9 research outputs found
Π’Π΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ Π°Π½ΠΈΠ·ΠΎΡΡΠΎΠΏΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»ΡΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π°, ΠΏΠΎΠ΄Π²ΠΈΠΆΠ½Π°Ρ Π³ΡΠ°Π½ΠΈΡΠ° ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ ΠΏΠΎΠ΄Π²Π΅ΡΠΆΠ΅Π½Π° Π»ΠΎΠΊΠ°Π»ΡΠ½ΠΎΠΌΡ ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΠΎ-ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΎΠΌΡ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌΡ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΠΈΡ Π² ΡΡΠ»ΠΎΠ²ΠΈΡΡ ΡΠ΅ΠΏΠ»ΠΎΠΎΠ±ΠΌΠ΅Π½Π° Ρ Π²Π½Π΅ΡΠ½Π΅ΠΉ ΡΡΠ΅Π΄ΠΎΠΉ
A noticeably raising interest in analytical research methods in the mathematical theory of the thermal conductivity of solids [1-3] was initiated by various causes, among which, as the most significant, special mention should go to the widespread practical engineering application of computer technology, mathematical modelling techniques and anisotropic materials of various origin. At present, the "anisotropic section" [3, 4] holds a most unique position in the mathematical theory of the thermal conductivity of solids, due both to the specificity of the mathematical models used in it, and to the fair-minded development need in fundamentally new high-performance and absolutely stable computational methods [4-6] to solve real, practically important engineering tasks.The spectrum of practical use of solutions to problems of the mathematical theory of the thermal conductivity, presented in an analytically closed form, is quite wide. In particular, such solutions are used to test new computational algorithms, and the problems generating these solutions are called test problems. And if in the traditional sections of the mathematical theory of the thermal conductivity a set of test problems is very extensive [1-3, 7], then test problems of the "anisotropic thermal conductivity" in regions with fixed and moving boundaries are inconsiderable in number [4, 8-14].The main objective of the research is to solve the problem of determining the temperature field of an anisotropic half-space, the boundary of which moves linearly and is subject to local pulse-periodic thermal action under conditions of heat exchange with the external environment.ΠΠ°ΠΌΠ΅ΡΠ½ΠΎΠ΅ ΠΏΠΎΠ²ΡΡΠ΅Π½ΠΈΠ΅ ΠΈΠ½ΡΠ΅ΡΠ΅ΡΠ° ΠΊ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΠΌΠ΅ΡΠΎΠ΄Π°ΠΌ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΉ Π² ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ ΡΠ²Π΅ΡΠ΄ΡΡ
ΡΠ΅Π» [1-3] ΠΈΠ½ΠΈΡΠΈΠΈΡΠΎΠ²Π°Π½ΠΎ ΡΠ°Π·Π»ΠΈΡΠ½ΡΠΌΠΈ ΠΏΡΠΈΡΠΈΠ½Π°ΠΌΠΈ, ΡΡΠ΅Π΄ΠΈ ΠΊΠΎΡΠΎΡΡΡ
, ΠΊΠ°ΠΊ Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ Π·Π½Π°ΡΠΈΠΌΡΡ
