American Institute of Mathematical Sciences

Advanced
July  2011, 10(4): 1183-1204. doi: 10.3934/cpaa.2011.10.1183

The bifurcation of interfacial capillary-gravity waves under O(2) symmetry

Mark Jones 1,

1. 

School of Mathematics, Kingston University, Kingston-on-Thames, KT1 2EE01103, United Kingdom

Received  March 2010 Revised  November 2010 Published  April 2011

The evolution of an interface between two fluids of different densities is considered. The particular case under examination is when the motion is due to an interaction between the Mth and Nth harmonics of the fundamental mode. By means of a hodograph transformation the problem is cast as an operator equation between two suitable function spaces. Classical techniques are used to reduce the problem to a finite system of algebraic equations. Solutions are found which exhibit a rich variety of behaviour including primary, secondary and multiple bifurcation.
Keywords: Bifurcation, fluid-fluid interface, group invariance., Lyapunov-Schmidt.
Mathematics Subject Classification: Primary: 37G40, 76B15; Secondary: 34G5.
Citation: Mark Jones. The bifurcation of interfacial capillary-gravity waves under O(2) symmetry. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1183-1204. doi: 10.3934/cpaa.2011.10.1183
References:
[1]

D. Armbruster and G. Dangelmayr, Coupled stationary bifurcations in nonflux boundary value problems,, Math. Proc. Camb. Phil. Soc., 101 (1987), 167.  doi: 10.1017/S0305004100066500.  Google Scholar

[2]

N. K. Bari, "A Treatise on Trigonometric Series,", Pergamon Press, (1964).   Google Scholar

[3]

B. Chen and P. G. Saffman, Steady capillary-gravity waves in deep water-1.weakly nonlinear waves,, Stud. App. Math., 60 (1979), 183.   Google Scholar

[4]

P. Christodoulides and F. Dias, Stability of capillary-gravity interfacial waves between two bounded fluids,, Phys. Fluids, 70 (1995), 3013.  doi: 10.1063/1.868678.  Google Scholar

[5]

P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves,, J. Fluid Mech., 265 (1994), 303.  doi: 10.1017/S0022112094000856.  Google Scholar

[6]

H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems,, Physica D, 5 (1982), 1.  doi: 10.1016/0167-2789(82)90048-3.  Google Scholar

[7]

M. Golubitsky and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol I,", Springer, (1985).   Google Scholar

[8]

M. Golubitsky, I Stewart and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol II,", Springer, (1988).   Google Scholar

[9]

M. Jones, Small amplitude capillary-gravity waves in a channel of finite depth,, Glasgow Math. J., 31 (1989), 465.   Google Scholar

[10]

M. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction,, J. Fluid Mech., 341 (1997), 295.  doi: 10.1017/S0022112097005624.  Google Scholar

[11]

M. Jones, A system of equations which predicts the nonlinear evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption,, Fluid Dyn. Res., 21 (1997), 455.  doi: 10.1016/S0169-5983(97)00024-5.  Google Scholar

[12]

M. Jones, Group invariances, unfoldings and the bifurcation of capillary-gravity waves,, Int. J. Bif. Chaos, 7 (1997), 1243.  doi: 10.1142/S021812749700100X.  Google Scholar

[13]

M. Jones and J. F. Toland, Symmetry and the bifurcation of capillary-gravity waves,, Arch. Rat. Mech. Anal., 96 (1986), 29.  doi: 10.1007/BF00251412.  Google Scholar

[14]

N. Kotchine, Determination rigoureuse des ondes permanentes d'ampleur finie a la surface de seperation deux liquides de profondeur finie,, Math. Ann., 98 (1928), 582.  doi: 10.1007/BF01451610.  Google Scholar

[15]

L. M. Milne-Thomson, "Theretical Hydrodynamics," 4th, edition, (1960).   Google Scholar

[16]

H. Okamoto, On the problem of water-waves of permanent configuration,, Nonlinear Analysis, 14 (1990), 469.  doi: 10.1016/0362-546X(90)90035-F.  Google Scholar

[17]

H. Okamoto, Interfacial progressive water waves- a singularity theoretic view,, Tohoku Math. J., 49 (1997), 33.   Google Scholar

[18]

H. Okamoto and M. Shoji, Remarks on the bifurcation of two dimensional capillary-gravity waves of finite depth,, Publ. Res. Inst. Math. Sci., 30 (1994), 611.  doi: 10.2977/prims/1195165792.  Google Scholar

[19]

J. F. Toland and M. C. W. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. R. Soc. Lond., 399 (1985), 391.  doi: 10.1098/rspa.1985.0063.  Google Scholar

show all references

References:
[1]

D. Armbruster and G. Dangelmayr, Coupled stationary bifurcations in nonflux boundary value problems,, Math. Proc. Camb. Phil. Soc., 101 (1987), 167.  doi: 10.1017/S0305004100066500.  Google Scholar

[2]

N. K. Bari, "A Treatise on Trigonometric Series,", Pergamon Press, (1964).   Google Scholar

[3]

B. Chen and P. G. Saffman, Steady capillary-gravity waves in deep water-1.weakly nonlinear waves,, Stud. App. Math., 60 (1979), 183.   Google Scholar

[4]

P. Christodoulides and F. Dias, Stability of capillary-gravity interfacial waves between two bounded fluids,, Phys. Fluids, 70 (1995), 3013.  doi: 10.1063/1.868678.  Google Scholar

[5]

