#### Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.23.202101.82–90

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 35K200

#### Mathematical modeling of heat transfer in the film-substrate-thermostat system during heating of an electrically conductive film by a high-density pulse current

N. D. Kuzmichev, M. A. Vasyutin, E. V. Danilova, E. A. Lapshina

National Research Mordovia State University (Saransk, Russian Federation)

Abstract. Mathematical modeling of heat transfer in the film-substrate-thermostat system with a pulsed flow of high-density current through an electrically conductive film has been carried out. On the basis of the simulation, the analysis of the heating of a niobium nitride film with a high resistivity near the critical temperature of the transition to the superconducting state is made. The inhomogeneous heat conduction equation which is solved numerically, simulates heat transfer in the film-substrate-thermostat system for the third on the left and the first on the right initial boundary value problem. Using the symmetry of the problem, the parameter $H$ is determined, which is equal to the ratio of the heat transfer of the film surface to its thermal conductivity; this parameter is necessary for effective heat removal. It is shown that effective heat removal from films can be provided by current-carrying and potential clamping contacts made, for example, of beryllium bronze. This makes possible to study the current-voltage characteristics of superconductors near the critical transition temperature to the superconducting state with high-density currents $(10^4 - 10^5 A/cm^2)$ without significant heating of the samples.

Key Words: inhomogeneous heat conduction equation, 1st initial-boundary value problem, 3rd initial-boundary value problem, niobium nitride membrane, pulsed heating by current

For citation: N. D. Kuzmichev, M. A. Vasyutin, E. V. Danilova, E. A. Lapshina. Mathematical modeling of heat transfer in the film-substrate-thermostat system during heating of an electrically conductive film by a high-density pulse current. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 23:1(2021), 82–90. DOI: https://doi.org/10.15507/2079-6900.23.202101.82–90

Submitted: 10.01.2021; Revised: 15.02.2021; Accepted: 20.02.2021

Information about the authors:

Nikolay D. Kuzmichev, Professor of Department of Computer Science and CAD-Technology, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Dr. Sci. (Phys.-Math.), ORCID: http://orcid.org/0000-0001-6707-4950, kuzmichevnd@yandex.ru

Mikhael A. Vasyutin, Associate Professor of Department of Computer Science and CAD-Technology, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph. D. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-4856-7407, vasyutinm@mail.ru

Ekaterina V. Danilova, Post-Graduate Student of the Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: https://orcid.org/0000-0003-0554-3795, danilova29-94@mail.ru

Elena A. Lapshina, Associate Professor of Department of Computer Science and CAD-Technology, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph. D. (Pedagogy), ORCID: http://orcid.org/0000-0002-8828-273X, e.lapshina2010@yandex.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

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