, ΡΠ»Π΅Π΄ΡΠ΅Ρ Π²ΡΠ΄Π΅Π»ΠΈΡΡ ΡΠΈΡΠΎΠΊΠΎΠ΅ Π²Π½Π΅Π΄ΡΠ΅Π½ΠΈΠ΅ Π² ΠΈΠ½ΠΆΠ΅Π½Π΅ΡΠ½ΡΡ ΠΏΡΠ°ΠΊΡΠΈΠΊΡ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΡΠ΅Ρ
Π½ΠΈΠΊΠΈ, ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΈ Π°Π½ΠΈΠ·ΠΎΡΡΠΎΠΏΠ½ΡΡ
ΠΌΠ°ΡΠ΅ΡΠΈΠ°Π»ΠΎΠ² ΡΠ°Π·Π»ΠΈΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠΎΠΈΡΡ
ΠΎΠΆΠ΄Π΅Π½ΠΈΡ. Π Π½Π°ΡΡΠΎΡΡΠ΅Π΅ Π²ΡΠ΅ΠΌΡ Π² ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ ΡΠ²Π΅ΡΠ΄ΡΡ
ΡΠ΅Π» Β«Π°Π½ΠΈΠ·ΠΎΡΡΠΎΠΏΠ½ΡΠΉ ΡΠ°Π·Π΄Π΅Π»Β» [3,4] Π·Π°Π½ΠΈΠΌΠ°Π΅Ρ ΠΎΡΠΎΠ±ΠΎΠ΅ ΠΏΠΎΠ»ΠΎΠΆΠ΅Π½ΠΈΠ΅, ΠΎΠ±ΡΡΠ»ΠΎΠ²Π»Π΅Π½Π½ΠΎΠ΅ ΠΊΠ°ΠΊ ΡΠΏΠ΅ΡΠΈΡΠΈΠΊΠΎΠΉ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΡ
Π² Π½Π΅ΠΌ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ, ΡΠ°ΠΊ ΠΈ ΠΎΠ±ΡΠ΅ΠΊΡΠΈΠ²Π½ΠΎΠΉ Π½Π΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎΡΡΡΡ ΡΠ°Π·ΡΠ°Π±ΠΎΡΠΊΠΈ ΠΏΡΠΈΠ½ΡΠΈΠΏΠΈΠ°Π»ΡΠ½ΠΎ Π½ΠΎΠ²ΡΡ
Π²ΡΡΠΎΠΊΠΎΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΈ Π°Π±ΡΠΎΠ»ΡΡΠ½ΠΎ ΡΡΡΠΎΠΉΡΠΈΠ²ΡΡ
Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΡΡ
ΠΌΠ΅ΡΠΎΠ΄ΠΎΠ² [4-6], ΠΎΡΠΈΠ΅Π½ΡΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
Π½Π° ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΡΠ΅Π°Π»ΡΠ½ΡΡ
, ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈ Π²Π°ΠΆΠ½ΡΡ
ΠΈΠ½ΠΆΠ΅Π½Π΅ΡΠ½ΡΡ
Π·Π°Π΄Π°Ρ.Π‘ΠΏΠ΅ΠΊΡΡ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΡ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π·Π°Π΄Π°Ρ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ, ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Π½ΡΡ
Π² Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈ Π·Π°ΠΌΠΊΠ½ΡΡΠΎΠΌ Π²ΠΈΠ΄Π΅, Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎ ΡΠΈΡΠΎΠΊ. Π ΡΠ°ΡΡΠ½ΠΎΡΡΠΈ, ΠΏΠΎΠ΄ΠΎΠ±Π½ΡΠ΅ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡ Π΄Π»Ρ ΡΠ΅ΡΡΠΈΡΠΎΠ²Π°Π½ΠΈΡ Π½ΠΎΠ²ΡΡ
Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΡΡ
Π°Π»Π³ΠΎΡΠΈΡΠΌΠΎΠ², Π° ΡΠ°ΠΌΠΈ Π·Π°Π΄Π°ΡΠΈ, ΠΏΠΎΡΠΎΠΆΠ΄Π°ΡΡΠΈΠ΅ ΡΡΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΡ, Π½Π°Π·ΡΠ²Π°ΡΡ ΡΠ΅ΡΡΠΎΠ²ΡΠΌΠΈ Π·Π°Π΄Π°ΡΠ°ΠΌΠΈ. Π Π΅ΡΠ»ΠΈ Π² ΡΡΠ°Π΄ΠΈΡΠΈΠΎΠ½Π½ΡΡ
ΡΠ°Π·Π΄Π΅Π»Π°Ρ
ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠ΅ΠΎΡΠΈΠΈ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ ΠΌΠ½ΠΎΠΆΠ΅ΡΡΠ²ΠΎ ΡΠ΅ΡΡΠΎΠ²ΡΡ
Π·Π°Π΄Π°Ρ Π²Π΅ΡΡΠΌΠ° ΠΎΠ±ΡΠΈΡΠ½ΠΎ [1-3, 7] ΡΠΎ ΡΠ΅ΡΡΠΎΠ²ΡΠ΅ Π·Π°Π΄Π°ΡΠΈ Β«Π°Π½ΠΈΠ·ΠΎΡΡΠΎΠΏΠ½ΠΎΠΉ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈΒ» Π² ΠΎΠ±Π»Π°ΡΡΡΡ
Ρ Π½Π΅ΠΏΠΎΠ΄Π²ΠΈΠΆΠ½ΡΠΌΠΈ ΠΈ Π΄Π²ΠΈΠΆΡΡΠΈΠΌΠΈΡΡ Π³ΡΠ°Π½ΠΈΡΠ°ΠΌΠΈ Π²Π΅ΡΡΠΌΠ° Π½Π΅ΠΌΠ½ΠΎΠ³ΠΎΡΠΈΡΠ»Π΅Π½Π½Ρ [4, 8-14].ΠΡΠ½ΠΎΠ²Π½Π°Ρ ΡΠ΅Π»Ρ ΠΏΡΠΎΠ²Π΅Π΄Π΅Π½Π½ΡΡ
ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΉ β ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ Π·Π°Π΄Π°ΡΠΈ ΠΎΠ± ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠΈ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ Π°Π½ΠΈΠ·ΠΎΡΡΠΎΠΏΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»ΡΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π°, Π³ΡΠ°Π½ΠΈΡΠ° ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ ΠΏΠ΅ΡΠ΅ΠΌΠ΅ΡΠ°Π΅ΡΡΡ ΠΏΠΎ Π»ΠΈΠ½Π΅ΠΉΠ½ΠΎΠΌΡ Π·Π°ΠΊΠΎΠ½Ρ ΠΈ ΠΏΠΎΠ΄Π²Π΅ΡΠΆΠ΅Π½Π° Π»ΠΎΠΊΠ°Π»ΡΠ½ΠΎΠΌΡ ΠΈΠΌΠΏΡΠ»ΡΡΠ½ΠΎ-ΠΏΠ΅ΡΠΈΠΎΠ΄ΠΈΡΠ΅ΡΠΊΠΎΠΌΡ ΡΠ΅ΠΏΠ»ΠΎΠ²ΠΎΠΌΡ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΠΈΡ Π² ΡΡΠ»ΠΎΠ²ΠΈΡΡ
ΡΠ΅ΠΏΠ»ΠΎΠΎΠ±ΠΌΠ΅Π½Π° Ρ Π²Π½Π΅ΡΠ½Π΅ΠΉ ΡΡΠ΅Π΄ΠΎΠΉ
ΠΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΌΠΎΠ΄Π΅Π»ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ ΠΏΡΠΎΡΠ΅ΡΡΠΎΠ² ΡΠ΅ΠΏΠ»ΠΎΠΏΠ΅ΡΠ΅Π½ΠΎΡΠ° Π² ΡΠ²Π΅ΡΠ΄ΠΎΠΌ ΡΠ΅Π»Π΅ Ρ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅ΠΌ Π² Π²ΠΈΠ΄Π΅ ΡΠ°ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ»ΠΎΡ, ΠΏΠΎΠ³Π»ΠΎΡΠ°ΡΡΠΈΠΌ ΠΏΡΠΎΠ½ΠΈΠΊΠ°ΡΡΠ΅Π΅ ΠΈΠ·Π»ΡΡΠ΅Π½ΠΈΠ΅
The paper deals with determining a temperature field of an isotropic solid with inclusion represented as a spherical layer that absorbing penetrating radiation. A hierarchy of simplified analogues of the basic model of the heat transfer process in the system under study was developed, including a βrefined model of concentrated capacityβ, a βconcentrated capacityβ model, and a βtruncated model of concentrated capacityβ. Each of the mathematical models of the hierarchy is a mixed problem for a second-order partial differential equation of the parabolic type with a specific boundary condition that actually takes into account the spherical layer available in the system under study.The use of the Laplace integral transform and the well-known theorems of operational calculus in analytically closed form enabled us to find solutions to the corresponding problems of unsteady heat conduction. The βconcentrated capacitanceβ model was in detail analysed with the object under study subjected to the radiation flux of constant density. This model is associated with a thermally thin absorbing inclusion in the form of a spherical layer. It is shown that it allows us to submit the problem solution of unsteady heat conduction in the analytical form, which is the most convenient in terms of both its practical implementation and a theoretical assessment of the influence, the spherical layer width has on the temperature field of the object under study.Sufficient conditions are determined under which the temperature field of the analysed system can be identified with a given accuracy through the simplified analogues of the basic mathematical model. For simplified analogues of the basic model, the paper presents theoretical estimates of the maximum possible error when determining the radiated temperature field.Π Π°ΡΡΠΌΠΎΡΡΠ΅Π½Π° Π·Π°Π΄Π°ΡΠ° ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ ΠΈΠ·ΠΎΡΡΠΎΠΏΠ½ΠΎΠ³ΠΎ ΡΠ²Π΅ΡΠ΄ΠΎΠ³ΠΎ ΡΠ΅Π»Π° Ρ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅ΠΌ Π² Π²ΠΈΠ΄Π΅ ΡΠ°ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ»ΠΎΡ, ΠΏΠΎΠ³Π»ΠΎΡΠ°ΡΡΠΈΠΌ ΠΏΡΠΎΠ½ΠΈΠΊΠ°ΡΡΠ΅Π΅ ΠΈΠ·Π»ΡΡΠ΅Π½ΠΈΠ΅. Π Π°Π·ΡΠ°Π±ΠΎΡΠ°Π½Π° ΠΈΠ΅ΡΠ°ΡΡ
