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Double Degeneracy in the Immiscible Wilton Ripple Phenomenon: A Three Perturbation Parameter Problem in Bifurcation Theory

Department of Mathematics, Kingston University, Kingston-on-Thames, KT1 2EE, Surrey, UK

E-mail Address: mark.jones@kingston.ac.uk

https://doi.org/10.1142/S021812742150190XCited by:0 (Source: Crossref)

Abstract

A study is made of the waves which may occur at the interface of two fluids and which arise from an interaction between the two lowest adjacent harmonics of the motion. The problem is studied by reformulating it, via a conformal mapping transformation, into a pair of coupled nonlinear functional differential equations which are invariant under the action of the group $O (2) .$ $O (2) .$ The classical method of Lyapunov–Schmidt is then employed to replace this formulation by a pair of algebraic equations, known as the bifurcation equations. Throughout our analysis we impose a physical condition which essentially means that the densities and phase speeds in each fluid are approximately equal. This means that the bifurcation equations assume a cubic form rather than the quadratic form which they generically possess in problems of this type. The bifurcation equations are then analyzed by means of singularity theory by which means we are able to draw the solution curves of the system. A large variety of solution diagrams are found to exhibit such behavior as multiple primary bifurcations, secondary bifurcations and bifurcation from infinity.

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References

Armbruster, D. & Dangelmayr, G. [1986] “ Corank-two bifurcations for the Brusselator with non-flux boundary conditions,” Dyn. Stab. Syst. 1, 187–200. Crossref, Google Scholar
Armbruster, D. & Dangelmayr, G. [1987] “ Coupled stationary bifurcations in non-flux boundary value problems,” Math. Proc. Camb. Phil. Soc. 101, 167–192. Crossref, Web of Science, Google Scholar
Barakat, R. & Houston, A. [1968] “ Nonlinear periodic capillary-gravity waves in a fluid of finite depth,” J. Geophys. Res. 73, 6545–6555. Crossref, Web of Science, Google Scholar
Buzano, E. & Russo, A. [1987] “ Bifurcation problems with $O (2) \oplus ℤ_{2}$ symmetry and the buckling of a cylindrical shell,” Annali di Matematica Pura ed Applicata (IV) 146, 217–262. Crossref, Web of Science, Google Scholar
Chen, B. & Saffman, P. G. [1979] “ Steady gravity-capillary waves on deep water-I. Weakly nonlinear waves,” Stud. Appl. Math. 60, 183–210. Crossref, Web of Science, Google Scholar
Crandall, M. G. & Rabinowitz, P. H. [1971] “ Bifurcation from simple eigenvalues,” J. Funct. Anal. 8, 321–340. Crossref, Google Scholar
Dancer, E. N. [1973] “ Global solution branches for positive mappings,” Arch. Rat. Mech. Anal. 52, 181–192. Crossref, Web of Science, Google Scholar
Dangelmayr, G. [1986] “ Steady state mode interactions in the presence of $O (2)$ symmetry,” Dyn. Stab. Syst. 1, 159–185. Crossref, Google Scholar
Dangelmayr, G. & Armbruster, D. [1986] “ Steady state mode interactions in the presence of $O (2)$ symmetry and in non-flux boundary value problems,” Contemp. Math. 56, 53–68. Crossref, Google Scholar
Golubistsky, M. & Schaeffer, D. [1979a] “ A theory for imperfect bifurcation theory via singularity theory,” Commun. Pure Appl. Math. 32, 21–98. Crossref, Web of Science, Google Scholar
Golubistsky, M. & Schaeffer, D. [1979b] “ Imperfect bifurcation in the presence of symmetry,” Commun. Math. Phys. 67, 205–232. Crossref, Web of Science, Google Scholar
Golubistsky, M. & Schaeffer, D. [1985] Singularities and Groups in Bifurcation Theory, Vol. I (Springer, NY). Crossref, Google Scholar
Golubistsky, M., Stewart, I. & Schaeffer, D. [1988] Singularities and Groups in Bifurcation Theory, Vol. II (Springer, NY). Crossref, Google Scholar
