Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection
The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with comple...
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Kazan Federal University
2024-02-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://uzakufismat.elpub.ru/jour/article/view/26 |
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| author | A. A. Azanov E. N. Lemeshkova |
| author_facet | A. A. Azanov E. N. Lemeshkova |
| author_sort | A. A. Azanov |
| collection | DOAJ |
| description | The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time. |
| format | Article |
| id | doaj-art-d25db83d3ab3455a9df252b6c7a5cf7c |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2024-02-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-d25db83d3ab3455a9df252b6c7a5cf7c2025-02-02T23:06:08ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982024-02-01165432634310.26907/2541-7746.2023.4.326-34324Qualitative Properties of the Solution of a Conjugate Problem of Thermal ConvectionA. A. Azanov0E. N. Lemeshkova1Siberian Federal UniversityInstitute of Computational Modelling, Siberian Branch, Russian Academy of SciencesThe joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.https://uzakufismat.elpub.ru/jour/article/view/26oberbeck–boussinesq modelthermal convectionthermocapillarityinterfaceinverse problema priori estimates |
| spellingShingle | A. A. Azanov E. N. Lemeshkova Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection Учёные записки Казанского университета: Серия Физико-математические науки oberbeck–boussinesq model thermal convection thermocapillarity interface inverse problem a priori estimates |
| title | Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection |
| title_full | Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection |
| title_fullStr | Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection |
| title_full_unstemmed | Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection |
| title_short | Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection |
| title_sort | qualitative properties of the solution of a conjugate problem of thermal convection |
| topic | oberbeck–boussinesq model thermal convection thermocapillarity interface inverse problem a priori estimates |
| url | https://uzakufismat.elpub.ru/jour/article/view/26 |
| work_keys_str_mv | AT aaazanov qualitativepropertiesofthesolutionofaconjugateproblemofthermalconvection AT enlemeshkova qualitativepropertiesofthesolutionofaconjugateproblemofthermalconvection |