ΠΈΡ ΡΠΏΡΠΎΡΠ΅Π½Π½ΡΡ
Π°Π½Π°Π»ΠΎΠ³ΠΎΠ² Π±Π°Π·ΠΎΠ²ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΏΡΠΎΡΠ΅ΡΡΠ° ΡΠ΅ΠΏΠ»ΠΎΠΏΠ΅ΡΠ΅Π½ΠΎΡΠ° Π² ΠΈΠ·ΡΡΠ°Π΅ΠΌΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΠ΅, Π²ΠΊΠ»ΡΡΠ°ΡΡΠ°Ρ Β«ΡΡΠΎΡΠ½Π΅Π½Π½ΡΡ ΠΌΠΎΠ΄Π΅Π»Ρ ΡΠΎΡΡΠ΅Π΄ΠΎΡΠΎΡΠ΅Π½Π½ΠΎΠΉ Π΅ΠΌΠΊΠΎΡΡΠΈΒ», ΠΌΠΎΠ΄Π΅Π»Ρ Β«ΡΠΎΡΡΠ΅Π΄ΠΎΡΠΎΡΠ΅Π½Π½Π°Ρ Π΅ΠΌΠΊΠΎΡΡΡΒ» ΠΈ Β«ΡΡΠ΅ΡΠ΅Π½Π½ΡΡ ΠΌΠΎΠ΄Π΅Π»Ρ ΡΠΎΡΡΠ΅Π΄ΠΎΡΠΎΡΠ΅Π½Π½ΠΎΠΉ Π΅ΠΌΠΊΠΎΡΡΠΈΒ». ΠΠ°ΠΆΠ΄Π°Ρ ΠΈΠ· ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΈΡ
ΠΌΠΎΠ΄Π΅Π»Π΅ΠΉ ΠΈΠ΅ΡΠ°ΡΡ
ΠΈΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»ΡΠ΅Ρ ΡΠΎΠ±ΠΎΠΉ ΡΠΌΠ΅ΡΠ°Π½Π½ΡΡ Π·Π°Π΄Π°ΡΡ Π΄Π»Ρ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Π² ΡΠ°ΡΡΠ½ΡΡ
ΠΏΡΠΎΠΈΠ·Π²ΠΎΠ΄Π½ΡΡ
Π²ΡΠΎΡΠΎΠ³ΠΎ ΠΏΠΎΡΡΠ΄ΠΊΠ° ΠΏΠ°ΡΠ°Π±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΡΠΈΠΏΠ° ΡΠΎ ΡΠΏΠ΅ΡΠΈΡΠΈΡΠ΅ΡΠΊΠΈΠΌ ΠΊΡΠ°Π΅Π²ΡΠΌ ΡΡΠ»ΠΎΠ²ΠΈΠ΅ΠΌ, ΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈ ΡΡΠΈΡΡΠ²Π°ΡΡΠΈΠΌ Π½Π°Π»ΠΈΡΠΈΠ΅ ΡΠ°ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ»ΠΎΡ Π² ΠΈΠ·ΡΡΠ°Π΅ΠΌΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΠ΅.Π‘ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ΠΌ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΏΡΠ΅ΠΎΠ±ΡΠ°Π·ΠΎΠ²Π°Π½ΠΈΡ ΠΠ°ΠΏΠ»Π°ΡΠ° ΠΈ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΡ
ΡΠ΅ΠΎΡΠ΅ΠΌ ΠΎΠΏΠ΅ΡΠ°ΡΠΈΠΎΠ½Π½ΠΎΠ³ΠΎ ΠΈΡΡΠΈΡΠ»Π΅Π½ΠΈΡ Π² Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΈ Π·Π°ΠΌΠΊΠ½ΡΡΠΎΠΌ Π²ΠΈΠ΄Π΅ Π½Π°ΠΉΠ΄Π΅Π½Ρ ΡΠ΅ΡΠ΅Π½ΠΈΡ ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΡ
Π·Π°Π΄Π°Ρ Π½Π΅ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠΉ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ. ΠΠΎΠ΄ΡΠΎΠ±Π½ΠΎ ΠΏΡΠΎΠ°Π½Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π° ΠΌΠΎΠ΄Π΅Π»Ρ Β«ΡΠΎΡΡΠ΅Π΄ΠΎΡΠΎΡΠ΅Π½Π½Π°Ρ Π΅ΠΌΠΊΠΎΡΡΡΒ» Π² ΡΠΈΡΡΠ°ΡΠΈΠΈ, ΠΊΠΎΠ³Π΄Π° Π½Π° ΠΎΠ±ΡΠ΅ΠΊΡ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΉ Π²ΠΎΠ·Π΄Π΅ΠΉΡΡΠ²ΡΠ΅Ρ ΠΏΠΎΡΠΎΠΊ ΠΈΠ·Π»ΡΡΠ΅Π½ΠΈΡ Ρ ΠΏΠΎΡΡΠΎΡΠ½Π½ΠΎΠΉ ΠΏΠ»ΠΎΡΠ½ΠΎΡΡΡΡ. ΠΡΠ° ΠΌΠΎΠ΄Π΅Π»Ρ Π°ΡΡΠΎΡΠΈΠΈΡΡΠ΅ΡΡΡ Ρ ΡΠ΅ΡΠΌΠΈΡΠ΅ΡΠΊΠΈ ΡΠΎΠ½ΠΊΠΈΠΌ ΠΏΠΎΠ³Π»ΠΎΡΠ°ΡΡΠΈΠΌ Π²ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅ΠΌ Π² ΡΠΎΡΠΌΠ΅ ΡΠ°ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ»ΠΎΡ. ΠΠΎΠΊΠ°Π·Π°Π½ΠΎ, ΡΡΠΎ ΠΎΠ½Π° ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΠΏΡΠ΅Π΄ΡΡΠ°Π²ΠΈΡΡ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΡΠ°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΠΌΠΎΠΉ Π·Π°Π΄Π°ΡΠΈ Π½Π΅ΡΡΠ°ΡΠΈΠΎΠ½Π°ΡΠ½ΠΎΠΉ ΡΠ΅ΠΏΠ»ΠΎΠΏΡΠΎΠ²ΠΎΠ΄Π½ΠΎΡΡΠΈ Π² Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠΌ Π²ΠΈΠ΄Π΅, Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ ΡΠ΄ΠΎΠ±Π½ΠΎΠΌ Ρ ΡΠΎΡΠΊΠΈ Π·ΡΠ΅Π½ΠΈΡ ΠΈ Π΅Π³ΠΎ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΎΠ³ΠΎ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΡ, ΠΈ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΎΡΠ΅Π½ΠΊΠΈ Π²Π»ΠΈΡΠ½ΠΈΡ ΡΠΈΡΠΈΠ½Ρ ΡΠ°ΡΠΎΠ²ΠΎΠ³ΠΎ ΡΠ»ΠΎΡ Π½Π° ΡΠΎΡΠΌΠΈΡΡΠ΅ΠΌΠΎΠ΅ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ ΠΎΠ±ΡΠ΅ΠΊΡΠ° ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠΉ.ΠΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Ρ Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΡΠ΅ ΡΡΠ»ΠΎΠ²ΠΈΡ, ΠΏΡΠΈ Π²ΡΠΏΠΎΠ»Π½Π΅Π½ΠΈΠΈ ΠΊΠΎΡΠΎΡΡΡ
ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ΅ ΠΏΠΎΠ»Π΅ Π°Π½Π°Π»ΠΈΠ·ΠΈΡΡΠ΅ΠΌΠΎΠΉ ΡΠΈΡΡΠ΅ΠΌΡ ΠΌΠΎΠΆΠ½ΠΎ ΠΈΠ΄Π΅Π½ΡΠΈΡΠΈΡΠΈΡΠΎΠ²Π°ΡΡ Ρ Π·Π°Π΄Π°Π½Π½ΠΎΠΉ ΡΠΎΡΠ½ΠΎΡΡΡΡ ΠΏΡΠΈ ΠΏΠΎΠΌΠΎΡΠΈ ΡΠΏΡΠΎΡΠ΅Π½Π½ΡΡ
Π°Π½Π°Π»ΠΎΠ³ΠΎΠ² Π±Π°Π·ΠΎΠ²ΠΎΠΉ ΠΌΠ°ΡΠ΅ΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ. ΠΠ»Ρ ΡΠΏΡΠΎΡΠ΅Π½Π½ΡΡ
Π°Π½Π°Π»ΠΎΠ³ΠΎΠ² Π±Π°Π·ΠΎΠ²ΠΎΠΉ ΠΌΠΎΠ΄Π΅Π»ΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½Ρ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΌΠ°ΠΊΡΠΈΠΌΠ°Π»ΡΠ½ΠΎ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΠΉ ΠΏΠΎΠ³ΡΠ΅ΡΠ½ΠΎΡΡΠΈ Π² ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠΈ ΠΈΠ·Π»ΡΡΠ°Π΅ΠΌΠΎΠ³ΠΎ ΡΠ΅ΠΌΠΏΠ΅ΡΠ°ΡΡΡΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ»Ρ
Anisotropic Half-Space Temperature Field with its Moving Boundary Being under Local Pulse-periodic Heat Action in Heat Exchange Conditions with External Environment
A noticeably raising interest in analytical research methods in the mathematical theory of the thermal conductivity of solids [1-3] was initiated by various causes, among which, as the most significant, special mention should go to the widespread practical engineering application of computer technology, mathematical modelling techniques and anisotropic materials of various origin. At present, the "anisotropic section" [3, 4] holds a most unique position in the mathematical theory of the thermal conductivity of solids, due both to the specificity of the mathematical models used in it, and to the fair-minded development need in fundamentally new high-performance and absolutely stable computational methods [4-6] to solve real, practically important engineering tasks.The spectrum of practical use of solutions to problems of the mathematical theory of the thermal conductivity, presented in an analytically closed form, is quite wide. In particular, such solutions are used to test new computational