Research of the Propagation of an Impurity in a Medium in One Applied Axisymmetric Problem

From the application point of view, the propagation of an admixture in a cylindrical volume filled with low-density air is of interest. It is associated with the evaporation of a substance from a small "glass" in which convective currents associated with the heating of its bottom take place. The propagation of an admixture is considered taking into account both diffusion and convective transfer due to thermal processes inside the "glass". The distribution of velocities in the main volume is sought by solving the Navier-Stokes equation, and the transfer equation with a diffusion term is solved for the admixture. A finite-difference numerical scheme implemented using our own program code is used. Solutions are obtained in cases corresponding to different heights of the "glass" walls, different ratios between the coefficients describing the processes of convective transfer and diffusion. It is shown that high walls significantly impede the process of admixture propagation into the main volume, and the substance is mainly concentrated inside the "glass" without moving beyond its limits. These results are similar to the data on the transfer of one of the components of the vector potential in the problem of amplification of the frozen magnetic field due to convection in the problem solved earlier. Just as there, structures are formed that repeat the features of the flow, and the maximum value is reached on the axis of symmetry. The issue of applying these results in practice and their experimental verification in laboratory conditions is discussed. It is noted that, in general, the propagation of the impurity corresponds to the data obtained in the course of experimental studies conducted earlier.

1. Chae D. // Handbook of Differential Equations: Evolutionary Equation, ed. by C. M. Dafermos and M. Pokorny. Amsterdam: Elsevier, 2008.

2. Gibbon J. D. The three-dimensional Euler equations: Where do we stand? // Physica D. 2008. V. 237, No. 14–17 P. 1 894–1 904. https://doi.org/10.1016/j.physd.2007.10.014.

3. Кузнецов Е. А., Рубан В. П. Коллапс вихревых линий в гидродинамике // ЖЭТФ. 2000. Т. 118, № 10. С. 893–905.

4. Wolibner, W. Un theorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math Z 1933, 37, 698–726.

5. Kuznetsov, E.A.; Naulin, V.; Nielsen, A.H.; Rasmussen, J.J. Effects of sharp vorticity gradients in two-dimensional hydrodynamic turbulence. Phys. Fluids 2007, 19, 105110.

6. Кузнецов Е. А., Михайлов Е. А. Заметки о коллапсе в магнитной гидродинамике // Журнал экспериментальной и теоретической физики. – 2020. – Т. 158, № 3 (9). – С. 561–572.

7. Кузнецов Е. А., Михайлов Е. А., Сердюков М. Г. Нелинейная динамика проскальзывающих течений // Известия высших учебных заведений. Радиофизика. – 2023. – Т. 66, № 2-3. – С. 145–160.

8. Белевич М.Ю. Гидромеханика. Основы классической теории. СПб., изд. РГГМУ, 2006.

9. Свешников А.Г., Боголюбов А.Н., Кравцов В.В. Лекции по математической физике. М., Изд-во МГУ, 1993.

10. Самарский А.А., Попов Ю.П. Разностные схемы задач газовой динамики. М., 1992.

11. Калиткин Н.Н. Численные методы. СПб., БХВ-Петербург, 2011.

12. https://gitflic.ru/project/eamsci/impurity_hd.

13. Каминский В.А., Обвинцева Н.Ю., Калачинская И.С., Дильман В.В. Моделирование конвекции Релея в нестационарном процессе испарения // Математическое моделирование. – 2007. – Т.19, № 11. С. 3–10.

14. В.А.Каминский, Н.Ю.Обвинцева. Испарение жидкости в условиях конвективной неустойчивости в газовой фазе // Журнал физической химии. – 2008. – Т.82, № 7. – С. 1368 – 1373.