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EFFECTS OF SURFACE TENSION ON TRAPPED MODES IN A TWO-LAYER FLUID

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  02 October 2015

S. SAHA  and
S. N. BORA Open the ORCID record for S. N. BORA [Opens in a new window]
S. SAHA
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India email s.sunanda@iitg.ernet.in, swaroop@iitg.ernet.in
S. N. BORA*
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India email s.sunanda@iitg.ernet.in, swaroop@iitg.ernet.in
*
✉swaroop@iitg.ernet.in
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Abstract

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We consider a two-layer fluid of finite depth with a free surface and, in particular, the surface tension at the free surface and the interface. The usual assumptions of a linearized theory are considered. The objective of this work is to analyse the effect of surface tension on trapped modes, when a horizontal circular cylinder is submerged in either of the layers of a two-layer fluid. By setting up boundary value problems for both of the layers, we find the frequencies for which trapped waves exist. Then, we numerically analyse the effect of variation of surface tension parameters on the trapped modes, and conclude that realistic changes in surface tension do not have a significant effect on the frequencies of these.

Keywords

trapped modesurface tensioncut-off frequencytwo-layer fluid

MSC classification

Primary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Secondary: 76B55: Internal waves
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 2 , October 2015 , pp. 189 - 203
Copyright
© 2015 Australian Mathematical Society 

References

Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, London, 2000).CrossRefGoogle Scholar
Bhattacharjee, J. and Sahoo, T., “Flexural gravity wave problems in two-layer fluids”, Wave Motion 45 (2008) 133–153; doi:10.1016/j.wavemoti.2007.04.006.CrossRefGoogle Scholar
Harter, R., Abrahams, I. D. and Simon, M. J., “The effect of surface tension on trapped modes in water-wave problems”, Proc. R. Soc. Lond. A 463 (2007) 147–180; doi:10.1098/rspa.2007.0063.Google Scholar
Harter, R., Simon, M. J. and Abrahams, I. D., “The effect of surface tension on localized free-surface oscillations about surface-piercing bodies”, Proc. R. Soc. Lond. A 464 (2008) 3039–3054; doi:10.1098/rspa.2008.0059.Google Scholar
John, F., “On the motion of floating bodies II”, Commun. Pure Appl. Math. 3 (1950) 45–101; doi:10.1002/cpa.3160030106.CrossRefGoogle Scholar
Jones, D. S., “The eigenvalues of ${\rm\nabla}^{2}u+{\it\lambda}u=0$ when the boundary conditions are given on semi-infinite domains”, Proc. Camb. Phil. Soc. 49 (1953) 668–684; doi:10.1017/S0305004100028875.CrossRefGoogle Scholar
Kassem, S. E., “Wave source potentials for two superposed fluids, each of finite depth”, Math. Proc. Camb. Phil. Soc. 100 (1986) 595–599; doi:10.1017/S0305004100066329.CrossRefGoogle Scholar
Kuznetsov, N., “Trapped modes of internal waves in a channel spanned by a submerged cylinder”, J. Fluid Mech. 254 (1993) 113–126; doi:10.1017/S0022112093002058.CrossRefGoogle Scholar
Linton, C. M. and Cadby, J. R., “Trapped modes in a two-layer fluid”, J. Fluid Mech. 481 (2003) 215–234; doi:10.1017/S002211200300377X.CrossRefGoogle Scholar
McIver, M., “An example of non-uniqueness in the two-dimensional linear water wave problem”, J. Fluid Mech. 315 (1996) 257–266; doi:10.1017/S0022112096002418.CrossRefGoogle Scholar
Motygin, O. V. and McIver, P., “Trapping of gravity-capillary water waves by submerged obstacles”, Proc. R. Soc. Lond. A 465 (2009) 1743–1761; http://www.iwwwfb.org/abstracts/iwwwfb24/iwwwfb24_40.pdf.Google Scholar
Mohapatra, S. C. and Sahoo, T., “On capillary gravity-wave motion in two-layer fluids”, J. Engrg. Math. 71 (2011) 253–277; doi:10.1016/j.apor.2010.12.001.CrossRefGoogle Scholar
Saha, S. and Bora, S. N., “Trapped modes in a two-layer fluid of finite depth bounded above by a rigid lid”, Wave Motion 50 (2013) 1050–1060; doi:10.1016/j.wavemoti.2013.04.009.CrossRefGoogle Scholar
Saha, S. and Bora, S. N., “Flexural gravity waves trapped in a two-layer fluid of finite depth”, Appl. Ocean Res. 44 (2014) 1–12; doi:10.1016/j.apor.2013.08.005.CrossRefGoogle Scholar
Stokes, G. G., “Report on recent researches in hydrodynamics”, Math. Phys. Pap. 1 (1880) 157–187; doi:10.1017/CBO9780511702242.011.Google Scholar
Ursell, F., “Surface waves on deep water in the presence of a submerged circular cylinder”, Proc. Camb. Phil. Soc. 46 (1950) 141–152; doi:10.1017/S0305004100025561.CrossRefGoogle Scholar
Ursell, F., “Trapping modes in the theory of surface waves”, Proc. Camb. Phil. Soc. 47 (1951) 347–358; doi:10.1017/S0305004100026700.CrossRefGoogle Scholar
Ursell, F., “Edge waves on a sloping beach”, Proc. R. Soc. Lond. A 214 (1952) 79–97; doi:10.1098/rspa.1952.0152.Google Scholar
Ursell, F., “The local expansion of a source of oblique water waves in the free surface”, Wave Motion 33 (2001) 109–116; doi:10.1016/S0165-2125(00)00067-6.CrossRefGoogle Scholar