Hyperbolic Heat Conduction Research Articles

Effects of heat wave propagation and initial stress on the thermo-elastic behavior of porous-auxetic FG circular plate with variable hybrid foundation

Engineering structures' heterogeneity and porous-auxetic features generate some prominent effects on their thermo-mechanical behavior. In addition, initial stress and heat wave propagation also play important roles in the mechanical behavior of FGM structures. Therefore, this paper investigates the effects of heat wave propagation and initial stress on the thermo-elastic behavior of porous-auxetic FG circular/annular plates with variable hybrid foundations. The one-dimensional transient hyperbolic heat conduction equation is modeled using the Cattaneo-Vernotte (C-V) theory and numerically solved using the Crank-sNicolson (C-N) finite difference scheme. Besides, a mathematical model of the other governing equations is extracted based on the classical thermoporoelasticity theory and semi-analytically solved by employing the state-space method and differential quadrature technique. Numerical examples are provided to illustrate the findings of this investigation graphically. The results reveal that the grading indices, and thermal relaxation time have a significant effect on heat wave propagation in the FG circular plate. Initial stress, porosity, and auxeticity also play an important role in its thermoelastic behavior.

Dynamic Thermal Response of Multiple Interface Cracks between a Half-Plane and a Coating Layer under General Transient Temperature Loading.

This paper explores the thermal behavior of multiple interface cracks situated between a half-plane and a thermal coating layer when subjected to transient thermal loading. The temperature distribution is analyzed using the hyperbolic heat conduction theory. In this model, cracks are represented as arrays of thermal dislocations, with densities calculated via Fourier and Laplace transformations. The methodology involves determining the temperature gradient within the uncracked region, and these calculations contribute to formulating a singular integral equation specific to the crack problem. This equation is subsequently utilized to ascertain the dislocation densities at the crack surface, which facilitates the estimation of temperature gradient intensity factors for the interface cracks experiencing transient thermal loading. This paper further explores how the relaxation time, loading parameters, and crack dimensions impact the temperature gradient intensity factors. The results can be used in fracture analysis of structures operating at high temperatures and can also assist in the selection and design of coating materials for specific applications, to minimize the damage caused by temperature loading.

Acceleration waves in thermoelastic complex media with temperature-dependent phase fields

AbstractWe analyze homothermal acceleration waves in complex materials (those with active microstructure) in the presence of internal constraints that link the temperature to a manifold-valued phase-field describing a generic material microstructure at a certain spatial scale. Such a constraint leads to hyperbolic heat conduction even in the absence of macroscopic strain; we show how it influences the way acceleration waves propagate. The scheme describes a thermoelastic behavior that is compatible with dependence of the free energy on temperature gradient (a dependence otherwise forbidden by the second law of thermodynamics in the traditional non-isothermal description of simple bodies). We eventually provide examples in which the general treatment that we develop applies.

Decay of the compressible Navier-Stokes equations with hyperbolic heat conduction

This paper is concerned with the global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction, where the heat flux is assumed to satisfy Cattaneo's law. When the initial perturbation is small and regular, we establish the global existence of the classical solution without any additional assumption on the relaxation time τ. Furthermore, assuming the initial perturbation belongs to a negative Sobolev space, we obtain the optimal time-decay rates of the high-order spatial derivatives of solutions by using a pure energy method instead of complicated spectral analysis. It is also observed that the heat flux, due to its damping structure, decays to the motionless state at a faster time-decay rate compared to density, velocity, and temperature.

Dynamic Thermal Response of Multiple Parallel Cracks in a Half Plane under General Transient Thermal Loading

Understanding the dynamic thermal response of materials is crucial for designing effective thermal protection systems, particularly in extreme thermal environments such as transient thermal loading, extremely low/high temperature, etc. This study investigates the dynamic thermal response of a half plane containing multiple parallel cracks under transient thermal loading using a non‐Fourier, hyperbolic heat conduction model. Our findings highlight significant deviations from traditional Fourier models, enhancing the predictive capabilities for designing thermal protection systems in extreme thermal environments. The cracks are modeled as distributions of thermal dislocations, with densities determined through Fourier and Laplace transforms. By solving the resulting singular integral equations, we calculate the temperature gradient intensity factors across multiple scenarios. Additionally, we examine the effects of thermal relaxation time, loading parameters, and crack spacing on the thermal intensity factors, both in transient and steady‐state conditions. This research confirms the robustness of the hyperbolic model and its practical implications for thermal analysis in engineering applications.

