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7Математическое моделирование в задачах физики атмосферы, океана, климата
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The reported study was funded by Novosibirsk State University of Architecture and Civil Engineering (Sibstrin), project N-4.
References
1. Yu. A. Chirkunov. Generalized Equivalence Transformations and group classification of systems of differential equations. J. Appl. Mech. Techn. Phys. 2012. V 53. No 2. P. 147-155.
2. Yu. A. Chirkunov. Invariant submodels and exact solutions of the generalization of the Leith model of the wave turbulence. Acta Mech. 2018. V. 229. No 10. P. 4045-4056.
Exact solutions of the nonlinear Khokhlova-Zabolotskaya-Kuznetsov hydroacoustics model
Yu. A. Chirkunov1, N. F. Belmetsev2
1Novosibirsk State University of Architecture and Civil Engineering (Sibstrin)
2Tyumen State University
Email: [email protected]
DOI: 10.24411/9999-017A-2019-10161
A three-dimensional Khokhlov-Zabolotskaya-Kuznetsov (KZK) model of the nonlinear hydroacoustics with dissipation is described by third order nonlinear differential equation. We obtained that the (KZK) equation admits an infinite Lie group of the transformations. We studied the submodels of rank 0 and 1, described by the invariant solutions of the (KZK) equation. These solutions are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the solutions that we called by "Ultrasonic knife" and "Ultrasonic destroyer". For the submodel "Ultrasonic knife" at each fixed moment of the time in the field of the existence of the solution near a some plane the pressure increases indefinitely and becomes infinite on this plane. The submodel "Ultrasonic destroyer" contains a countable number of "Ultrasonic knives". With a help of the invariant solutions we researched a propagation of the intensive acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change, or the acoustic pressure, speed and acceleration of its change in the radial direction, or the acoustic pressure, speed and acceleration of its change in the direction of one of the axes are specified at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solutions of boundary value problems, describing these wave processes.
The reported study was funded by Novosibirsk State University of Architecture and Civil Engineering (Sibstrin), project N-4.
References
1. Yu. A. Chirkunov, N. F. Belmetsev. Invariant submodels and exact solutions of Khohlov-Zabolotskaya- Kuznetsov model of nonlinear hydroacoustics with dissipation. Int. J. Non-Linear Mech. 2017. V 95. P. 216-223.
A nonlinear model of the motion of a liquid or gas in a porous medium in the presence of a source or absorption with maximum symmetry
Yu. A. Chirkunov, Yu. L. Skolubovich
Novosibirsk State University of Architecture and Civil Engineering
Email: [email protected]
DOI: 10.24411/9999-017A-2019-10162
For a three-dimensional nonlinear diffusion model of a porous medium with a non-stationary source or absorption admitting a 9-parameter Lie group of transformations, all invariant solutions of rank 0 were obtained. In particular, we obtained the solutions, which we called "a layered circular pie", "a layered spiral pie", "a layered plane pie" and "a layered spherical pie". The solution "a layered circular pie" describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each circle from the family of concentric circles a pressure is the same. The solution "a layered spiral pie" describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each logarithmic spiral, from the obtained family of logarithmic spirals a pressure is the same . The solution "a layered spherical pie" describes a motion of the liquid or gas in a porous medium, for which at each fixed moment of a time at all points of each sphere , from the family of concentric spheres a pressure is the same . A set of the solutions "a layered circular pie", "a layered spiral pie" and "a layered spherical pie" contains the solutions describing a distribution of the pressure in a porous medium after a point blast or a point hydraulic shock. Also this set
