Effects Of Thermal Relaxation Time Research Articles

Impact of an MHD Cattaneo-Christov model for a Williamson fluid flow across a wedge with a convective boundary condition: Homotopy Analysis Method

The present study aims to scrutinise the influence of the Cattaneo-Christov (CC) heat flux model on magnetohydrodynamic (MHD) laminar boundary-layer flow through a wedge in a non-Newtonian Williamson fluid along with convective boundary condition embedded in a porous medium. Significantly, the presence of the CC model explores the heat transfer characteristics which forecast the effect of thermal relaxation time on the boundary-layer phenomenon. The governing equations are formulated and modified into non-linear, coupled ODEs by utilising similarity transformations and then tackled by Homotopy Analysis Method (HAM). The obtained results are discussed and presented graphically for various values of emerging parameters on velocity and temperature distributions. The results reveal that the velocity is an enhancing function of the magnetic parameter, permeability parameter and Hartree pressure gradient parameter whereas decreasing with larger values of the Williamson fluid parameter. It is evident that the temperature upsurges due to the Biot number and decays with the relaxation parameter. The numerical values of skin-friction and Nusselt number are tabulated. Comparative reviews between the formerly published literature and the current data are made for specific cases, which are inspected to be in tremendous agreement. Abbreviations: BLF: Boundary layer flow; MHD Magnetohydrodynamic; CC model: Cattaneo-Christov model; NNF: non-Newtonian fluid; ODEs: Ordinary differential equations; PDEs: Partial differential equations

Mathematical Modelling for Studying the Influence of the Initial Stress and Relaxation Time on Reflection Waves in Thermo-piezoelectric Half-space

The aim of this study is to illustrate of the influence of the thermal relaxation time according the Lord-Shulman model and the initial stress on the reflection of plane waves in an elastic transversely isotropic thermo-piezoelectric half-space. The governing equation and suitable boundary conditions of a transversely isotropic thermo-piezoelectric medium are solved to obtain reflection coefficients when incident waves are falling at the interface between vacuum and half-space. The effects of thermal relaxation time and the initial stress are calculated numerically and shown graphically on these coefficients

Non-Fourier Heat Conduction and Thermal-Stress Analysis of a Spherical Ice Particle Subjected to Thermal Shock in PEM Fuel Cell at Quick Cold Start-Up

Quick start at low temperature is one of the key bottleneck technologies that restrict the large-scale commercialization of proton exchange membrane (PEM) fuel cell vehicles. Ice and devices in battery systems based on thermal deicing are inevitably subjected to thermal shock. In order to study this problem, a non-Fourier heat conduction model is established to study the temperature response of a spherical ice particle subjected to thermal shock with different boundary conditions on the surface. Furthermore, distribution of thermal stress in the particle is obtained using the calculated temperature field, and the effect of thermal relaxation time and boundary conditions on the temperature response as well as the thermal stress field are also analyzed. The results, which are significantly different from that obtained using Fourier law of heat conduction, show that the mechanical stresses and the serious expansion of the ice particle may lead to the severe deformation of the devices connected to the ice during the cold start-up, imposing a great challenge in controlling structural reliability of PEM fuel cells. The numerical results are expected to provide a scientific theoretical basis for PEM fuel cell design and low-temperature start-up control.

Effects of thermal relaxation times and porosity in a Lord-Shulman and refined multi-phase lags model of generalized thermoelasticity

The aim of this article is to build a mathematical model to study the problem of generalized thermoelastic porous medium according to the Lord–Shulman model and refined multi-phase lags model. The medium is considered to be homogeneous, isotropic and porous, such that changes in the size of the fracture field are in two dimensions. Four ordinary differential equations of the modified couple stress of thermoelastic porous media are obtained instead of partial differential equations and they solved precisely. The harmonic wave analysis procedure is proposed to find systematically the appropriate solutions to the variables of stresses, displacements, the change in the temperature, and in the volume fraction field. The outcomes of the physical variables have been computed numerically and clarified graphically. The comparison between both refined multi-phase lags and Lord–Shulman models are carried out for particular values of distance and various values of thermal relaxation times. This article’s findings may be useful to many researchers and scientists in the subject of materials science, especially porous media, as well as those working on creating modern technological materials.

