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Bifurcation of a partially immersed plate between two parallel plates

Published online by Cambridge University Press:  15 March 2017

Xinping Zhou Open the ORCID record for Xinping Zhou [Opens in a new window]  and
Fei Zhang Open the ORCID record for Fei Zhang [Opens in a new window]
Xinping Zhou
Affiliation:
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Wuhan 430074, PR China
Fei Zhang*
Affiliation:
Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, PR China Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Wuhan 430074, PR China
*
†Email addresses for correspondence: feizhang11@hust.edu.cn, xpzhou08@hust.edu.cn
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Abstract

The three-plate system in which a vertical plate is located between two spaced parallel plates partially immersed in an infinite water bath in a downward gravity field is considered. With different contact angles and distance between the plates on both sides, the force profiles of the middle plate in this three-plate system are investigated using the Young–Laplace equation in two dimensions, and five non-trivial qualitative force profiles are found to possibly depend on the contact angles and the distance. The study is then extended to the qualitative changes of stability and behaviours in the system, and the striking properties related to the bifurcation theory come to light. Results show that, for different contact angles, there are at most eight possible bifurcation diagrams where the distance between the plates on both sides is chosen as the bifurcation parameter. By analysing the force profile of the middle plate in each of the eight bifurcation diagrams, the stabilities of the equilibria of the plate can be obtained. The number and the stabilities of equilibria will change when the bifurcation parameter passes the critical value.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Interfacial Flows (free surface): Capillary flows Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 817 , 25 April 2017 , pp. 122 - 137
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Copyright
© 2017 Cambridge University Press 

