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 AM  Vol.3 No.4 , April 2012
A Similarity Technique for Solving Two-Layer Shallow-Water Equations
Abstract: This paper is devoted to the analysis of the two-layer shallow-water equations representing gravity currents. A similarity technique which is the characteristic function method is applied for this study. The application of the characteristic function method makes it possible to obtain the similarity forms depending on a group of infinitesimal transformations. Thus, the number of independent variables is reduced by one and the governing partial differential equations with the auxiliary conditions reduce to a system of ordinary differential equations with the appropriate auxiliary conditions. Numeric solutions are presented and discussed.
Cite this paper: M. Kassem, M. Helal, M. Mekky and E. Mohamed, "A Similarity Technique for Solving Two-Layer Shallow-Water Equations," Applied Mathematics, Vol. 3 No. 4, 2012, pp. 315-321. doi: 10.4236/am.2012.34047.
References

[1]   J. E. Simpson, “Gravity Currents: In the Environment and Laboratory,” Cambridge University Press, Cambridge, 1997.

[2]   C. B. Vreugdenhil, “Numerical Methods for ShallowWater Flow,” Kluwer, Dordrecht, 1994.

[3]   R. Leveque, “Numerical Methods for Conservation Laws,” Birkhauser, Basel, 1992. doi:10.1007/978-3-0348-8629-1

[4]   E. Godlewski and P. A. Raviart, “Numerical Approximation of Hyperbolic Systems of Conservation Laws,” Springer, New York, 1996.

[5]   S. Xin and Z. Xin, “The Relaxation Schemes for Systems of Conservation Laws in Arbitrary Space Dimensions,” Communications on Pure and Applied Mathematics, Vol. 48, No. 3, 1955, pp. 235-276.

[6]   P. J. Montgomery and T. B. Moodie, “Generalization of a Relaxation Scheme for Systems of Forced Nonlinear Hyperbolic Conservation Laws with Spatially Dependent Flux Functions,” Studies in Applied Mathematics, Vol. 110, No. 1, 2003, pp. 1-19. doi:10.1111/1467-9590.00228

[7]   S. J. D’Alessio, T. B. Moodie, J. P. Pascal and G. E. Swaters, “Gravity Currents Produced by Sudden Releases of a Fixed Volume of Heavy Fluid”, Studies in Applied Mathematics, Vol. 96, No. 4, 1996, pp. 359-385..

[8]   P. Glaister, “Similarity Solutions of the Shallow-Water Equations”, Journal of Hydraulic Research, Vol. 29, No. 1, 1991, pp. 107-116. doi:10.1080/00221689109498995

[9]   M. S. Velan and M. Lakshmanan, “Lie Symmetries and Invariant Solutions of the Shallow-Water Equation,” International Journal of Non-Linear Mechanics, Vol. 31, No. 3, 1996, pp. 339-344. doi:10.1016/0020-7462(95)00063-1

[10]   T. ?zer, “Symmetry Group Analysis of Benney System and Application for the Shallow-Water Equations,” Mechanics Research Communications, Vol. 32, No. 3, 2005, pp. 241-254. doi:10.1016/j.mechrescom.2004.10.002

[11]   T. ?zer, “On Symmetry Group Properties and General Similarity Forms of the Benney Equations in the Lagrangian Variables”, Journal of Computational and Applied Mathematics, Vol. 169, No. 2, 2004, pp. 297-313. doi:10.1016/j.cam.2003.12.027

[12]   T. ?zer and N. Antar, “The Similarity Forms and Invariant Solutions of the Two-Layer Shallow-Water Equations,” Nonlinear Analysis: Real World Applications, Vol. 9, No. 3, 2008, pp. 791-810. doi:10.1016/j.nonrwa.2006.12.010

[13]   B. J. Cantwell, “An Introduction to Symmetry Analysis,” Cambridge University Press, Cambridge, 2002.

[14]   R. Seshadri and T. Y. Na, “Group Invariance in Engineering Boundary Value Problems,” Springer-Verlag, New York, 1985. doi:10.1007/978-1-4612-5102-6

[15]   M. B. Abd-el-Malek and M. M. Helal, “Characteristic Function Method for Classification of Equations of Hydrodynamics of a Perfect Fluid,” Journal of Computational and Applied Mathematics, Vol. 182, No. 1, 2005, pp. 105-116. doi:10.1016/j.cam.2004.11.042

[16]   M. B. Abd-el-Malek and M. M. Helal, “The Characteristic Function Method and Exact Solutions of Nonlinear Sheared Flows with Free Surface under Gravity,” Journal of Computational and Applied Mathematics, Vol. 189, No. 1-2, 2006, pp. 2-21. doi:10.1016/j.cam.2005.04.038

[17]   J. W. Rottman and R. E. Grundy, “Self-Similar Solutions of the Shallow-Water Equations Representing Gravity Currents with Variable Inflow,” Journal of Fluid Mechanics, Vol. 169, 1986, pp. 337-351. doi:10.1017/S0022112086000678

[18]   S. J. D. D’Alessio, T. B. Moodie, J. P. Pascal and G. E. Swaters, “Gravity Currents Produced by Sudden Release of a Fixed Volume of Heavy Fluid,” Studies in Applied Mathematics, Vol. 96, No. 4, 1996, pp. 359-385.

[19]   B. J. Cantwell, “Introduction to Symmetry Analysis,” Cambridge University Press, Cambridge, 2002.

 
 
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