A Similarity Technique for Solving Two-Layer Shallow-Water Equations

Affiliation(s)

Department of Engineering Physics and Mathematics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

Department of Engineering Physics and Mathematics, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

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.

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.

KEYWORDS

The Characteristic Function Method; The Two-Layer Shallow-Water Equations; Gravity Currents

The Characteristic Function Method; The Two-Layer Shallow-Water Equations; Gravity Currents

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.

M. Kassem, M. Helal, M. Mekky and E. Mohamed, "A Similarity Technique for Solving Two-Layer Shallow-Water Equations,"

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.

[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.