The effects of thermal relaxation time in a two-dimension porous medium due to thermal loading

ABSTRACT In this article, the generalized model for thermoelastic waves with one relaxation time is utilized to compute the increment of temperature, the components of displacement, the changes in volume fraction field and the stress components in a two-dimension porous medium. By using Fourier–Laplace transforms with the eigenvalue approach, the physical quantities are analytically obtained. The derived method is evaluated with numerical results which are applied to the porous medium in simplified geometry. Numerical outcomes for all the physical quantities considered are implemented and represented graphically. The effects of thermal relaxation time in the temperature, the changes in volume fraction field, the displacement components and the stress components have been depicted graphically.

Study of deformation due to thermal shock in porous thermoelastic material with reference temperature dependent properties

The present study is focused on the numerical investigation in time domain of the deformation in thermoelastic material with one relaxation time (Lord and Shulman theory) with voids under dependence of modulus of elasticity and thermal conductivity on reference temperature. For dependency on reference temperature, we have taken two cases: (i) linear case and (ii) quadratic case. For numerical investigation, the finite element method with eight-/nine-noded isoparametric finite elements has been used. The results demonstrate that finite element method can reliably predict the deformation of the medium. Finite element method has an advantage over Laplace/Fourier transformation as the equations are solved directly in the time domain, but in Laplace/Fourier transformation, numerical inversion is quite complicated with possible errors. The comparison of these two cases has been discussed in result and discussion section for the temperature distribution, the displacement component, volume fraction field and the stresses. A comparison of Newmark method and Galerkin’s method for time domain has been given though three-dimensional graphs. An appreciable effect of thermal relaxation times t 0 is observed from the various results.

Cattaneo–Christov heat flux model for three-dimensional flow of a viscoelastic fluid on an exponentially stretching surface

ABSTRACT In this article, we explore the three-dimensional boundary-layer flow over an exponentially stretching surface in two parallel ways. Constitutive equations of a second-grade fluid are used. Instead of classical Fourier’s law, Cattaneo–Christov heat flux model is employed for the formulation of the energy equation. This model can predict the effects of thermal relaxation time on the boundary layer. The resulting partial differential equations are reduced into ordinary differential equations by similarity transformations. Homotopy Analysis Method (HAM) is employed to solve the non-linear problem. Physical impact of emerging parameters on the momentum and thermal boundary-layer thickness are studied.

A GL Model on Thermo-Elastic Interaction in a Poroelastic Material Using Finite Element Method

The purpose of this study is to provide a method to investigate the effects of thermal relaxation times in a poroelastic material by using the finite element method. The formulations are applied under the Green and Lindsay model, with four thermal relaxation times. Due to the complex governing equation, the finite element method has been used to solve these problems. All physical quantities are presented as symmetric and asymmetric tensors. The effects of thermal relaxation times and porosity in a poro-thermoelastic medium are studied. Numerical computations for temperatures, displacements and stresses for the liquid and the solid are presented graphically.