Harrison, W. J. [1909] “ The influence of viscosity and capillarity on waves of finite amplitude,” Proc. Lond. Math. Soc. 7, 107–121. Crossref, Google Scholar
Jones, M. C. W. & Toland, J. F. [1986] “ Symmetry and the bifurcation of capillary-gravity waves,” Arch. Rat. Mech. Anal. 96, 29–53. Crossref, Web of Science, Google Scholar
Jones, M. C. W. [1997a] “ Group invariances, unfoldings and the bifurcation of capillary-gravity waves,” Int. J. Bifurcation and Chaos 7, 1243–1266. Link, Web of Science, Google Scholar
Jones, M. C. W. [1997b] “ Universal unfoldings of group invariant equations which model second and third resonant capillary-gravity waves,” Comp. Math. Appl. 32, 59–89. Crossref, Web of Science, Google Scholar
Jones, M. C. W. [2000] “ A co-dimension unfolding of a third harmonic resonant interaction between two contiguous continua,” Zeit. Angew. Math. Phys. 51, 351–373. Crossref, Web of Science, Google Scholar
Jones, M. C. W. [2011] “ The bifurcation of interfacial capillary-gravity waves under $O (2)$ symmetry,” Comm. Pure Applied Anal. 10, 1183–1204. Crossref, Google Scholar
Jones, M. C. W. [2015] “ Group equivariant singularity theory in the bifurcation of interfacial Wilton ripples,” Int. J. Bifurcation and Chaos 25, 1550147-1–19. Link, Web of Science, Google Scholar
Kotchine, N. [1928] “ Détermination rigoreuse de ondes permanentes d’ampleur finie a la surface de séparation deux liquides de profondeur finie,” Math. Ann. 98, 582–615. Crossref, Google Scholar
McGoldrick, L. F. [1965] “ Resonant interactions between capillary-gravity waves,” J. Fluid Mech. 21, 305–331. Crossref, Web of Science, Google Scholar
McGoldrick, L. F. [1970a] “ An experiment on second order capillary-gravity wave resonant wave interactions,” J. Fluid Mech. 40, 251–271. Crossref, Web of Science, Google Scholar
McGoldrick, L. F. [1970b] “ On Wilton’s ripples: A special case of resonant interactions,” J. Fluid Mech. 42, 193–200. Crossref, Web of Science, Google Scholar
McGoldrick, L. F. [1972] “ On the rippling of small waves: A harmonic nonlinear nearly resonant interaction,” J. Fluid Mech. 52, 725–751. Crossref, Web of Science, Google Scholar
Nayfeh, A. H. [1970] “ Finite amplitude surface waves in a liquid layer,” J. Fluid Mech. 40, 671–684. Crossref, Web of Science, Google Scholar
Nayfeh, A. H. & Saric, W. S. [1972] “ Nonlinear waves in Kelvin–Helmholtz flow,” J. Fluid Mech. 55, 311–327. Crossref, Web of Science, Google Scholar
Nayfeh, A. H. [1973] “ Second harmonic resonance in the interaction of an air stream with capillary-gravity waves,” J. Fluid Mech. 59, 803–816. Crossref, Web of Science, Google Scholar
Okamoto, H. [1990] “ On the problem of water waves of permanent configuration,” Nonlin. Anal. 14, 469–481. Crossref, Web of Science, Google Scholar
Okamoto, H. & Shoji, M. [1996] “ The resonance of modes in the problem of two-dimensional capillary-gravity waves,” Physica D 95, 336–350. Crossref, Web of Science, Google Scholar
Okamoto, H. [1997] “ Interfacial progressive water waves — A singularity theoretic view,” Tôhoku Math. J. 49, 33–57. Crossref, Web of Science, Google Scholar
Pierson, W. J. & Fife, P. C. [1961] “ Some nonlinear properties of long-crested periodic waves with lengths near 2.44 centimeters,” J. Geophys. Res. 66, 163–179. Crossref, Web of Science, Google Scholar
Rabinowitz, P. H. [1971] “ Some global results for nonlinear eigenvalue problems,” J. Funct. Anal. 7, 487–513. Crossref, Google Scholar
Reeder, J. & Shinbrot, M. [1981] “ On Wilton’s ripples II. Rigorous results,” Arch. Rat. Mech. Anal. 77, 321–347. Crossref, Web of Science, Google Scholar
Toland, J. F. & Jones, M. C. W. [1985] “ The bifurcation and secondary bifurcation of capillary-gravity waves,” Proc. R. Soc. Lond. 399, 391–417. Google Scholar
Wilton, J. R. [1915] “On ripples,” Phil. Mag. 29, 688–700. Google Scholar