algorithms, and the problems generating these solutions are called test problems. And if in the traditional sections of the mathematical theory of the thermal conductivity a set of test problems is very extensive [1-3, 7], then test problems of the "anisotropic thermal conductivity" in regions with fixed and moving boundaries are inconsiderable in number [4, 8-14].The main objective of the research is to solve the problem of determining the temperature field of an anisotropic half-space, the boundary of which moves linearly and is subject to local pulse-periodic thermal action under conditions of heat exchange with the external environment
Mathematical Modeling of Heat Transfer Processes in a Solid With Spherical Layer-type Inclusion to Absorb Penetrating Radiation
The paper deals with determining a temperature field of an isotropic solid with inclusion represented as a spherical layer that absorbing penetrating radiation. A hierarchy of simplified analogues of the basic model of the heat transfer process in the system under study was developed, including a βrefined model of concentrated capacityβ, a βconcentrated capacityβ model, and a βtruncated model of concentrated capacityβ. Each of the mathematical models of the hierarchy is a mixed problem for a second-order partial differential equation of the parabolic type with a specific boundary condition that actually takes into account the spherical layer available in the system under study.The use of the Laplace integral transform and the well-known theorems of operational calculus in analytically closed form enabled us to find solutions to the corresponding problems of unsteady heat conduction. The βconcentrated capacitanceβ model was in detail analysed with the object under study subjected to the radiation flux of constant density. This model is associated with a thermally thin absorbing inclusion in the form of a spherical layer. It is shown that it allows us to submit the problem solution of unsteady heat conduction in the analytical form, which is the most convenient in terms of both its practical implementation and a theoretical assessment of the influence, the spherical layer width has on the temperature field of the object under study.Sufficient conditions are determined under which the temperature field of the analysed system can be identified with a given accuracy through the simplified analogues of the basic mathematical model. For simplified analogues of the basic model, the paper presents theoretical estimates of the maximum possible error when determining the radiated temperature field
THE EFFECT OF TEMPERATURE ON VISCOPLASTIC PORE COLLAPSE
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