P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves,, J. Fluid Mech., 265 (1994), 303.  doi: 10.1017/S0022112094000856.  Google Scholar

[6]

H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems,, Physica D, 5 (1982), 1.  doi: 10.1016/0167-2789(82)90048-3.  Google Scholar

[7]

M. Golubitsky and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol I,", Springer, (1985).   Google Scholar

[8]

M. Golubitsky, I Stewart and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol II,", Springer, (1988).   Google Scholar

[9]

M. Jones, Small amplitude capillary-gravity waves in a channel of finite depth,, Glasgow Math. J., 31 (1989), 465.   Google Scholar

[10]

M. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction,, J. Fluid Mech., 341 (1997), 295.  doi: 10.1017/S0022112097005624.  Google Scholar

[11]

M. Jones, A system of equations which predicts the nonlinear evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption,, Fluid Dyn. Res., 21 (1997), 455.  doi: 10.1016/S0169-5983(97)00024-5.  Google Scholar

[12]

M. Jones, Group invariances, unfoldings and the bifurcation of capillary-gravity waves,, Int. J. Bif. Chaos, 7 (1997), 1243.  doi: 10.1142/S021812749700100X.  Google Scholar

[13]

M. Jones and J. F. Toland, Symmetry and the bifurcation of capillary-gravity waves,, Arch. Rat. Mech. Anal., 96 (1986), 29.  doi: 10.1007/BF00251412.  Google Scholar

[14]

N. Kotchine, Determination rigoureuse des ondes permanentes d'ampleur finie a la surface de seperation deux liquides de profondeur finie,, Math. Ann., 98 (1928), 582.  doi: 10.1007/BF01451610.  Google Scholar

[15]

L. M. Milne-Thomson, "Theretical Hydrodynamics," 4th, edition, (1960).   Google Scholar

[16]

H. Okamoto, On the problem of water-waves of permanent configuration,, Nonlinear Analysis, 14 (1990), 469.  doi: 10.1016/0362-546X(90)90035-F.  Google Scholar

[17]

H. Okamoto, Interfacial progressive water waves- a singularity theoretic view,, Tohoku Math. J., 49 (1997), 33.   Google Scholar

[18]

H. Okamoto and M. Shoji, Remarks on the bifurcation of two dimensional capillary-gravity waves of finite depth,, Publ. Res. Inst. Math. Sci., 30 (1994), 611.  doi: 10.2977/prims/1195165792.  Google Scholar

[19]

J. F. Toland and M. C. W. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. R. Soc. Lond., 399 (1985), 391.  doi: 10.1098/rspa.1985.0063.  Google Scholar

[1]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[2]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
[3]

Jeffrey J. Early, Juha Pohjanpelto, Roger M. Samelson. Group foliation of equations in geophysical fluid dynamics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1571-1586. doi: 10.3934/dcds.2010.27.1571
[4]

Chiu-Ya Lan, Chi-Kun Lin. Asymptotic behavior of the compressible viscous potential fluid: Renormalization group approach. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 161-188. doi: 10.3934/dcds.2004.11.161
[5]

B. Wiwatanapataphee, Theeradech Mookum, Yong Hong Wu. Numerical simulation of two-fluid flow and meniscus interface movement in the electromagnetic continuous steel casting process. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1171-1183. doi: 10.3934/dcdsb.2011.16.1171
[6]

George Avalos, Roberto Triggiani. Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 817-833. doi: 10.3934/dcds.2008.22.817
[7]

Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838
[8]

Takayuki Kubo, Yoshihiro Shibata, Kohei Soga. On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3741-3774. doi: 10.3934/dcds.2016.36.3741
[9]

Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086
[10]

Tian Ma, Shouhong Wang. Boundary layer separation and structural bifurcation for 2-D incompressible fluid flows. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 459-472. doi: 10.3934/dcds.2004.10.459
[11]

Quan Wang. Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 543-563. doi: 10.3934/dcdsb.2014.19.543
[12]

Boris Andreianov, Frédéric Lagoutière, Nicolas Seguin, Takéo Takahashi. Small solids in an inviscid fluid. Networks & Heterogeneous Media, 2010, 5 (3) : 385-404. doi: 10.3934/nhm.2010.5.385
[13]

R. H.W. Hoppe, William G. Litvinov. Problems on electrorheological fluid flows. Communications on Pure & Applied Analysis, 2004, 3 (4) : 809-848. doi: 10.3934/cpaa.2004.3.809
[14]

Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028
[15]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133
[16]

Horst Heck, Gunther Uhlmann, Jenn-Nan Wang. Reconstruction of obstacles immersed in an incompressible fluid. Inverse Problems & Imaging, 2007, 1 (1) : 63-76. doi: 10.3934/ipi.2007.1.63
[17]

T. Tachim Medjo, Louis Tcheugoue Tebou. Robust control problems in fluid flows. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 437-463. doi: 10.3934/dcds.2005.12.437
[18]

Seung-Yeal Ha, Doron Levy. Particle, kinetic and fluid models for phototaxis. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 77-108. doi: 10.3934/dcdsb.2009.12.77
[19]

Vincent Giovangigli. Persistence of Boltzmann entropy in fluid models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 95-114. doi: 10.3934/dcds.2009.24.95
[20]

Ciro D'Apice, Rosanna Manzo. A fluid dynamic model for supply chains. Networks & Heterogeneous Media, 2006, 1 (3) : 379-398. doi: 10.3934/nhm.2006.1.379

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences