution. In contrast, the nonlinear numerical results clearly show that among three disturbance modes, the two-dimensional annular disturbance one yields the longest rupture time, the two-dimensional axisymmetric disturbance one yields the second longest, and the three-dimensional disturbance one does the shortest. Accordingly, it can be concluded that the rupture status of thin films on cylinder is most likely evolved in the threedimensional disturbance mode as predicted. C 2002 Elsevier Science (USA) Key Words: nonlinear; three-dimensional; rupture theory; thin liquid film; cylinder; stability theory.
INTRODUCTION The dynamic stability and the related rupture behavior of thin liquid films on a solid substrate is an important research topic in the fields of biological science (e.g., the action of healing, converging, and coalescence during the rupture process of cell membranes [1–3], mechanical engineering (e.g., the flowing phenomenon of films on a working surface during coating, laser cutting, and aggregation of condensed vapor on the interior wall of a tube [4, 5], and chemical engineering (e.g., the study of wetting and floating-sieving and the application of foams and emulsions [6, 7]). Recently, the application of film rupture theory has been extended to the fields of aerospace and medication (8, 9). The stability of an extremely thin liquid film is influenced substantially by the van der Waals potential, such that a tiny disturbance may drive it to be unstable or even lead to film rupturing (10). The earliest study of the stability and rupture theory of liquid films was initiated by Ruckenstein and Jain (11), who modeled a thin liquid film as a continuum and used a linear stability theory to examine its dynamic stability. Williams and Davis (12) adopted a long wavelength approximation to derive the corresponding nonlinear evolution equation of a film surface. Burelbach et al. (13) further studied the accuracy of numerical solution to the nonlinear evolution equation. On the other hand, Sharma and Ruckenstein (14) used a perturbation method to obtain the approximate solution to nonlinear rupture time, and this solution scheme was later modified by Hwang et al. (15). Hwang et al. (16) also used the long wavelength approximation to derive the three-dimensional nonlinear evolution equation and numerically calculated the corresponding rupture time. 0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
All rights reserved.
Because the geometry of a cylinder is relatively complex, the research on the rupture problem of liquid films on it remains quite limited (17–19). Chen and Hwang (20) examined the nonlinear rupture theory of liquid films on a cylinder, and their numerical results indicated that the lateral capillarity would accelerate film rupturing. However, the nonlinear rupture studies of such a subject performed by the above researchers are mainly limited to the two-dimensional simplified model, and the corresponding truly three-dimensional nonlinear rupture theory is scant in the literature and definitely requires a further investigation.
FORMULATION In this section, the physical model of a rupture theory on a cylinder is mathematically presented, the long wavelength approximation is used to derive the three-dimensional nonlinear rupture evolution equation associated with two simplified two-dimensional forms, and the linear stability theory is adopted to yield the corresponding linear solution. The physical model describing a thin liquid film on a cylinder (radius = a) is illustrated in Fig. 1. Here, h is the local dimensionless thickness of the film. The aforementioned long wavelength evolution equation derived by Williams and Davis (12) is given by h T + ∇ · (h −1 ∇h) + ∇ · (h 3 ∇ 2 ∇h) = 0,
[1]
where the subscript T denotes the normalized time parameter. The above equation also governs the behavior of three-dimensional long-wave interfacial disturbances on a static film (of an average dimensionless thickness h¯ = 1) subjected 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 introducing H = h − 1 and linearizing the equation in H (here H is the perturbation denoting the three-dimensional disturbance imposed on the static film), the linearized system of Eq. [1] is derivated as follows: HT + (Hx x x + Hx yy /r 2 + Hx /r 2 + Hx )x + (Hyyy /r 2 + Hx x y + Hy /r 2 + Hy ) y /r 2 = 0,
[2]
where the subscript x, subscript y, and r denote the normalized z coordinate, normalized θ coordinate, and normalized cylinder radius, respectively. If we use normal mode for the disturbance amplitude, it can be written as H = H0 exp(ωT + ikx + imy) + c.c.,
[3]
Where H0 is the initial disturbance amplitude, ω is the growth rate of the disturbance, and k and m are the wavenumbers. If we substitute Eq. [3] into Eq. [2], then the characteristic equation can be obtained as ω = (k 2 + m 2 /r 2 )[(1 + 1/r 2 ) − (k 2 + m 2 /r 2 )].