Space-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction

We consider compressible Navier-Stokes equations with hyperbolic heat conduction in R3. A space-time description of the classical solution is given when the initial perturbation is suitable small. The result implies that all of the unknowns obey the generalized Huygens’ principle as the classical compressible Navier-Stokes equations in Liu and Wang [Commun. Math. Phys. 196, 145–173 (1998)]. Additionally, we show that the decay of the flux q is faster than the other unknowns.

Causal explicit algorithm for heat conduction in a plasma

Hyperbolic heat conduction extends standard Spitzer-Harm heat conduction by including a term proportional to the time derivative of the heat flux. The new term arises from a kinetic derivation of the heat flux that includes higher order corrections. We present a causal explicit numerical algorithm for solving the nonlinear hyperbolic heat conduction equation in an unmagnetized plasma. The maximum stable timestep for the causal explicit algorithm scales linearly with the cell size, owing to the hyperbolic nature of the problem. This is in contrast to the quadratic scaling of the maximum stable timestep with the cell size for the parabolic forward time centered space algorithm. The favorable scaling of the timestep with the cell size enables a practical explicit implementation of heat conduction in high-performance massively parallel plasma codes. In particular, we have implemented the causal explicit algorithm in the laser plasma interaction code pF3D. We verify the CE algorithm and analyze its convergence rate by simulating a harmonic mode, which has an analytic solution within the context of the HHC model. We also compare simulations using the CE algorithm to those using the forward time centered space algorithm on a pair of test problems: evolution in time of a Gaussian temperature perturbation in a uniform plasma and heat transport in the presence of inverse bremsstrahlung heating by a Gaussian laser speckle.

Hyperbolic Conduction and Radiation Heat Transfer in Axisymmetric Cylinders with Variable Properties

This work investigates the combined mode of hyperbolic heat conduction and radiation transfer in a one-dimensional axisymmetric cylinder filled with absorbing, emitting, and scattering media. The volumetric radiation is investigated thanks to the semianalytic solution of the matrix formulation of the spherical harmonics equations PN. The governing hyperbolic energy equation is solved using the finite volume method (FVM) with Roe’s correction of interface fluxes in order to enhance the performances of the method, and the lattice Boltzmann method (LBM) has been designed for comparisons. The effects of the parameters such as constant and spatial-dependent scattering albedos, temperature-dependent thermal conductivity, heat-generated sources, extinction, and the conduction–radiation parameter on both the temperature and heat flux distributions for steady and transient states within the medium are examined. The results of the present work are in excellent agreement with those available in the literature. The PN−FVM results are also compared to those obtained with the PN−LBM combination, and excellent agreement is obtained. These results show that the mentioned parameters have a significant effect on both the temperature profiles and the hyperbolic sharp wave front. This study also shows that the proposed layered approach is an efficient, fast, and accurate solution method for radiative analysis in inhomogeneous media, whereas the Roe’s correction of interface fluxes in the FVM is suitable to accommodate a thermal wave front in non-Fourier analysis.

Heat transfer analysis in a longitudinal porous trapezoidal fin by non-Fourier heat conduction model: An application of artificial neural network with Levenberg–Marquardt approach

Thermal critical issues commonly occur in advanced electrical devices as a response to excessive heat generation or a loss in efficient surface area for heat exclusion. This issue can be addressed by utilizing the extended surface to improve the heat transfer performance of the devices. Thus, the present analysis is devoted to scrutinizing the non-Fourier unsteady heat transference of a trapezoidal porous fin. Levenberg–Marquardt technique of backpropagation artificial neural network (LMT-BANN) is employed here to analyze the thermal variation in the fin. The developed governing equation is the hyperbolic heat conduction equation (HHCE), which is transformed into a dimensionless partial differential equation (PDE) using dimensionless variables. LMT-BANN is employed on thermal numerical data and is developed to trace numerical approximation of fin problem using a methodology that includes testing, training, and validation. A graphical visualization of the consequences of thermal variables on the temperature field is presented. The notable evidence of this study reveals that as the magnitude of the convection factor increases, thermal dissipation through the fin gradually decreases. Further, the LMT-BANN technique has been determined to be an effective, reliable, and rapidly convergent stochastic computational solver that can be used effectively for examining the thermal model.

Laser-Induced Thermoelastic Response in an Isotropic Medium Having Variable Material Moduli

Thermal displacement and stress distribution are analysed in the domain of plane-strain geometry of an isotropic thermoelastic medium having variable material moduli. In this context of the analysis, laser induced heat sources are planted on the reference plane. To analyse the thermal responses due to the injected heat sources into the medium, hyperbolic type heat conduction model with three phase-lags incorporated. Analytical solutions are expressed in-terms of Laplace–Fourier integral transform domain and the physical behaviour of displacement and stresses are depicted through a discretised form of inverse-integral transformation technique. Finally, parameterized characteristic analysis are presented graphically and the salient features of thermal displacements are highlighted.

Two collinear cracks in a transversely isotropic medium under the hyperbolic heat conduction law

Two collinear cracks in a transversely isotropic medium under the hyperbolic heat conduction law

Two-Dimensional C-V Heat Conduction Investigation of an FG-Finite Axisymmetric Hollow Cylinder

In the present work, we implement a graded finite element analysis to solve the axisymmetric 2D hyperbolic heat conduction equation in a finite hollow cylinder made of functionally graded materials using quadratic Lagrangian shape functions. The graded FE method is verified, and the simple rule of the mixture with power-law volume fraction is found to enhance the effective thermal properties’ gradation along the radial direction, including the thermal relaxation time. The effects of the Vernotte numbers and material distributions on temperature waves are investigated in depth, and the results are discussed for Fourier and non-Fourier heat conductions, and homogeneous and inhomogeneous material distributions. The homogeneous cylinder wall made of SUS304 shows faster temperature wave velocity in comparison to the ceramic-rich cylinder wall, which demonstrates the slowest one. Furthermore, the temperature profiles along the radial direction when n = 2 and n = 5 are almost the same in all Ve numbers, and by increasing the Ve numbers, the temperature waves move slower in all the material distributions. Finally, by tuning the material distribution which affects the thermal relaxation time, the desirable results for temperature distribution can be achieved.

Application of hyperbolic heat conduction model in thermal lens spectroscopy

Between the techniques directed to studying the thermal properties of materials, we can cite the photothermal methods, including the thermal lens technique in this group. The technique of thermal lens spectroscopy has, in most cases, its temperature profile described by Fourier’s Law of heat conduction. In this work, the temperature distribution was successfully represented through the hyperbolic heat conduction model. However, with the application of the hyperbolic model, the differential equation that describes the problem has no analytical solution; therefore, we used two techniques for their solution. The method of lines (purely numerical method) and the generalized integral transform technique (analytical-numerical method). Both approaches have excellent rates of convergence in their solutions. They also present the new agreement proposed solution with experimental data, thus, providing a new perspective for future work in this spectroscopic technique.

A practical jointed approach to transient hyperbolic heat conduction of FGM cylinders and spheres

A practical jointed approach to transient hyperbolic heat conduction of FGM cylinders and spheres

Decay of the Compressible Navier-Stokes Equations with Hyperbolic Heat Conduction

Decay of the Compressible Navier-Stokes Equations with Hyperbolic Heat Conduction

Analytical solution of non-Fourier heat conduction in a 3-D hollow sphere under time-space varying boundary conditions

