Existence criterion for the equations solution of ideal gas motion at given helical velocity

V. V. Markov ; G. B. Sizykh

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 28 (2020) no. 6, pp. 643-652.

Résumé

Purpose of the study is to obtain a criterion for existence of stationary solution of the complete system of equations describing the flow of ideal perfect gas for a given non-solenoidal helical velocity field. Conditions of such a criterion should contain only the components of this velocity and their derivatives. The fulfillment of conditions must be necessary and sufficient for the existence of such fields of density and pressure, which, together with the considered velocity, satisfy the complete system of equations. Methods. Without using asymptotic, numerical, and other approximate methods, the analysis of the complete system of equations of the classical model of the flow of ideal perfect gas with constant heat capacities is carried out. Results. A criterion for the existence of a solution to the complete system of equations for stationary motion of ideal perfect gas for a non-solenoidal helical velocity field is proposed, consisting of a system of equations and inequalities containing only velocity components and their derivatives. An example of a non-solenoidal helical velocity field is presented, for which, according to the proposed criterion, there is no solution to the complete system of equations. The study demonstrates that the justification of the correspondence of the velocity field to any model of fluid motion is a meaningful problem, without which this field cannot be associated with the fluid flow velocity. Conclusion. The problem of the existence of an exact solution of the complete system of equations for a given velocity field has been proposed and the solution one has been obtained for the simplest model of stationary fluid motion and a nonsolenoidal helical velocity field. It is shown that not every non-solenoidal helical velocity can be considered the velocity of a compressible fluid. The relevance of the problem posed is confirmed by an example of research (Morgulis A. et al. Comm. On Pure and Applied Math, 1995), in which the non-solenoidal helical velocity presented by the authors is unlawfully attributed to the flow of compressible fluid since the proof of the existence of corresponding solution of the complete system of equations of any model of compressible liquid is not given.

Détail

Keywords: flow velocity of compressible fluid, helical velocity, exact solution, flow of ideal perfect gas, non-solenoidal helical velocity field, Beltrami solutions.