The new exact solution of the compressible 3D Navier–Stokes equations

Chefranov, S. G.; Chefranov, Artem S.

doi:10.1016/j.cnsns.2019.105118

Cited by 12 publications

(15 citation statements)

References 15 publications

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“…Axisymmetric flow in 3D is also two-dimensional. Recently, an exact solution to the compressible 3D Navier-Stokes equations was obtained by using a relation between the pressure and the divergence of the velocity, which was in turn derived from the positive-definiteness of the entropy production rate, a weaker form of MEPP [9,10].…”

Section: Navier-stokes and Euler Equations: Prior Resultsmentioning

confidence: 99%

Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations

Glimm

Lazarev

2020

SN Appl. Sci.

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Section: Navier-stokes and Euler Equations: Prior Resultsmentioning

confidence: 99%

Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations

Glimm

Lazarev

2020

SN Appl. Sci.

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“…It is also of interest to further develop the proposed approach for already dynamically active impurities, in particular, in connection with the possibility of mathematical modeling of threshold processes of carcinogenesis based on the analogy with threshold phenomena in hydrodynamics [ 3 , 11 , 20 – 22 ].…”

Section: Discussionmentioning

confidence: 99%

Estimation of the stable frozen zone volume and the extent of contrast for a therapeutic substance

Korpan

Chefranov

2020

PLoS ONE

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Background In biomedical science and clinical practice, an estimation of the stable frozen zone volume and distribution of concentration fields of injected diagnostic and healing solutions in the tissues of living organisms is of great importance and does not currently have any mathematical solution aimed at its precise evaluation. Objective The aim of this research is the estimation of the stable frozen zone volume at ultra-low temperatures as well as the distribution of temperature areas and concentration fields of injected diagnostic and healing substances in vitro. The results can improve our understanding of the stable frozen zone volume and the extent of contrast for a therapeutic substance. Materials and methods A cryogenic zone (ice ball) was generated at-180˚C using liquid nitrogen without any difficulties in vitro. The effects of freeze-thaw processes using ultra-low temperature and the cryogenic response of a 1.5% gelatin solution in water (%g/v) kept at a constant temperature of 20˚C and continuously stirred were mathematically analyzed. The stable frozen zone volume was illustrated in vitro and measured in terms of its length, depth and cryogenic margin using a standard medical ruler and Vernier caliper after a freezing period at-180˚C, using liquid nitrogen to provide cooling and freezing of a small portion of this solution in the vessel at room temperature (20˚C). Round-shaped cryoprobes with diameters of 15 mm and 50 mm were applied to create a frozen zone volume in vitro. A single cryoprobe was used per procedure. The sample exposure time was 3 min. After this time, the volume of the frozen region remains unchanged, which indicates that the equilibrium stationary state has been reached. The experimental design, cryogenic procedure and freeze-thaw processes of the hemisphere were described and illustrated in vitro item by item. The statistical analysis

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“…Meanwhile, at identically zero viscosity the Burgers equation exactly coincides with the Hopf equation describing the motion of liquid particles by inertia. For the multi-dimensional Hopf equation an exact explicit, closed analytical general solution to the compressible turbulence problem which corresponds exactly to the smooth initial velocity field is obtained in the Euler variables [4]- [6].…”

Section: Introductionmentioning

confidence: 99%

“…In our paper, a solution to all this problems is given based on the exact solution for 1-D compressible Euler equations in the explicit form, which is more useful than Riemann's solution. By using the generalized functions approach developed in [4][5][6], we obtain an explicit analytical solution of the 1-D compressible Euler equations, which gives an explicit analytical description of a nonlinear simple wave and its time of collapse for arbitrary smooth initial conditions. It is important to note that now the role of a pressure or the local speed of sound is accurately taken into account, in contrast to considerations which are provided in the frame of the Burgers equation in [1], [18], [19] and in the frame of the Hopf equation [2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

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Exact solution to the main turbulence problem for a compressible medium and the universal −8/3 law turbulence spectrum of breaking waves

Chefranov

2021

Physics of Fluids

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An exact analytical solution to the one-dimensional compressible Euler equations in the form of a nonlinear simple wave is obtained. Instead of the well-known Riemann solution, the explicit dependence on the arbitrary initial fields for that solution and for its timeis matched with the observed turbulence spectrum in the magnetosheath and solar wind. The observed plasma mirror modes nonlinear mechanism arising out corresponding to this spectrum, is established. This mechanism, in turn, is associated with the formation of localized clusters of charged particles whose motion is described by a slow simple wave.

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The new exact solution of the compressible 3D Navier–Stokes equations

Cited by 12 publications

References 15 publications

Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations

Maximum entropy production as a necessary admissibility condition for the fluid Navier–Stokes and Euler equations

Estimation of the stable frozen zone volume and the extent of contrast for a therapeutic substance

Exact solution to the main turbulence problem for a compressible medium and the universal −8/3 law turbulence spectrum of breaking waves

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