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The purpose of this paper is to frame a dual-phase-lag model using the fractional theory of thermoelasticity with relaxation time. The generalized Fourier law of heat conduction based upon Tzou model that includes temperature gradient, the thermal displacement and two different translations of heat flux vector and temperature gradient has been used to formulate the heat conduction model. The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration.

The work presented in this manuscript proposes a dual-phase-lag mathematical model of a thick circular plate in a finite cylindrical domain subjected to axis-symmetric heat flux. The model has been designed in the context of fractional thermoelasticity by considering two successive terms in Taylor’s series expansion of fractional Fourier law of heat conduction in the two different translations of heat flux vector and temperature gradient. The analytical results have been obtained in Laplace transform domain by transforming the original problem into eigenvalue problem using Hankel and Laplace transforms. The numerical inversions of Laplace transforms have been achieved using the Gaver−Stehfast algorithm, and convergence criterion has been discussed. For illustrative purpose, the dual-phase-lag model proposed in this manuscript has been applied to a periodically varying heat source. The numerical results have been depicted graphically and compared with classical, fractional and generalized thermoelasticity for various fractional orders under consideration.

The microstructural interactions and corresponding thermal changes have been studied due to the involvement of relaxation time and delay time translations. This results in achieving the finite speed of thermal wave. Classical coupled and generalized thermoelasticity theories are recovered by considering the various special cases for different order of fractional derivatives and two different translations under consideration. This model has been applied to study the thermal effects in a thick circular plate subjected to a periodically varying heat source.

A dual-phase-lag model can effectively be incorporated to study the transient heat conduction problems for an exponentially decaying pulse boundary heat flux and/or for a short-pulse boundary heat flux in long solid tubes and cylinders. This model is also applicable to study the various effects of the thermal lag ratio and the shift time. These dual-phase-lag models are also practically applicable in the problems of modeling of nanoscale heat transport problems of semiconductor devices and accordingly semiconductors can be classified as per their ability of heat conduction.

To the authors’ knowledge, no one has discussed fractional thermoelastic dual-phase-lag problem associated with relaxation time in a finite cylindrical domain for a thick circular plate subjected to an axis-symmetric heat source. This is the latest and novel contribution to the field of thermal mechanics.

In this work, a modified thermoelastic model of heat conduction, including higher order of time derivative, is constructed by extending the Roychoudhuri model (TPL) (Choudhuri, 2007). In this new model, Fourier’s law of heat conduction is replaced by using Taylor series expansions, including three different phase lags for the heat flux, the thermal displacement and the temperature gradient. The generalized thermoelasticity models of Lord–Shulman (Lord and Shulman, 1967), Green and Naghdi (1991), dual-phase lag (Tzou, 1996) and three-phase lag (TPL) (Choudhuri, 2007) are obtained as special cases. The paper aims to discuss these issues.

The aim of this work is to establish a new generalized mathematical model of thermoelasticity that includes TPL in the vector of heat flux, and in the thermal displacement and temperature gradients extending TPL model (Li et al., 2019e). In this model, Fourier law of heat conduction is replaced by using Taylor series expansions to a modification of the Fourier law with introducing three different phase lags for the heat flux vector, the temperature gradient, and the thermal displacement gradient and keeping terms up with suitable higher orders.

The established high-order three-phase-lag heat conduction model reduces to the previous models of thermoelasticity as special cases.

In this paper, a TPL thermoelastic model is developed by extending the Roychoudhuri (Sherief and Raslan, 2017) model (TPL) considering the Taylor series approximation of the equation of heat conduction. This model is an alternative construction to the TPL model. The new model includes high order of TPL in the vector of heat flux, and in the thermal displacement and temperature gradients.

The purpose of current flow configuration is to spotlights the thermophysical aspects of magnetohydrodynamics (MHD) viscoinelastic fluid flow over a stretching surface.

The fluid momentum problem is mathematically formulated by using the Prandtl–Eyring constitutive law. Also, the non-Fourier heat flux model is considered to disclose the heat transfer characteristics. The governing problem contains the nonlinear partial differential equations with appropriate boundary conditions. To facilitate the computation process, the governing problem is transmuted into dimensionless form via appropriate group of scaling transforms. The numerical technique shooting method is used to solve dimensionless boundary value problem.

The expressions for dimensionless velocity and temperature are found and investigated under different parametric conditions. The important features of fluid flow near the wall, i.e. wall friction factor and wall heat flux, are deliberated by altering the pertinent parameters. The impacts of governing parameters are highlighted in graphical as well as tabular manner against focused physical quantities (velocity, temperature, wall friction factor and wall heat flux). A comparison is presented to justify the computed results, it can be noticed that present results have quite resemblance with previous literature which led to confidence on the present computations.

The computed results are quite useful for researchers working in theoretical physics. Additionally, computed results are very useful in industry and daily-use processes.

