Two-dimensional model of solid media to describe fluid waves

About the authors:

Sergey L. Deryabin, Dr. Sci. (Phys.-Math.), Professor, Department of Higher and Applied Mathematics, Ural State University of Railway Transport (Ekaterinburg)
Alexey V. Mezentsev, Senior Lecturer, Department of Higher and Applied Mathematics, Ural State University of Railway Transport (Ekaterinburg)

Abstract:

To describe the propagation of long waves many models of equations of shallow water are used. It should be mentioned, that the models of shallow water cannot provide us with depth distributions of velocity and density of the fluid. This research describes the parameters of the wave model of two-dimensional gas dynamics for the polytropic gas with gas politropy rate equal to 7. Solutions for the two of the initial-boundary value problems describing the current of the fluid from the surface of the bottom to the surface of water are provided. The current has a weak discontinuity within itself and it is, therefore, a piecewise component. Boundary conditions are found: on the bottom surface, on the water surface and on the weak discontinuity. The boundary conditions can be used for numerical calculations.

References:

1. Khakimzianov, G.S., Shokin, Iu.I., Barakhnin, V.B., Shokina, N.Iu. Chislennoe modelirovanie techenii zhidkosti s poverkhnostnymi volnami [Numerical simulation of fluid flows with surface waves]. Novosibirsk, 2001. 394 p. (in Russian).

2. Ovsiannikov, L.V. Lektsii po osnovam gazovoi dinamiki [Lectures on the basics of gas dynamics]. Izhevsk, 2003. 336 p. (in Russian).

3. Kovenia, V.M., Ianenko, N.N. Metod rasshchepleniia v zadachakh gazovoi dinamiki [The splitting method in the problems of gas dynamics]. Novosibirsk, 1983. 319 p. (in Russian).

4. Bautin, S.P. Kharakteristicheskaia zadacha Koshi i ee prilozheniia v gazovoi dinamike [Characteristic Cauchy problem and its applications in gas dynamics]. Novosibirsk: Nauka, 2009. 368 p. (in Russian).

5. Bautin, S.P., Deriabin, S.L., Sommer, A.F., Khakimzianov, G.S. Investigation of solutions of the shallow water equations in the neighborhood of the moving line edge. Vychislitel'nye tekhnologii — Computational Technologies. 2010. V. 15. № 6. Pp. 19–41. (in Russian).

6. Fedotova, Z.I., Khakimzianov, G.S. Non-linear dispersive equations of shallow water on the unsteady bottom. Vychislitel'nye tekhnologii — Computational Technologies. 2008. V. 13. № 4. Pp. 114–126. (in Russian).

7. Nigmatullin, R.I., Bolotnova, R.Kh. Wide-range equation of state for water and steam. Method of constructing. Teplofizika vysokikh temperatur — Thermophysics of high temperatures. 2008. V. 46. № 2. Pp. 206–218. (in Russian).

8. Bautin, S.P., Deriabin, S.L. Matematicheskoe modelirovanie istecheniia ideal'nogo gaza v vakuum [Mathematical modeling of the ideal gas outflow into vacuum]. Novosibirsk: Nauka, 2005. 390 p. (in Russian).

9. Rashevskii, P.K. Kurs differentsial'noi geometrii [A course of differential geometry]. Moscow, 1950. 428 p. (in Russian).

10. Kurant, R. Uravneniia s chastnymi proizvodnymi [Partial differential equations]. Moscow, 1964. 830 p. (in Russian).