Forced capillary gravity surface waves over a bump – Critical surface tension case

Choi, Jeongwhan ; Whang, SungIm ; Kim, Junseok

ANZIAM Journal, Tome 59 (2018), / Harvested from Australian Mathematical Society

Résumé

This paper concerns forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with nonzero surface tension on the free surface and a small bump on a horizontal rigid flat bottom. It is known that if non-dimensional wave speed, called Froude number, is near 1 and a non-dimensional surface tension, called Bond number, is near 1/3, the KdV theory fails and a time dependent fifth order KdV equation, called the Kawahara equation, can be derived to study the wave motion on the free surface. In this paper both time independent and time dependent forms of the Kawahara equation with a forcing are studied numerically and theoretically and various numerical results are presented. References B. Buffoni. Infinitely many large amplitude homoclinic orbits for a class of autonomous hamiltonian systems. J. Differential Equations (121):109–120, 1995. doi:10.1006/jdeq.1995.1123 B. Buffoni and J. F. Toland. Global existence of homoclinic and periodic orbits for a class of autonomous hamiltonian systems. J. Differential Equations, (118):104–120, 1995. doi:10.1006/jdeq.1995.1068 A. V. Buryak and A. R. Champneys. On the stability of solitary wave solutions of the fifth-order kdv equation. Phys. Lett. A, (233):58–62, 1997. doi:10.1016/S0375-9601(97)00453-2 A. R. Champneys and J. F. Toland. Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous hamiltonian systems. Nonlinearity, (6):665–721, 1993. J. W. Choi, S. M. Sun, and M. C. Shen. Steady capillary-gravity waves on the interface of two-layer fluid over an obstruction-forced modified k-dv equation. J. Eng. Math., (28):193–210, 1994. doi:10.1007/BF00058436 J. W. Choi, S. M. Sun, and S. I. Whang. Steady surface waves on water over a bump with critical surface tension. accepted. J. W. Choi, S. M. Sun, and S. I. Whang. Supercritical surface gravity waves generated by a positive forcing. European J. Mech. B/Fluids, (27):750–770, 2008. doi:10.1016/j.euromechflu.2008.01.006 L. K. Forbes. Critical free surface flow over a semi-circular obstruction. J. Eng. Math., (22):3–13, 1988. doi:10.1007/BF00044362 J. Hunter and J. Scheurle. Existence of perturbed solitary wave solutions to a model equation for water waves. Phys. D, (32):253–268, 1988. doi:10.1016/0167-2789(88)90054-1 J.K Hunter and J.M. Vanden-Broeck. Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech., (134):205–219, 1983. doi:10.1017/S0022112083003316 T. Iguchi. A mathematical justification of the forced korteweg-de vries equation for capillary-gravity waves. Kyushu Journal of Mathematics, 60(2):267–303, 2006. doi:10.2206/kyushujm.60.267 J. W. Miles. Stationary, transcritical channel flows. J. Fluid Mech., (162):489–499, 1986. doi:10.1017/S0022112086002136 Y. Pomeau, A. Ramani, and B. Grammaticos. Structural stability of the korteweg-de vries solitons under a singular perturbation. Phys. D, (31):127–134, 1988. doi:10.1016/0167-2789(88)90018-8 J. J. Stoker. Water Waves: The Mathematical Theory with Applications. Pure and Applied Mathematics, Interscience Publishers, Inc., New York, 4 edition, 1957. S. M. Sun and M. C. Shen. Exponentially small estimate for a generalized solitary wave solution to the perturbed k-dv equation. Nonlinear Anal., (23):545–564, 1994. doi:10.1016/0362-546X(94)90093-0 L. N. Trefethen. Spectral Methods in MATLAB. SIAM, Philadelphia, PA, 2003. J. M. Vanden-Broeck. Free surface flow over a semi-circular obstruction in a channel. Phys. Fluids, (30):2315–2317, 1987.

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Publié le : 2018-01-01
DOI : https://doi.org/10.21914/anziamj.v59i0.12634