A formal expansion procedure for the internal solitary wave problem in a two-fluid system of constant topography. (English) Zbl 0736.76011
Summary: Equations are derived for a two-dimensional internal solitary wave between two immiscible superimposed, inviscid and incompressible liquids bounded by two rigid planes in a channel of infinite length analogous to those derived by B. Epstein [Q. Appl. Math. 32, 89-95 (1974; Zbl 0282.76012)] for the case of two-dimensional free surface solitary water waves. The effect of surface tension at the surface of separation has been neglected. A pair of ordinary differential equations, of an “infinite” order, as well as an algebraic equation has been obtained and solved by a power series expansion in terms of a parameter “\(a\)” which depends mainly on the reciprocal of Froude number. Comparing similar powers of “\(a\)” leads to a set of nonlinear, second-order differential equations. The analytical form of the wave profile has been found, up to a fourth order of the parameter “\(a\)”, in terms of the Froude number, density ratio between the two liquids, and the distance between the two planes bounding the two fluids. The effect of the Froude number, density ratio and the distance between the two planes, on the wave profile, maximum amplitude of the wave and the speed of a particle just above the interface, has been studied and illustrated.
MSC:
| 76B55 | Internal waves for incompressible inviscid fluids |
| 76V05 | Reaction effects in flows |
| 76B25 | Solitary waves for incompressible inviscid fluids |
Keywords:
two-dimensional internal solitary wave; immiscible superimposed, inviscid and incompressible liquids; power series expansionCitations:
Zbl 0282.76012References:
| [1] | Benjamin, T. B.: Internal waves of permanent form in fluids of great depth. J. Fluid Mech.29, 559-592 (1967). · Zbl 0147.46502 · doi:10.1017/S002211206700103X |
| [2] | Benny, D. J.: Long non-linear waves of fluid flows. J. Math. and Phys.45, 52-63 (1966). · Zbl 0151.42501 |
| [3] | Davis, R. E., Acrivos, A.: Solitary internal waves in deep water. J. Fluid Mech.29, 593-607 (1967). · Zbl 0147.46503 · doi:10.1017/S0022112067001041 |
| [4] | Dodd, R. K., Eilbeck, C. J., Gibbon, J. D., Morris, H. C.: Solitons and nonlinear wave equations. London: Academic Press 1982. · Zbl 0496.35001 |
| [5] | Darzin, P. G.: Solitons, 2nd ed. Cambridge: University Press 1984. |
| [6] | Epstein, B.: A formal expansion procedure for the solitary wave problem. Quart. Appl. Math.32, 89-95 (1974). · Zbl 0282.76012 |
| [7] | Friedrichs, K. O., Hyres, D. H.: The existence of solitary waves. Comm. Pure Appl. Math.7, 517-550 (1954). · Zbl 0057.42204 · doi:10.1002/cpa.3160070305 |
| [8] | Grimshaw, R.: Long nonlinear internal waves in channels of arbitrary cross-section. J. Fluid Mech.86, 415-431 (1978). · Zbl 0374.76023 · doi:10.1017/S0022112078001214 |
| [9] | Holyer, J. H.: Large amplitude progressive interfacial waves. J. Fluid Mech.93, 433-448 (1979). · Zbl 0426.76021 · doi:10.1017/S0022112079002585 |
| [10] | Kakutani, T., Yamasaki, N.: Solitary waves on a two-layer fluid. J. Phys. Soc. Jpn.45, 674-679 (1978). · doi:10.1143/JPSJ.45.674 |
| [11] | Kassem, S. E.: Wave source potentials for two superposed fluids, each of finite depth. Math. Proc. Camb. Phil. Soc.100, 595-599 (1986). · Zbl 0621.76023 · doi:10.1017/S0305004100066329 |
| [12] | Keulegan, G. H.: Characteristics of internal solitary waves. J. Res. Natl. Bur. Stand.51, 133-140 (1953). · Zbl 0052.21601 |
| [13] | Lamb, H.: Hydrodynamics, 6th ed. Cambridge University Press. (Dover edition 1945). · JFM 26.0868.02 |
| [14] | Long, R. R.: Solitary waves in one- and two-fluid systems. Tellus8, 460-471 (1956). · doi:10.1111/j.2153-3490.1956.tb01249.x |
| [15] | Miles, J. W.: Solitary waves. Ann. Rev. Fluid Mech.12, 11-43 (1980). · Zbl 0463.76026 · doi:10.1146/annurev.fl.12.010180.000303 |
| [16] | Newell, A. C.: The history of the soliton. J. Appl. Mech.50, 1127-1138 (1983). · Zbl 0553.76016 · doi:10.1115/1.3167195 |
| [17] | Ono, H.: Wave propagation in an inhomogeneous anharmonic lattice. J. Phys. Soc. Jpn.32, 332-336 (1983). · doi:10.1143/JPSJ.32.332 |
| [18] | Peters, A. S., Stoker, J. J.: Solitary waves in liquids having non-constant density. Comm. Pure Appl. Math.13, 115-164 (1960). · Zbl 0090.43301 · doi:10.1002/cpa.3160130110 |
| [19] | Scott, A. C., Chu, F. Y. F., Mclaughlin, D. W.: The soliton: a new concept in applied science. Proc. IEEE.61 (10), 1443-1483 (1973). · doi:10.1109/PROC.1973.9296 |
| [20] | Shen, M. C.: Solitary waves in compressible media. New York Univ. Inst. Sci., Rep. IMM-NYV325. 1973. |
| [21] | Speigel, M. R.: Theory and problems of complex variables. McGraw-Hill 1964. · Zbl 0148.22504 |
| [22] | Yang, W. H., Yih, C. S.: Internal waves in a circular channel J. Fluid Mech.74, 183-192 (1976). · Zbl 0326.76022 · doi:10.1017/S0022112076001754 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