Thermal response of cylindrical tissue induced by laser irradiation with experimental study

PurposeThe purpose of this paper is to provide a method for determining the numerical solutions of the thermal damage of cylindrical living tissues using hyperbolic bioheat model. Due to the complex governing equation, the finite element approach has been adopted to solve these problems. To approve the accuracy of the numerical solution, the numerical outcomes obtained by the finite element approach are compared with the existing experimental study. In addition, the comparisons between the numerical outcomes and the existing experimental data displays that the present mathematical models are efficient tools to evaluate the bioheat transfer in the cylindrical living tissue. Numerical computations for temperatures and thermal damage are presented graphically.Design/methodology/approachIn this section, the complex equation of bioheat transfer based upon one relaxation time in cylindrical living tissue is summarized by using the finite element method. This method has been used here to get the solution of equation (8) with initial conditions (9) and boundary conditions (10). The finite element technique is a strong method originally advanced for numerical solutions of complex problems in many fields, and it is the approach of choice for complex systems. Another advantage of this method is that it makes it possible to visualize and quantify the physical effects independently of the experimental limits. Abbas and his colleagues [26-34] have solved several problems under generalized thermoelastic theories.FindingsIn this study, the different values of blood perfusion and thermal relaxation time of the dermal part of cylindrical living tissue are used. To verify the accuracy of the numerical solutions, the numerical outcomes obtained by the finite element procedure and the existing experimental study have been compared. This comparison displays that the present mathematical model is an effective tool to evaluate the bioheat transfer in the living tissue.Originality/valueThe validation of the obtained results by using experimental data the numerical solution of hyperbolic bioheat equation is presented. Due to the nonlinearity of the basic equation, the finite element approach is adopted. The effects of thermal relaxation times on the thermal damage and temperature are studied.

Investigation of the one-dimensional transient heat conduction problem of a coated high strength steel plate

In this paper, the one-dimensional transient heat conduction problem is investigated of a coated high strength steel (HSS) plate which is composed of two coating layers and a HSS layer. As the coating is extremely thin, non-Fourier heat conduction is applied to this part, while the steel part is analyzed by Fourier conduction. Then the temperature increment equations are obtained, which can be calculated by the Newmark method. The effects of thermal relaxation time, temperature boundary conditions, and coating parameters on temperature increment distribution of the coated HSS plate are also presented. Thus, the one-dimensional transient heat conduction problem of a coated HSS plate can be solved, which contributes to practical application and engineering design.

Hyperchaos in convection with the Cattaneo-Christov heat-flux model

In this paper, we report on the nonlinear dynamics of a five-variable, four-parameter system, which models the effects of thermal relaxation time on Rayleigh-Benard convection of Boussinesq fluid layer heated underneath. Six cross-sections of the four-dimensional parameter-space are considered. By using Lyapunov exponents spectra to characterize the dynamical behavior at each point of each these diagrams, we show that different parameter regions are allowed, from equilibrium points to chaos and hyperchaos regions.

Spectral Quasi-Linearization Method for Entropy Generation Using the Cattaneo–Christov Heat Flux Model

In this paper, we studying the entropy generation in Sakiadis nanofluid flowing along a moving plate subject to magnetic field and a Cattaneo–Christov heat flux model that may predict the effects of thermal relaxation time on the boundary layer flow. The nonlinear transport equations are solved using a spectral quasi-linearization method. An analysis of the convergence of the method is presented, and the importance of various fluid and physical parameters concerning the behavior of the solutions is explored. Numerical analysis of the residual error and convergence properties of the method are also discussed. One of the benefits of the proposed method is that it is computationally fast and gives very accurate results after only a few iterations using very few grid points in the numerical discretization process. It is shown that the method converges fast and gives accurate results. The results show that entropy generation increases with an increase in the Reynolds number. The Bejan number is strongly affected by variations in the magnetic parameter, Brinkman number and temperature difference parameter.

Thermo diffusion effects on MHD Casson fluid flow over non-flatness stretching surface: Keller box method

ABSTRACT This paper numerically analyses thermo diffusion effects on MHD Casson fluid flow over a stretching sheet with variable thickness. Cattaneo–Christov heat flux model is imposed to predict the effects of thermal relaxation time. Similarity transformation is employed to normalise the governing boundary layer equations. Keller box method is applied to solve the system of equations. The behaviour of emerging parameters, such as Casson parameter (), Magnetic parameter (), power index (), variable viscosity parameter (), Prandtl number (), Soret number ( ) and Schmidt number () on velocity, temperature and concentration distributions, are discussed and analysed graphically. It is found that temperature profile decreases for large entries of thermal relaxation parameter. The concentration boundary layer thickness increases for large entries of Soret number.