[4]
When ω = 0, the cutoff wave-number kc is determined to be: kc = [(1 + 1/r 2 ) − m 2 /r 2 ]1/2 . 480
[5]
NOTE
FIG. 1.
Illustrative view of three-dimensional surface profile of a liquid film on a cylinder.
Moreover, setting dω/dk = 0 yields the maximum growth rate of disturbance ωm and the corresponding wave-number of maximum growth rate km as given below: km = [(1 + 1/r 2 )/2 − m 2 /r 2 ]1/2
481
and
ωm = (1 + 1/r 2 )1/2 /4.
[6]
From these, the linear solution of shortest rupture time TRL is obtained as follows: TRL = −ln(H0 )/ωm .
[7]
If the annular effect around the cylindrical perimeter is neglected, then the purely three-dimensional disturbance mode becomes the two-dimensional axisymmetric disturbance mode (i.e., m = 0). On the other hand, if the axial effect along the cylindrical long axis is neglected (i.e., liquid film evolves regardless of the axial capillarity), then the three-dimensional mode becomes the twodimensional annular disturbance mode (i.e., k = 0).
set to be −π/k ≤ x ≤ π/k and 0 ≤ y ≤ 2π/m. The Newton–Raphson iterative method is introduced to solve Eq. [2], and the convergence tolerance is set to be 10−6 . Whenever the local thickness of the film becomes zero, the calculation is terminated. And such a termination time is regarded as the nonlinear rupture time TRN . For all nonlinear numerical results presented in this short note, H0 = 0.025, S = 1.0, and A = 0.001 are assigned. Table 1 summarizes the relationship of the maximum disturbance growth rate ωm , linear rupture time TRL , and nonlinear rupture time TRN for the twodimensional annular mode at cylinder radius a = 80–150 and annular wavenumber m = 1–4. From this table, it is found that (1) ωm increases with increasing a, so it does with m at small m; (2) both TRL and TRN increase with increasing a; (3) TRN decreases with increasing m; and (4) TRN is always shorter than TRL . The computed values of TRN for the three-dimensional disturbance mode a = 150, m = 1, 2, 3 are 1.33, 1.29, and 1.26, respectively. By comparing these values with the last row of Table 1, it is observed that (1) like the two-dimensional
RESULTS AND DISCUSSION The nonlinear three-dimensional evolution equation of free surface of a liquid film on a cylinder, Eq. [2] is a strongly nonlinear partial differential equation, which is only solvable through a numerical scheme. The initial condition is h(X, Y, T ) = 1 + H0 exp(ωT ) cos(kx) cos(my). The finite difference method is adopted in this note, where the central finite difference scheme is employed for the spatial coordinates and the Crank–Nicholson implicit method is used for the time domain. Periodical boundaries along the x and y coordinates are
TABLE 1 Relationship of ωm , TRL , and TRN for the Two-Dimensional Annular Mode at a = 80 to 150 and m = 1 to 4 A
m
ωm
TRL
TRN
80
1 2 1 2 3 1 2 3 4
0.1563 0.3320 0.1000 0.2800 0.1800 0.0444 0.1541 0.278 0.2370
8.8722 4.1752 13.8629 4.9511 7.7016 31.1916 8.9976 5.3779 5.8443
3.1753 1.3120 4.7200 1.5300 1.2750 10.064 2.8741 1.5927 1.2960
100
150
FIG. 2. Variation of TRN with k for the three-dimensional disturbance mode at m = 0–3, a = 150.
482
NOTE
annular disturbance mode, TRN of the three-dimensional disturbance mode also decreases with increasing m; (2) under the same condition (a, m, k), TRN of the three-dimensional disturbance mode is much smaller than that of the twodimensional annular disturbance mode. Additional analysis results (which are not listed herein) also indicate that TRN for the three-dimensional disturbance mode decreases with decreasing a because a smaller a implies a higher lateral capillarity as well as a more unstable film system. Figure 2 shows the relationship between TR and k at a = 150 for the twodimensional axisymmetric disturbance mode (m = 0) and the three-dimensional disturbance mode (m = 1–3). From this figure, it is observed that for both modes TRN initially decreases with increasing k, but it starts to increase when it passes its minimum. TRN for the three-dimensional disturbance mode is also smaller than that for the two-dimensional axisymmetric disturbance mode. In summary, under the same condition, TRN for the three-dimensional disturbance mode is the smallest among three modes. This can be well explained by the fact that the rupture of liquid film for the three-dimensional disturbance mode initiates at a point within a trough region but not along a line for the two-dimensional case (16).