In the current research, a comprehensive analytical technique is presented to evaluate the non-Fourier thermal behavior of a 3-D hollow sphere subjected to arbitrarily-chosen space and time dependent boundary conditions. The transient hyperbolic thermal diffusion equation is driven based upon the Cattaneo-Vernotte (C-V) model and nondimensionalized using proper dimensionless parameters. The conventional procedure of separation of variables is applied for solving the 3-D hyperbolic heat conduction equation with general boundary conditions. In order to handle the time dependency of the boundary conditions, Duhamel's theorem is employed. Furthermore, for the purpose of demonstrating the applicability of the obtained general solution, two particular cases with different time-space varying boundary conditions are considered. Subsequently, their respective non-Fourier thermal characteristics are elaborately discussed and compared with the Fourier case. The quantitative analysis is carried out, including the profiles of the time-dependent temperature and 3-D distributions of temperature at different time frames. Eventually, the influences of the controlling factors such as Fourier number and Vernotte number on the temperature field distributions within a hollow sphere for both cases are assessed. The findings reveal that the lag time in the hyperbolic thermal propagation diminishes with a decrement of Vernotte number, and it asymptotically vanishes for the Fourier case. Also, the number and severity of the jump points that occurred in the non-Fourier cases decrease by increasing the Fourier number, and these points finally vanish at particular Fourier numbers.

Investigation of the fracture problem of functionally graded materials with an inclined crack under strong transient thermal loading

In this paper, a special method for solving transient thermal stress intensity factors (TSIFs) in functionally graded materials (FGMs) with inclined cracks under strong, transient thermal shock loading is established based on the hyperbolic heat conduction equation. First, the Newmark method is employed to obtain the strong transient temperature field of FGMs containing inclined cracks. Then, the transient interaction energy integral method (IEIM) is used to solve the TSIFs for inclined cracks. In addition, the influences of the distribution patterns of the thermal relaxation parameter, the crack length and the crack angle on the TSIFs, and the crack growth angle are investigated. Moreover, we discuss the difference between using the hyperbolic heat conduction theory and the classical, Fourier heat conduction theory in calculating the TSIFs of FGMs with inclined cracks. The results show that the present, combined IEIM-Newmark method can be used to solve strong transient thermal shock crack problems with high efficiency. This work can be used for the thermal design, evaluation, and engineering applications of FGM structures.

Finite element method for hyperbolic heat conduction model with discontinuous coefficients in one dimension

Finite element method for hyperbolic heat conduction model with discontinuous coefficients in one dimension

Thermal characteristics of longitudinal fin with Fourier and non-Fourier heat transfer by Fourier sine transforms

The quest for high-performance of heat transfer components on the basis of accommodating shapes, smaller weights, lower costs and little volume has significantly diverted the industries for the enhancement of heat dissipation with variable thermal properties of fins. This manuscript proposes the fractional modeling of Fourier and non-Fourier heat transfer of longitudinal fin via non-singular fractional approach. The configuration of longitudinal fin in terms of one dimension is developed for the mathematical model of parabolic and hyperbolic heat transfer equations. By considering the Fourier and non-Fourier heat transfer from longitudinal fin, the mathematical techniques of Fourier sine and Laplace transforms have been invoked. An analytic approach is tackled for handling the governing equation through special functions for the fractionalized parabolic and hyperbolic heat transfer equations in longitudinal fin. For the sake of comparative analysis of parabolic verses hyperbolic heat conduction of fin temperature, we depicted the distinct graphical illustrations; for instance, 2-dimensional graph, bar chart, contour graphs, heat graph, 3-dimensional graphs and column graphs on for the variants of different rheological impacts of longitudinal fin.

Space‐time stabilized FEM applied on the hyperbolic heat conduction equation for high speed frictional contact

AbstractFrictional contact results in dissipation of energy, wherein one part is transformed into thermal energy. This leads to heating under the surface, where the resulting transient temperature fields at short times and high Peclet numbers are of interest. Therefore, the numerical method of finite elements with space‐time stabilization technique is applied on the 2D hyperbolic heat conduction equation of Cattaneo‐Vernotte with respect to a moving frame. It is shown, that the combination of the θ‐Method with Streamline‐Upwind‐Petrov‐Galerkin‐FEM gives satisfying solutions for a wide broadband of Peclet numbers.