Generalized photo-thermo-elastic interaction in a semiconductor plate with two relaxation times

This study focuses on the effect of thermal relaxation times in a two-dimensional semiconductor medium by using eigenvalue approach. The formulation is applied under Green and Lindsay theory with two relaxation times. The bounding surface of the half-space is subjected to a heat flux with an exponentially decaying pulse and taken to be traction free. The variables solutions are obtained analytically by using Laplace-Fourier transformations and the eigenvalue approach methodology. A semiconducting material like as silicon is considered. The numerical computations are obtained and the outcomes are displayed graphically to assess the effects of the thermal relaxation times on the plasma, thermal, and elastic waves.

A non-Fourier approach towards the analysis of heat transfer enhancement with water based nanofluids through a channel

In this article, a non-Fourier approach to model the heat transfer phenomenon in nanofluids having application to automotive industry is studied. In this respect, a recently proposed hyperbolic heat flux equation is embedded into the heat energy equation and thereby incorporating the effect of thermal relaxation time. Nanofluids are formed by considering copper oxide (CuO), Titanium dioxide (TiO2) and Aluminum oxide (Al2O3) nano-solid particles in the base fluid. The flow governing system of PDEs along with boundary conditions is transformed into its respective coupled system of nonlinear ODEs using suitable similarity functions. Runge-Kutta-Fehlberg (RK-5) numerical scheme embedded with shooting method is implemented and used to solve the obtained boundary value problem. Numerical simulations are performed and tabulated to analyze the influence of solid volume fraction on local coefficient of skin-friction and Nusselt number. A comparison is made between the results by Fourier and present heat flux model. We conclude that the presented new approach is more general and thus allows predicting the influence of thermal relaxation time on the heat transfer characteristics. Moreover, consideration of present model over the Fourier model helps to predict the actual temporal behavior of solution.

MHD stagnation point flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model

This article presents an analysis of stagnation point of coupled flow and heat transfer of an upper-convected Maxwell fluid over a stretching sheet along with magnetic effects and slip condition at the boundary. The recently proposed Cattaneo–Christov model is employed in the energy equation to investigate the effects of thermal relaxation time. Similarity transformations are adopted to convert the governing partial differential equations into ordinary differential equations. Numerical solution of the system of ODEs is achieved by shooting method together with Runge–Kutta method of order four. The effects of stretching ratio parameter (0 ≤ e ≤ 0.5), elasticity number (0 ≤ β ≤ 1.5), heat flux relaxation time (0 ≤ γ ≤ 1.5), magnetic parameter (0 ≤ M ≤ 1.5), slip coefficient (1 ≤ b ≤ 4) and Prandtl number (0 ≤ Pr ≤ 1.5) on velocity and temperature are investigated graphically and numerically. It is observed that temperature boosts up with an increase in thermal relaxation time.

Temperature Response in a Living Tissue with Different Heating Source at the Skin Surface Under Relaxation Time

In this paper we have analytically characterizes the temperature distribution in a living tissue for more effective thermal treatment of tumor cells based on the modified Pennes bioheat transfer equation subjected to constant, oscillatory, cosine heat flux as well as sinusoidally heating criteria. A thorough parametric study has been made and determined optimal values of the parameters that lead to reach the temperature above therapeutic value \(43^\circ \mathrm{C}\). We assumed that heat mainly propagates in the direction perpendicular to the skin surface. The analytical solutions of the thermal wave model of one-dimensional bioheat transfer equation with different heating conditions are obtained by using Laplace transform technique. The effects of thermal relaxation time and heat flux on the skin-surface, the blood perfusion rate, thermal conductivity and frequency of heating on temperature distribution are analyzed qualitatively as well as quantitatively. The study shows that the temperature distribution has an enhancing effect on the thermal relaxation time and heat flux at the skin-surface. Further, the blood perfusion rate has reducing effect on the temperature of the living tissues.