We first provide an overview of some predominant theoretical methods currently used for predicting thermal conductivity of thin dielectric films: the equation of radiative transfer, the temperature‐dependent thermal conductivity theories based on the Callaway model, and the molecular dynamics simulation. This overview also highlights temporal and spatial scale issues by looking at a unified theory that bridges physical issues presented in the Fourier and Cattaneo models. This newly developed unified theory is the so‐called C‐ and F‐processes constitutive model. This model introduces the notion of a new dimensionless heat conduction model number, which is the ratio of the thermal conductivity of the fast heat carrier F‐processes to the total thermal conductivity comprised of both the fast heat carriers F‐processes and the slow heat carriers C‐processes. Illustrative numerical examples for prediction of thermal conductivity in thin films are primarily presented.

This study aims to attempt to construct a new mathematical model of the generalized thermoelasiticity theory based on the memory-dependent derivative (MDD) considering three-phase-lag effects. The governing equations of the problem associated with kernel function and time delay are illustrated in the form of vector matrix differential equations. Implementing Laplace and Fourier transform tools, the problem is sorted out analytically by an eigenvalue approach method. The inversion of Laplace and Fourier transforms are executed, incorporating series expansion procedures. Displacement component, temperature and stress distributions are obtained numerically and illustrated graphically and compared with the existing literature.

This study is to analyze the influence of MDD of three-phase-lag heat conduction interaction in an isotropic semi-infinite medium. The current model has been connected to generalize two-dimensional (2D) thermoelasticity problem. The governing equations are shown in vector matrix form of differential equation concerning Laplace-transformed domain and solved by using the eigenvalue technique. The combined Laplace Fourier transform is applied to find the analytical interpretations of temperature, stresses, displacement for silicon material in a non-dimensional form. Inverse Laplace transform has been found by applying Fourier series expansion techniques introduced by Honig and Hirdes (1984) after performing the inverse Fourier transform.

The main conclusion of this current study is to demonstrate an innovative generalized concept for heat conducting Fourier’s law associated with moderation of time parameter, time delay variable and kernel function by applying the MDDs. However, an important role is played by the time delay parameter to characterize the behavioral patterns of the physical field variables. Further, a new categorization for materials may be created rendering to this new idea along MDD for the time delay variables to develop a new measure of its potential to regulate heat in the medium.

Generalized thermoelasticity is hastily undergoing modification day-by-day from basic thermoelasticity. It has been progressed to get over from the limitations of fundamental thermoelasticity, for instance, infinite velocity components of thermoelasticity interference, in the adequate thermoelastic response of a solid to short laser pulses and deprived illustrations of thermoelastic performance at low temperature. In the past few decades, the fractional calculus is used to change numerous existing models of physical procedure, and its applications are used in various fields of physics, continuum mechanics, fluid mechanics, biology, viscoelasticity, biophysics, signal and image processing, control theory, engineering fields, etc.

This study aims to at numerically retrieve five constant dimensional thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements.

The thermal-wave model of bio-heat transfer is used as an appropriate model because of its realism in situations in which the heat flux is extremely high or low and imposed over a short duration of time. For the numerical discretization, an unconditionally stable finite difference scheme used as a direct solver is developed. The sensitivity coefficients of the dimensionless boundary temperature measurements with respect to five constant dimensionless parameters appearing in a non-dimensionalised version of the governing hyperbolic model are computed. The retrieval of those dimensionless parameters, from both exact and noisy measurements, is successfully achieved by using a minimization procedure based on the MATLAB optimization toolbox routine lsqnonlin. The values of the five-dimensional parameters are recovered by inverting a nonlinear system of algebraic equations connecting those parameters to the dimensionless parameters whose values have already been recovered.

Accurate and stable numerical solutions for the unknown thermo-physical properties of a biological tissue from dimensionless boundary temperature measurements are obtained using the proposed numerical procedure.

The current investigation is limited to the retrieval of constant physical properties, but future work will investigate the reconstruction of the space-dependent blood perfusion coefficient.

As noise inherently present in practical measurements is inverted, the paper is of practical significance and models a real-world situation.

The findings of the present paper are of considerable significance and interest to practitioners in the biomedical engineering and medical physics sectors.

In comparison to Alkhwaji et al. (2012), the novelty and contribution of this work are as follows: considering the more general and realistic thermal-wave model of bio-heat transfer, accounting for a relaxation time; allowing for the tissue to have a finite size; and reconstructing five thermally significant dimensional parameters.

The purpose of this paper is to determine the thermo-gravitational convective state of a non-Fourier fluid layer of the single-phase-lagging type, heated from below. Unlike existing methodologies, the spectral modes are not imposed arbitrarily. They are systematically identified by expanding the spectral coefficients in terms of the relative departure in the post-critical Rayleigh number (perturbation parameter). The number and type of modes is determined to each order in the expansion. Non-Fourier effects become important whenever the relaxation time (delay in the response of the heat flux with respect to the temperature gradient) is of the same order of magnitude as process time.

