scholarly journals Time-Fractional Fourier Law in a finite hollow cylinder under Gaussian-distributed heat flux

2018 ◽  
Vol 180 ◽  
pp. 02008 ◽  
Author(s):  
Slawomir Blasiak
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Hollow Cylinder ◽  
Integral Transformation ◽  
Temperature Fields ◽  
Radial Coordinate ◽  
Finite Hollow Cylinder ◽  
The Laplace Transform ◽  
Transformation Methods ◽  
Fractional Heat Conduction

This paper presents the solution of the theoretical model of heat conduction based on timefractional Fourier equation for a finite hollow cylinder treated with heat flux on one of the front surfaces. A derivative of fractional order in the Caputo sense was applied to record the temperature derivative in time. The distributions of temperature fields in the hollow cylinder were determined with the use of Fourier-Bessel series, as surface functions of two variables (r, θ) . The distributions of temperature fields were determined using analytical methods and applying integral transformation methods. The Laplace transform with reference to time, the Fourier finite cosine transform with reference to axial coordinate z and Marchi-Zgrablich transform for radial coordinate r. The fractional heat conduction equation was analysed for 0 < α ≤ 2

Thermoelastic Interaction in an Infinite Long Hollow Cylinder with Fractional Heat Conduction Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 378-392
Author(s):  
Ahmed. E. Abouelregal
Keyword(s):  
Heat Conduction ◽  
Laplace Transform ◽  
Hollow Cylinder ◽  
Heat Conduction Equation ◽  
Thermal Stresses ◽  
Generalized Thermoelasticity ◽  
Constant Heat Flux ◽  
Conduction Equation ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

AbstractIn this work, we introduce a mathematical model for the theory of generalized thermoelasticity with fractional heat conduction equation. The presented model will be applied to an infinitely long hollow cylinder whose inner surface is traction free and subjected to a thermal and mechanical shocks, while the external surface is traction free and subjected to a constant heat flux. Some theories of thermoelasticity can extracted as limited cases from our model. Laplace transform methods are utilized to solve the problem and the inverse of the Laplace transform is done numerically using the Fourier expansion techniques. The results for the temperature, the thermal stresses and the displacement components are illustrated graphically for various values of fractional order parameter. Moreover, some particular cases of interest have also been discussed.

scholarly journals Time-Fractional Heat Conduction in a Plane with Two External Half-Infinite Line Slits under Heat Flux Loading

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 689 ◽  
Author(s):  
Yuriy Povstenko ◽  
Tamara Kyrylych
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Heat Conduction Equation ◽  
Graphical Representation ◽  
Integral Transform ◽  
Conduction Equation ◽  
Infinite Plane ◽  
Conservation Of Energy ◽  
Infinite Line ◽  
Fractional Heat Conduction

The time-fractional heat conduction equation follows from the law of conservation of energy and the corresponding time-nonlocal extension of the Fourier law with the “long-tail” power kernel. The time-fractional heat conduction equation with the Caputo derivative is solved for an infinite plane with two external half-infinite slits with the prescribed heat flux across their surfaces. The integral transform technique is used. The solution is obtained in the form of integrals with integrand being the Mittag–Leffler function. A graphical representation of numerical results is given.

scholarly journals Heat Conduction in Hollow Cylinders with Radiation

1964 ◽  
Vol 14 (2) ◽  
pp. 159-164 ◽  
Author(s):  
E. Marchi ◽  
G. Zgrablich
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Integral Transformation ◽  
Finite Integral Transformation ◽  
Finite Integral ◽  
Hollow Cylinders ◽  
Radiation Type

AbstractA new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.

scholarly journals MATHEMATIC SIMULATION OF GENERALIZED PROBLEM OF HEAT-EXCHANGE IN THE BODIES WITH HEMISPHERICAL SHAPE

2019 ◽  
Vol 6 (125) ◽  
pp. 55-67
Author(s):  
Mykhailo Berdnyk
Keyword(s):  
Heat Conduction ◽  
Dirichlet Boundary ◽  
Integral Transformation ◽  
Dirichlet Boundary Conditions ◽  
Temperature Fields ◽  
Finite Velocity ◽  
Drilling Operations ◽  
Equation Of Heat Conduction ◽  
Finite Space ◽  
Drilling Bit

The article presents the first mathematical model for calculating temperature fields of hemispherical bodies, which approximately simulates operation of the diamond-drilling bit and takes into account angular velocity of drilling operations and finite velocity of heat conduction, and which was created as a physicomathematical boundary problem for hyperbolic equation of heat conduction with the Dirichlet boundary conditions. Besides, a new integral transformation was formulated for the two-dimensional finite space, with the help of which and with the help of finite element method and Galerkin method a temperature field was found in the form of convergence series.

scholarly journals The mixed unsteady heat conduction problem for a half-infinite hollow cylinder

2019 ◽  
pp. 222-225
Author(s):  
I. M. Turchyn ◽  
G. V. Vasylko ◽  
O. Ya. Ivaskevych
Keyword(s):  
Heat Conduction ◽  
Integral Transformation ◽  
Mixed Boundary ◽  
Heat Conduction Problem ◽  
Laguerre Polynomials ◽  
Operating Conditions ◽  
Temperature Fields ◽  
Algebraic Equations ◽  
Dual Integral Equations ◽  
Unsteady Heat Conduction

Analysis of temperature fields is important for many engineering applications. The account of actual operating conditions of these structures frequently leads to mixed heating condition. The authors of this paper developed a new effective method of solutions derivation for mixed boundary-value unsteady heat conduction problems. This paper considers the cylinder with at the part of surface of which the temperature distribution is known. Outside this area the heat transfer by Newton's law is performed. To the heat conductivity problem it is applied the Laguerre integral transformation in time variables and integral Fourier transformation in spatial variable. As a result the triangular sequence of ordinary differential equations is obtained. The general solution of these sequences is obtained in the form of algebraic convolution. Taking into account the mixed boundary conditions leads to dual integral equations. For solution of this problem it is proposed the method of Neumann's series. By this method the problem is reduced to the infinite system of algebraic equations, for which the convergence of reduction procedure is proved. Finally, the unknown temperature is submitted as a series of Laguerre polynomials. The coefficient of these series is Fourier integrals.

scholarly journals Heat conduction in a composite sphere - the effect of fractional derivative order on temperature distribution

2018 ◽  
Vol 157 ◽  
pp. 08008
Author(s):  
Urszula Siedlecka ◽  
Stanisław Kukla
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Laplace Transform ◽  
Time Derivative ◽  
Spherical Layer ◽  
Solid Sphere ◽  
Liouville Derivative ◽  
The Laplace Transform ◽  
Fractional Heat Conduction

The aim of the contribution is an analysis of time-fractional heat conduction in a sphere with an inner heat source. The object of the consideration is a solid sphere with a spherical layer. The heat conduction in the solid sphere and spherical layer is governed by fractional heat conduction equation with a Caputo time-derivative. Mathematical (classical) or physical formulations of the Robin boundary condition and the perfect contact of the solid sphere and spherical layer is assumed. The boundary condition and the heat flux continuity condition at the interface are expressed by the Riemann-Liouville derivative. An exact solution of the problem under mathematical conditions is determined. A solution of the problem under physical boundary and continuity conditions using the Laplace transform method has been obtained. The inverse of the Laplace transform by using the Talbot method are numerically determined. Numerical results show the effect of the order of the Caputo and the Riemann-Liouville derivatives on the temperature distribution in the sphere.

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