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References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, pp. 589–607. Dover.Google Scholar
Aspley, A., He, C. & McCuan, J. 2015 Force profiles for parallel plates partially immersed in a liquid bath. J. Math. Fluid Mech. 17, 87–102.Google Scholar
Bhatnagar, R. & Finn, R. 2006 Equilibrium configurations of an infinite cylinder in an unbounded fluid. Phys. Fluids 18, 047103.Google Scholar
Bhatnagar, R. & Finn, R. 2013 Attractions and repulsions of parallel plates partially immersed in a liquid bath: III. Bound. Value Probl. 2013, 277.Google Scholar
Bhatnagar, R. & Finn, R. 2016a On the capillarity equation in two dimensions. J. Math. Fluid Mech. 18, 731–738.Google Scholar
Bhatnagar, R. & Finn, R. 2016b The force singularity for partially immersed parallel plates. J. Math. Fluid Mech. 18, 739–755.CrossRefGoogle Scholar
Bowden, N., Terfort, A., Carbeck, J. & Whitesides, G. M. 1997 Self-assembly of mesoscale objects into ordered two-dimensional arrays. Science 276, 233–235.CrossRefGoogle ScholarPubMed
Bullard, J. W. & Garboczi, E. J. 2009 Capillary rise between planar surfaces. Phys. Rev. E 79, 011604.Google Scholar
Chan, D. Y. C., Henry, J. D. & White, L. R. 1981 The interaction of colloidal particles collected at fluid interfaces. J. Colloid Interface Sci. 79, 410–418.CrossRefGoogle Scholar
Concus, P. 1968 Static menisci in a vertical right circular cylinder. J. Fluid Mech. 34, 481–495.Google Scholar
Concus, P. & Finn, R. 1991 Exotic containers for capillary surfaces. J. Fluid Mech. 224, 383–394.Google Scholar
Concus, P., Finn, R. & Weislogel, M. 1999 Capillary surfaces in an exotic container: results from space experiments. J. Fluid Mech. 394, 119–135.CrossRefGoogle Scholar
Danov, K. D. & Kralchevsky, P. A. 2010 Capillary forces between particles at a liquid interface: general theoretical approach and interactions between capillary multipoles. Adv. Colloid Interface Sci. 154, 91–103.CrossRefGoogle Scholar
Finn, R. 1988 Non-uniqueness and uniqueness of capillary surfaces. Manuscr. Math. 61, 347–372.CrossRefGoogle Scholar
Finn, R. 2010 On Young’s paradox, and the attractions of immersed parallel plates. Phys. Fluids 22, 017103.Google Scholar
Finn, R. 2013 Capillary forces on partially immersed plates. In Differential and Difference Equations with Applications, pp. 13–25. Springer.CrossRefGoogle Scholar
Finn, R. & Lu, D. 2013 Mutual attractions of partially immersed parallel plates. J. Math. Fluid Mech. 15, 273–301.CrossRefGoogle Scholar
Forsythe, G. E., Moler, C. B. & Malcolm, M. A. 1977 Solution of nonlinear equations. In Computer Methods for Mathematical Computations, pp. 156–171. Prentice Hall.Google Scholar
Fortes, M. A. 1982 Attraction and repulsion of floating particles. Can. J. Chem. 60, 2889–2895.CrossRefGoogle Scholar
Gifford, W. A. & Scriven, L. E. 1971 On the attraction of floating particles. Chem. Engng Sci. 26, 287–297.Google Scholar
Joseph, D. D., Wang, J., Bai, R., Yang, B. H. & Hu, H. H. 2003 Particle motion in a liquid film rimming the inside of a partially filled rotating cylinder. J. Fluid Mech. 496, 139–163.Google Scholar
Kralchevsky, P. A., Paunov, V. N., Denkov, N. D., Ivanov, I. B. & Nagayama, K. 1993 Energetical and force approaches to the capillary interactions between particles attached to a liquid–fluid interface. J. Colloid Interface Sci. 155, 420–437.CrossRefGoogle Scholar
Kralchevsky, P. A., Paunov, V. N., Ivanov, I. B. & Nagayama, K. 1992 Capillary meniscus interaction between colloidal particles attached to a liquid–fluid interface. J. Colloid Interface Sci. 151, 79–94.Google Scholar
Liu, J. L., Feng, X. Q. & Wang, G. F. 2007 Buoyant force and sinking conditions of a hydrophobic thin rod floating on water. Phys. Rev. E 76, 066103.Google Scholar
Loudet, J. C., Alsayed, A. M., Zhang, J. & Yodh, A. G. 2005 Capillary interactions between anisotropic colloidal particles. Phys. Rev. Lett. 94, 018301.Google Scholar
Majumdar, S. R. & Michael, D. H. 1976 The equilibrium and stability of two dimensional pendent drops. Proc. R. Soc. Lond. A 351, 89–115.Google Scholar
McCuan, J. 2007 A variational formula for floating bodies. Pac. J. Maths 231, 167–191.Google Scholar
McCuan, J. & Treinen, R. 2013 Capillarity and Archimedes’ principle. Pac. J. Maths 265, 123–150.CrossRefGoogle Scholar
Miersemann, E.2015 Lecture Notes, ‘Liquid Interfaces’, Version November 2015, pp. 54–56. See http://www.math.uni-leipzig.de/∼miersemann.Google Scholar
Nicolson, M. M. 1949 The interaction between floating particles. Proc. Camb. Phil. Soc. 45, 288–295.Google Scholar
Padday, J. F. 1971 The profiles of axially symmetric menisci. Phil. Trans. R. Soc. Lond. A 269, 265–293.Google Scholar
Paunov, V. N., Kralchevsky, P. A., Denkov, N. D., Ivanov, I. B. & Nagayama, K. 1992 Capillary meniscus interaction between a microparticle and a wall. Colloids Surf. 67, 119–138.Google Scholar
Peruzzo, P., Defina, A., Nepf, H. M. & Stocker, R. 2013 Capillary interception of floating particles by surface-piercing vegetation. Phys. Rev. Lett. 111, 164501.CrossRefGoogle ScholarPubMed
Saif, T. A. 2002 On the capillary interaction between solid plates forming menisci on the surface of a liquid. J. Fluid Mech. 473, 321–347.Google Scholar
Seydel, R. 2009 Practical Bifurcation and Stability Analysis, vol. 5. Springer.Google Scholar
Stamou, D., Duschl, C. & Johannsmann, D. 2000 Long-range attraction between colloidal spheres at the air–water interface: the consequence of an irregular meniscus. Phys. Rev. E 62, 5263–5272.Google Scholar
Starov, V. M., Velarde, M. G. & Radke, C. J. 2007 Capillary interaction between solid bodies. In Wetting and Spreading Dynamics, Surfactant Science Series, vol. 138, pp. 144–152. CRC Press.Google Scholar
Treinen, R. 2016 Examples of non-uniqueness of the equilibrium states for a floating ball. Adv. Mater. Phys. Chem. 6, 177–194.CrossRefGoogle Scholar
Vassileva, N. D., Van Den Ende, D., Mugele, F. & Mellema, J. 2005 Capillary forces between spherical particles floating at a liquid–liquid interface. Langmuir 21, 11190–11200.CrossRefGoogle Scholar
Vella, D. 2015 Floating versus sinking. Annu. Rev. Fluid Mech. 47, 115–135.Google Scholar
Vella, D., Metcalfe, P. D. & Whittaker, R. J. 2006 Equilibrium conditions for the floating of multiple interfacial objects. J. Fluid Mech. 549, 215–224.Google Scholar
Wente, H. C. 2011 Exotic capillary tubes. J. Math. Fluid Mech. 13, 355–370.Google Scholar