In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A perturbation approach is developed to solve the nonlinear spectral system in the post-critical range.

The Nusselt number increases with non-Fourier effect as suggested in experiments in microscale and nanofluid convection.

Unlike existing nonlinear formulations for RB thermal convection, the present combined spectral-perturbation approach provides a systematic method for mode selection.

The purpose of this paper is to provide an overview and some recent advances in the models, analysis and simulation of thermal transport of phonons as related to the field of microscale/macroscale heat conduction in solids. The efforts focus upon a fairly comprehensive overview of the subject matter from a unified standpoint highlighting the various approximations inherent in the thermal models. Subsequently, the numerical formulations and illustrations using the current state-of-the-art are provided.

This paper is dedicated to the approximate solution to the relaxation time phonon Boltzmann equation (BE). While original contributions are pointed out and addressed appropriately, the efforts and contributions will be focussed on a relatively complete overview highlighting the field from one unified standpoint and clearly stating all assumptions that go into the approximations inherent to existing models. The contents will be divided as follows: In the first section the authors will give an overview of semi-classical phonon transport physics. Then the authors will discuss the equation of phonon radiative transport (EPRT) and its approximations—the ballistic-diffusive approximation (BDA) and the new heat equation (NHE). Next the authors derive and discuss the C-F model. A numerical discretization method valid for all models is then presented followed by results to numerical simulations and discussion.

From a unified treatment based on the introduction of an energy distribution function, the authors have derived the EPRT and its two well-known approximations: BDA and NHE. For completeness and to provide a vehicle for a general numerical discretization approach, the authors have also included analysis of the C-F model and the parabolic and hyperbolic descriptions of heat transfer along with it. The approximation of angular dependence of phonons in radiation-like descriptions of transport has been given special attention. The assumption of isotropy was found to be of paramount importance in the formulation of position space models for phononic thermal transport. For the thin film problem considered here, the NHE along with the proper boundary condition appears to be the best choice to approximate the phonon BE. Not only does it provide predictions that are in excellent agreement with EPRT, it does not require the discretization of phase space making it far more computationally efficient.

The authors hope this work will help dispel the idea that since Fourier’s law describes diffusion (under limiting assumptions) and it has shown to be ineffective in describing heat transfer for very thin films, that diffusion cannot describe heat transfer in thin films and one should look to a radiative description instead. If one considers diffusion in the sense of random motion, as invisaged by the original builders of the subject (Smoluchowski, Einstein, Ornstein et al.), instead of a temperature gradient, the idea that diffusion can govern thermal transport at this scale is not surprising. Indeed, the NHE is essentially a diffusion equation that describes the motion of particles up to the point of true randomness (isotropy) as well as thereafter.

The purpose of this paper is to investigate the heat transportation rate by using Cattaneo–Christov heat flux model. Furthermore, homogeneous-heterogeneous reaction is also deliberated in the modeling of concentration expression.

The nonlinear PDEs are reduced to ODEs via implementation of applicable transformations. Numerical scheme bvp4c is used to obtain convergent solutions.

The main findings are to characterize the generalized Fourier’s heat flux and homogeneous-heterogeneous reactions in 3D flow of non-Newtonian cross fluid.

It is to certify that this paper is neither published earlier nor submitted elsewhere.

To achieve material-invariant formulation for heat transfer of Carreau nanofluid, the effect of Cattaneo–Christov heat flux is studied on a natural convective flow of Carreau nanofluid past a vertical plate with the periodic variations of surface temperature and the concentration of species. Buongiorno model is considered for nanofluid transport, which includes the relative slip mechanisms, Brownian motion and thermophoresis.

The governing equations are non-dimensionalized using suitable transformations, further reduced to non-similar form using stream function formulation and solved by local non-similarity method with homotopy analysis method. The numerical computations are validated and verified by comparing with earlier published results and are found to be in good agreement.

The effects of varying the physical parameters such as Prandtl number, Schmidt number, Weissenberg number, thermophoresis parameter, Brownian motion parameter and buoyancy ratio parameter on velocity, temperature and species concentration are discussed and presented through graphs. The results explored that the velocity of shear thinning fluid is raised by increasing the Weissenberg number, while contrary response is seen for the shear thickening fluid. It is also found that heat transfer in Cattaneo–Christov heat conduction model is less than that in Fourier’s heat conduction model. Furthermore, the temperature and thermal boundary layer thickness expand with the increase in thermophoresis and Brownian motion parameter, whereas nanoparticle volume fraction increases with increase in thermophoresis parameter, but reverse trend is observed with increase in Brownian motion parameter.

The present investigation is relatively original as very little research has been reported on Carreau nanofluids under the effect of Cattaneo–Christov heat flux model.