Models in Thermomechanics with a Finite and Infinite Velocity of Heat Propagation

Кудинов, И. В.; Eremin, Alexey; Trubitsyn, K. V.; Stefanyuk, E. V.

doi:10.31085/9785392292516-2019-224

Cited by 3 publications

(4 citation statements)

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“…Соотношение (11) удовлетворяет условиям (8), (9). Коэффициенты () k bq будем находить из (10) и ДГУ, общие формулы для рых имеют вид [7]:…”

Section: дополнительные граничные условия в задачах теплопроводности ...unclassified

“…Таким образом, из основных ( 8), (9) и ДГУ (12)-( 14) невыполненными соотношением (11) будут лишь ДГУ (14), которые совместно с условием (10) Постоянная C1 находится из условия (7). Потребуем его выполнения в 10 точках области ( = 1,0; 0,2; 0,3;…, 0,9; 0,95, 10 , 1  i…”

Section: дополнительные граничные условия в задачах теплопроводности ...unclassified

“…Решение (29) точно удовлетворяет уравнению (6) и граничным условиям (8), ( 9) и приближенно -начальному условию (7). Причина точного выполнения уравнения (6) объясняется выполнением ДГУ (12)-( 14), что эквивалентно его выполнению на границах области (в предельном смысле -правая и левая части уравнения при подстановке в него решения (29) равны нулю).…”

Section: дополнительные граничные условия в задачах теплопроводности ...unclassified

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Additional boundary conditions in heat conduction problems with coordinate variable initial condition

Trubitsyn,

Gavrilova,

Kotova

et al. 2023

Vestnik IGEU

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It is exceedingly difficult to obtain mathematically accurate analytical solutions of heat conduction problems with a variable initial condition. Known solutions of these problems are expressed by cumbersome functional series that converge poorly in the range of small values of time and space variables. Thus, to obtain simpler and more effective solutions of these problems is an urgent issue. The authors have used an additional required function and additional boundary conditions to obtain solutions of the problem. Application of the additional required function allows us to reduce the original partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are in such a form that their fulfillment using the resulting solution is equivalent to the fulfillment of the equation at the boundary points. The authors have developed a technique to obtain an analytical solution of the heat conduction problem under a linear change of the initial condition, based on an additional required function and additional boundary conditions. Solution of an ordinary differential equation with respect to the additional required function determines the eigenvalues. In classical methods these eigenvalues are found in the solution of the Sturm–Liouville boundary value problem. The authors have proposed another, simpler solution to determine eigenvalues. An accurate analytical solution of the heat conduction problem for an unbounded plate with a coordinate-variable initial condition is obtained. The scientific and practical value of the proposed analytical solution is the development of an innovative approach to determine eigenvalues, as well as elimination of complex integrals when we solve the equation and initial conditions of the boundary value problem. It makes possible to simplify the use of the solution obtained in engineering applications.

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Section: дополнительные граничные условия в задачах теплопроводности ...unclassified

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Additional boundary conditions in heat conduction problems with coordinate variable initial condition

Trubitsyn,

Gavrilova,

Kotova

et al. 2023

Vestnik IGEU

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“…Biological systems are open thermodynamic systems which strive to become thermally balanced and in which heat transfer occurs continuously. [8][9][10]. Assessment of heat exchange processes is based on temperature and heat flux measurements.…”

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confidence: 99%

Heat exchange in the human eye: a review

Zadorozhnyy¹,

Korol²,

Naumenko³

et al. 2022

Oftalmol Zh

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Thermal homeostasis is required in order to ensure that the normal function of the human body is maintained under various environmental conditions. Various pathological processes impacting metabolism in tissues and organs (e.g., the human eye) are accompanied by changes in relative internal heat balance. Although numerous relevant studies have been conducted, heat exchange processes in the human eye have not been yet sufficiently investigated. Further research on the features of heat exchange in the eye is required not only to improve our knowledge in the field of physiology of the eye, but also to use the data obtained for developing novel advanced techniques for eye disease diagnosis and treatment.

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Method for determining eigen numbers in heat conduction problems for a cylinder

Kotova,

Klebleev,

Kudinov

2023

«Izvestiya vysshikh uchebnykh zavedenii. PROBLEMY ENERGETIKI»

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RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

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Models in Thermomechanics with a Finite and Infinite Velocity of Heat Propagation

Cited by 3 publications

References 0 publications

Additional boundary conditions in heat conduction problems with coordinate variable initial condition

Additional boundary conditions in heat conduction problems with coordinate variable initial condition

Heat exchange in the human eye: a review

Method for determining eigen numbers in heat conduction problems for a cylinder

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