CONCLUSIONS The objective of this short note is to theoretically study the nonlinear threedimensional rupture behavior of liquid film on a cylinder. From the analysis results, the following conclusions can be drawn. First, from the solutions of the linear stability analysis, it can be seen that the maximum disturbance growth rate ωm predicted by all of the disturbance modes are the same (referring to Eqs. [4] and [6]). This clearly shows that the linear theory can neither distinguish the three-dimensional effect from the two-dimensional one nor point out which disturbance mode is the most critical. Besides, the linear rupture time is always larger than the nonlinear solution under the same condition. Second, from the results of nonlinear analysis, it is found that the rupture time decreases with increasing annular wave-number m. Among three disturbance modes, the two-dimensional annular disturbance one yields the longest rupture time, the two-dimensional axisymmetric disturbance one does the second longest, and the three-dimensional disturbance one does the shortest. Therefore, the rupture status of liquid film on a cylinder is more likely evolved in the three-dimensional disturbance mode as predicted.
ACKNOWLEDGMENT The authors acknowledge the financial support (Grant No. NSC 89-2212-E231-005) provided by the National Science Council of the Republic of China.
3. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 106, 12 (1985). 4. Hunter, R. J., “Foundations of Colloids Science,” Oxford Univ. Press, Oxford, UK, 1986. 5. Derjaguin, B. V., “Theory of Stability of Colloids and Thin Films,” Consultants Bureau, New York, 1989. 6. Vrij, A., and Overbeck, J. Th. G., J. Am. Chem. Soc. 90, 3074 (1968). 7. Edwards, D. A., Brenner, H., and Wasan, D. T., “Interfacial Transport Processes and Rheology,” Butterworth–Heinemann, London, 1991. 8. Jain, R. K., and Ruckenstein, E., J. Colloid Interface Sci. 54, 108 (1976). 9. Scarpelli, E. M., “Surfactants and the Liquid Lining of the Lung,” Johns Hopkins Univ. Press, Baltimore, MD, 1988. 10. Sheludko, A., Adv. Colloid Interface Sci. 1, 391 (1967). 11. Ruckenstein, E., and Jain, R. K., J. Chem. Soc. Faraday Trans. II 70, 132 (1974). 12. Williams, M. B., and Davis, S. H., J. Colloid Interface Sci. 90, 220 (1982). 13. Burelbach, J. P., Bankoff, S. G., and Davis, S. H., J. Fluid Mech. 195, 463 (1988). 14. Sharma, A., and Ruckenstein, E., J. Colloid Interface Sci. 113, 456 (1986). 15. Hwang, C. C., Chang, S. H., and Chen, J. L., J. Colloid Interface Sci. 159, 184 (1993). 16. Hwang, C. C., Lin, C. K., and Uen, W. Y., J. Colloid Interface Sci. 190, 250 (1997). 17. Hammond, P. S., J. Fluid Mech. 137, 363 (1983). 18. Reisfeld, B., and Bankoff, S. G., J. Fluid Mech. 236, 167 (1992). 19. Herdt, G. C., J. Colloid Interface Sci. 160, 72 (1993). 20. Chen, J. L., and Hwang, C. C., J. Colloid Interface Sci. 182, 564 (1996). Chaur-Kie Lin∗ Chi-Chuan Hwang†,1 Te-Chih Ke‡ ∗ Department of Mechanical Engineering Ching Yun Institute of Technology Chung-Li, Taiwan 32020 Republic of China †Department of Engineering Science National Cheng Kung University Tainan, Taiwan 70101 Republic of China ‡Department of Construction Engineering National Yunlin University of Science and Technology Touliu, Yunlin, Taiwan 640 Republic of China Received September 5, 2001; accepted May 20, 2002
REFERENCES 1. Poste, G., and Nicolson, G. L., “Membrane Fusion,” North-Holland, Amsterdam, 1978. 2. Gallez, D., and Coakely, W. T., Progr. Biophys. Mol. Biol. 48, 155 (1986).
1 To whom correspondence should be addressed. Fax: 886-6-2766549. E-mail: [email protected].