Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-19T09:28:45.991Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  20 January 2015

Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India email,
Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar 140001, Punjab, India email,
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India email
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

Research Article
© 2015 Australian Mathematical Society 


Batchelor, G. K., An introduction to fluid dynamics (Cambridge University Press, Cambridge, UK, 1967).Google Scholar
Belward, S. R. and Forbes, L. K., “Fully non-linear two-layer flow over arbitrary topography”, J. Engrg. Math. 27 (1993) 419432; doi:10.1007/BF00128764.Google Scholar
Bhattacharjee, J. and Sahoo, T., “Flexural gravity wave problems in two-layer fluids”, Wave Motion 45 (2008) 133153; doi:10.1016/j.wavemoti.2007.04.006.Google Scholar
Brown, J. W., Churchill, R. V. and Lapidus, M., Complex variables and applications (McGraw-Hill, New York, 1996).Google Scholar
Candela, J., “Mediterranean water and global circulation”, in: Ocean circulation and climate—observing and modelling the global ocean, Volume 77 of Int. Geophys. Ser. (Academic Press–Elsevier, New York, 2001), 419–429.Google Scholar
Chakrabarti, A. and Martha, S. C., “A review on the mathematical aspects of fluid flow problems in an infinite channel with arbitrary bottom topography”, J. Appl. Math. Inform. 29 (2011) 15831602; Scholar
Chen, M. J. and Forbes, L. K., “Steady periodic waves in a threelayer fluid with shear in the middle layer”, J. Fluid Mech. 594 (2008) 157181; doi:10.1017/S0022112007008877.Google Scholar
Dias, F. and Vanden-Broeck, J.-M., “Open channel flows with submerged obstructions”, J. Fluid Mech. 206 (1989) 155170; doi:10.1017/S0022112089002260.Google Scholar
Dias, F. and Vanden-Broeck, J.-M., “Steady two-layer flows over an obstacle”, Philos. Trans. R. Soc. Lond. A 360 (2002) 21372154; doi:10.1098/rsta.2002.1070.Google Scholar
Dias, F. and Vanden-Broeck, J.-M., “Generalised critical free-surface flows”, J. Engrg. Math. 42 (2002) 291301; doi:10.1023/A:1016111415763.Google Scholar
Dongqiang, L., Shiqiang, D. and Baoshan, Z., “Hamiltonian formulation of nonlinear water waves in a two-fluid system”, Appl. Math. Mech. (English Ed.) 20 (1999) 343349 ;doi:10.1007/BF02458559.CrossRefGoogle Scholar
Duchêne, V., “On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast”, J. Nonlinear Sci. 24 (2014) 579632; doi:10.1007/s00332-014-9200-2.Google Scholar
Forbes, L. K., “Critical free-surface flow over a semi-circular obstruction”, J. Engrg. Math. 22 (1988) 313; doi:10.1007/BF00044362.Google Scholar
Forbes, L. K., “Two-layer critical flow over a semi-circular obstruction”, J. Engrg. Math. 23 (1989) 325342; doi:10.1007/BF00128906.Google Scholar
Forbes, L. K. and Hocking, G. C., “An intrusion layer in stationary incompressible fluids. Part 2: A solitary wave”, European J. Appl. Math. 17 (2006) 577595; doi:10.1017/S0956792506006711.Google Scholar
Forbes, L. K., Hocking, G. C. and Farrow, D. E., “An intrusion layer in stationary incompressible fluids. Part 1. Periodic waves”, European J. Appl. Math. 17 (2006) 557575 ;doi:10.1017/S0956792506006693.Google Scholar
Forbes, L. K. and Schwartz, L. W., “Free-surface flow over a semicircular obstruction”, J. Fluid Mech. 114 (1982) 299314; doi:10.1017/S0022112082000160.Google Scholar
Grimshaw, R. H. J. and Smyth, N., “Resonant flow of a stratified fluid over topography”, J. Fluid Mech. 16 (1986) 429464; doi:10.1017/S002211208600071X.CrossRefGoogle Scholar
Higgins, P. J., Read, W. W. and Belward, S. R., “A series-solution method for free-boundary problems arising from flow over topography”, J. Engrg. Math. 54 (2006) 345358 ;doi:10.1007/s10665-006-9039-0.Google Scholar
Kim, J., Moin, P. and Moser, R., “Turbulence statistics in fully developed channel flow at low Reynolds number”, J. Fluid Mech. 177 (1987) 133166; doi:10.1017/S0022112087000892.Google Scholar
Lamb, H., Hydrodynamics (Cambridge University Press, Cambridge, 1932).Google Scholar
Long, R. R., “Some aspects of the flow of stratified fluids: I. A theoretical investigation”, Tellus 5 (1953) 4258; doi:10.1111/j.2153–34901953.tb01035.x.Google Scholar
Lord Kelvin (W. Thomson), “On stationary waves in flowing water”, Philos. Mag. Ser. 5 22 (1886) 353357; doi:10.1080/14786448608627944.Google Scholar
Milewski, P. and Vanden-Broeck, J.-M., “Time dependent gravity capillary flows past an obstacle”, Wave Motion 29 (1999) 6379; doi:10.1016/S0165-2125(98)00021-3.Google Scholar
Shen, S. P., Shen, M. C. and Sun, S. M., “A model equation for steady surface waves over a bump”, J. Engrg. Math. 23 (1989) 315323; doi:10.1007/BF00128905.Google Scholar
Stokes, T. E., Hocking, G. C. and Forbes, L. K., “Unsteady flow induced by a withdrawal point beneath a free surface”, ANZIAM J. 47 (2005) 185202; doi:10.1017/S1446181100009986.Google Scholar
Titchmarsh, E. C., The theory of functions, 2nd edn (Oxford University Press, Oxford, UK, 1976).Google Scholar
Vanden-Broeck, J.-M., “Free-surface flow over a semi-circular obstruction in a channel”, Phys. Fluids 30 (1987) 23152317; doi:10.1063/1.866121.Google Scholar
Williams, M. J. M., Jenkins, A. and Determann, J., “Physical controls on ocean circulation beneath ice shelves revealed by numerical models”, Antarct. Res. Ser. 75 (1998) 285299 ;doi:10.1029/AR075p0285.Google Scholar
Winant, C. D., Dorman, C. E., Friehe, C. A. and Beardsley, R. C., “The marine layer off Northern California: an example of supercritical channel flow”, J. Atmos. Sci. 45 (1988) 35883605 ; doi:10.1175/1520-0469(1988)045<3588:TMLONC>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Yong, Z., “Resonant flow of a fluid past a concave topography”, Appl. Math. Mech. (English Ed.) 18 (1997) 479482; doi:10.1007/BF02453743.CrossRefGoogle Scholar