Numerical Study of Cattaneo-Christov Heat Flux Model for Viscoelastic Flow Due to an Exponentially Stretching Surface.

This work deals with the flow and heat transfer in upper-convected Maxwell fluid above an exponentially stretching surface. Cattaneo-Christov heat flux model is employed for the formulation of the energy equation. This model can predict the effects of thermal relaxation time on the boundary layer. Similarity approach is utilized to normalize the governing boundary layer equations. Local similarity solutions are achieved by shooting approach together with fourth-fifth-order Runge-Kutta integration technique and Newton’s method. Our computations reveal that fluid temperature has inverse relationship with the thermal relaxation time. Further the fluid velocity is a decreasing function of the fluid relaxation time. A comparison of Fourier’s law and the Cattaneo-Christov’s law is also presented. Present attempt even in the case of Newtonian fluid is not yet available in the literature.

Wave propagation in a generalized piezothermoelastic rotating bar of circular cross section

Purpose – The purpose of this paper is to study the wave propagation in a generalized piezothermoelastic rotating bar of circular cross-section using three-dimensional linear theory of elasticity. Design/methodology/approach – A mathematical model is developed to study the wave propagation in a generalized piezothermelastic rotating bar of circular cross-section by using Lord-Shulman (LS) and Green-Lindsay (GL) theory of thermoelasticity. After developing the formal solution of the mathematical model consisting of partial differential equations, the frequency equations have been derived by using the thermally insulated/isothermal and electrically shorted/charge free boundary conditions prevailing at the surface of the circular cross-sectional bar. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots. Findings – In order to include the time requirement for the acceleration of the heat flow and the coupling between the temperature and strain fields, the analytical terms have been derived for the non-classical thermo-elastic theories, LS and GL theory. The computed physical quantities such as thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency have been presented in the form of dispersion curves. From the graphical patterns of the structure, the effect of thermal relaxation times and the rotational speed as well as the anisotropy of the of the material on the various considered wave characteristics is more significant and dominant in the flexural modes of vibration. The effect of such physical quantities provides the foundation for the construction of temperature sensors, acoustic sensor and rotating gyroscope. Originality/value – In this paper, the influence of thermal relaxation times and rotational speed on the wave number with thermo-mechanical coupling, electro-mechanical coupling, frequency shift, specific loss and frequency has been observed and are presented as dispersion curves. The effect of thermal relaxation time and rotational speed on wave number for the case of generalized piezothermoelastic material of circular cross-section was never reported in the literature. These results are new and original.

Flexural response of thermoelastic thin beam resonators due to thermal and mechanical loads

The flexural vibrations of homogeneous, transversely isotropic, generalized (non-Fourier) thermoelastic simply-supported thin beam resonators due to thermal and mechanical loads have been investigated under Euler–Bernoulli hypothesis. The model equation governing the transverse vibrations has been solved by using double Laplace transform technique (LTT) with respect to space and time. A mesh-less computational method, Adomian decomposition method (ADM), has also been employed to solve the model equation in the space domain for the comparison purpose. The inversion of the transformed solution has been carried out by using Durbin׳s technique in order to obtain the results in physical time domain. The natural frequencies and the corresponding mode shapes of the beam have been computed. The effects of thermal variations and other parameters on the deflection and temperature profiles of the thin beam have been studied. The effect of thermal relaxation time (second sound) on the vibration characteristics of beam has also been observed. The results computed by using LTT and ADM have been compared and are found to be in good agreement. It is noticed that ADM is an accurate, computer friendly and provides a unified, systematic powerful mathematical tool for solving the static and dynamic vibration problems for the considered beams.