An Exact Stationary Decision for the Equation System of Gas Dynamics under Gravity and Coriolis Force

About the author:

Sergei P. Bautin, Dr. Sci. (Phys.-Math), Professor, Department of Higher and Applied Mathematics, Snezhinsk Physical-Technical Institute, National Research Nuclear University MEPhI (Snezhinsk); eLibrary AuthorID, sbautin@usurt.ru

Abstract:

This article considers the isentropic flow case for a system of gas equations, when the energy equation written for entropy is satisfied identically and a system of four nonlinear partial differential equations is obtained for the four unknown functions. The square of the sound velocity of the gas and the Cartesian components of the gas velocity vector are used as the unknown functions. In addition to the system of equations of gas dynamics, the author considers the complete system of Navier — Stokes equations, the solutions of which describe the flows of a compressible viscous heat conducting gas and satisfy the laws of conservation of mass, momentum, energy, and thermodynamic laws. This system is a system of five nonlinear partial differential equations for five unknown functions, in which the energy equation is written for temperature.

This work presents exact solutions for each of the considered systems. In the cases of both systems of partial differential equations in each exact solution, the Cartesian components of the gas velocity vector are constant, and the third component along the vertical axis is zero. In the case of a system of equations of gas dynamics, the desired function is the square of the speed of sound, and in the case of a complete system of Navier — Stokes equations, the unknown function-temperature is a linear function of the spatial variables. In these linear functions, the coefficients in front of independent variables depend on the modulus of the angular velocity vector of the Earth’s rotation, on the latitude of the point at which the flow is considered, and on the constant non-zero components of the gas velocity vector in the horizontal direction. Consequently, the constructed exact solutions explicitly accounts for the rotation of the Earth around its axis.

References:

1. Bautin S. P. 2009. Kharakteristicheskaya zadacha Koshi i eye prilozheniya v gazovoy dinamike [The Characteristic Cauchy Problem and Its Applications in Gas Dynamics]. Nauka. Novosibirsk.
2. Bautin S. P., Krutova I. Yu., Obukhov A. G., Bautin K. V. 2013. Razrushitel’nyye atmosfernyye vikhri: teoremy, raschety, eksperimenty [Destructive Atmospheric Vortices: The Theorems, Calculations, and Experiments]. Novosibirsk: Nauka.
3. Bautin S. P., Deraybin S. L., Krutova I. Yu., Obukhov A. G. 2017. Razrushitel’nyye atmosfernyye vikhri i vrashcheniye Zemli vokrug svoyey osi [Destructive Atmospheric Vortices and the Earth’s Rotation around Its Axis]. Yekaterinburg: Ural State University of Railway Transport.
4. Bautin S. P. 2008. Tornado i sila Koriolisa [Tornado and the Coriolis Force]. Novosibirsk: Nauka.