Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

Mechanical Study of Nano⁃ceramic Thermal Barrier Coatings by the Equation of Phonon Radiative Transfer

Temperature filed,thermal stress,especially tensile stress and J⁃integral are important for thermal barrier coatings(TBCs)under thermal shock.At the micro⁃and nano⁃scale,the energy transport mechanisms are significantly different from those at the macro⁃scale.The temperature fields,which are obtained by combining the Equation of Phonon Radiative Transport(EPRT)(for the nano⁃scale ceramic TBCs)and the Fourier law(for the substrate),are used as the thermal loading in the thermal stress and J⁃integral of an edge in the TBCs analysis by the finite element method.The temperature field and thermal stresses as well as J⁃integral are compared with those which are calculated by applying the Fourier law to both the TBCs and the substrate.The influence of the physical heat properties of the TBCs on the temperature field and thermal stress and J⁃integral have been analyzed in this paper.It is concluded that the temperature,thermal stress,including the tensile and compressive components,and J⁃integral which are calculated with the EPRT,are lower than that calculated with the Fourier law in the TBCs.Moreover,thermal stress in the TBCs increase with increasing phonon speed and relaxation time,but J⁃integral at the crack tip is in the opposite.

Best Determined Position of Vents Based on Jet Cooling Model

In some data centers,cold air is required to act on the cabinet to achieve cooling requirements,and the mixing of cold air and hot air reduces the utilization efficiency of cold air.In order to solve this problem,a jet cooling model is established to solve the optimal position of the outlet through the movement of cold air.

Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping

We study the well-posedness and decay properties of a one-dimensional thermoelastic laminated beam system either with or without structural damping,of which the heat conduction is given by Fourier's law effective in the rotation angle displacements.We show that the system is well-posed by using the Lumer-Philips theorem,and prove that the system is exponentially stable if and only if the wave speeds are equal,by using the perturbed energy method and Gearhart-Herbst-Prüss-Huang theorem.Further-more,we show that the system with structural damping is polynomially stable provided that the wave speeds are not equal,by using the second-order energy method.When the speeds are not equal,whether the system without structural damping may has polynomial stability is left as an open problem.