Dual Phase Lag Research Articles

Non-Fourier Heat Conduction of Nano-Films under Ultra-Fast Laser.

The ultra-fast laser heating process of nano-films is characterized by an ultra-short duration and ultra-small space size, in which the classical Fourier law based on the hypothesis of local equilibrium is no longer applicable. Based on the Cattaneo-Vernotte (CV) model and the dual-phase-lag (DPL) model, the two-dimensional analytical solutions of heat conduction in nano-films under ultra-fast laser are obtained using the integral transformation method. The results show that there is a thermal wave phenomenon inside the film, which becomes increasingly evident as the elapse of the lag time of the temperature gradient. Moreover, the wave amplitude in the vertical direction is much larger than that in the horizontal direction of the nano-film. By comparing the numerical result of the two models, it is found that the temperature distribution inside the nano-film based on the DPL model is gentler than that of the CV model. Additionally, the temperature distribution in the two-dimensional solution is lower than that in the one-dimensional solution under the same Knudsen number. In the comparison results of the CV model, the maximum peak difference in the thermal wave reaches 75.08 K when the Knudsen number is 1.0. This demonstrates that the horizontal energy carried by the laser source significantly impacts the temperature distribution within the film.

Thermodynamic stability analysis of non‐Fourier model‐based bio‐heat transfer in laser‐irradiated cylindrical‐shaped multilayered biological tissue

AbstractThe present research focuses on examining the thermic response of living tissue in the form of a triple‐layered cylindrical structure when subjected to laser light and the compatibility analysis of non‐Fourier heat transfer with thermodynamics second law. The temperature field in the triple‐layered cylindrical living tissue subjected to laser light is determined by numerically solving the transient radiative transfer equation in conjunction with the dual phase lag (DPL) based bioheat equation. Once the temperature field is known, the equilibrium and nonequilibrium entropy production rate (EPR) is calculated based on the hypothesis of classical irreversible thermodynamics and extended irreversible thermodynamics, respectively. The present results are verified against the data from the literature and found a good match between them. A comparative analysis of the Fourier and non‐Fourier models is accomplished. The equilibrium and nonequilibrium EPR values for the Fourier model are found to be positive. While the equilibrium EPR is negative for non‐Fourier heat conduction and does not satisfy the thermodynamics second law, nonequilibrium EPR is always a positive value for Fourier, DPL, and hyperbolic models and satisfies thermodynamics second law. It has been investigated how thermal relaxation times affect the temperature field and EPRs in tissue are subjected to laser light.

Study of micro‐elongated thermoelastic medium loaded with a piezoelectric layer under the influence of gravity using the dual‐phase‐lag model

AbstractThis paper aims to examine the influence of gravity on a thermoelastic micro‐elongated layer when a piezoelectric layer is above it, utilizing the theory of Lord–Shulman (L–S) and also the model of dual‐phase‐lag (DPL). A partial differential equation was transformed into an ordinary differential equation using the normal mode analysis. Aluminum epoxy numerical computations are carried out, and the results are presented in graphical format. The L–S theory and the model of DPL are compared in the presence and absence of gravity and it is found that gravity has quite a massive influence on all the physical quantities.

Study the effects of temperature and strain rates on transient thermomechanical responses on multilayer skin tissue

The present study is focused on understanding the effects of strain and temperature rates on heat transport mechanisms and the related thermo-mechanical interaction with skin tissue, which are essential for the effective application of thermal therapeutic techniques. The goal of this article is to develop a mathematical framework to investigate the thermo-mechanical phenomena occurring on triple-layered skin tissue using the Modified Green-Lindsay model (MGL) by taking into consideration of two thermal relaxation parameters. This model takes into consideration the effect of strain and temperature rate factors in thermoelastic behaviour which was neglected in several other thermoelastic models. The problem is formulated in a unified way to derive the governing equations and constitutive relations under the present model and also the thermoelasticity with two phase-lag parameters. A comprehensive assessment of solution is discussed for MGL model and compared the results with the corresponding results predicted by dual phase lag generalized thermoelastic model that involves two phase-lags. The effect of strain rate term in heat transport on skin tissue are especially highlighted.

Thermal responses of human tissues during nano-assisted hyperthermia for subcutaneous tumors

Non-invasive thermal therapy for subcutaneous tumors with the assistance of gold nanoparticles and surface cooling measures was proposed in the present study. To achieve real-time monitoring of 3-D temperature distribution and accurate measurement of thermal response, we developed a theoretical approach in which the dual-phase-lag (DPL) biological heat transfer model and Henriques' burn estimation model were employed. A closed-form solution for temperature distribution in the skin tissue can be obtained from this new model by employing the Green's function method. In addition, the influences of the embedment of gold nanoparticles, surface cooling, and non-Fourier effects on the temperature, thermal damage, and size of the burnt region were discussed in detail. The present model performs well in addressing the difficult issue of temperature and burn prediction inside human tissue. It is found that the combination of gold nanoparticles and surface cooling can concentrate thermal empyrosis in the subcutaneous tumor to enhance tumor elimination and protect both skin and deeper healthy tissue. This study provides theoretical support for the improvement and development of laser thermotherapy for subcutaneous tumors.

An Investigation into Thermal Vibrations Caused by a Moving Heat Supply on a Spinning Functionally Graded Isotropic Piezoelectric Bounded Rod

By laminating piezoelectric and flexible materials, we can increase their performance. Therefore, the electrical and mechanical properties of layered piezoelectric materials subjected to electromechanical loads and heat sources must be analyzed theoretically and mechanically. Since the problem of infinite wave propagation cannot be addressed using classical thermoelasticity, extended thermoelasticity models have been derived. The thermo-mechanical response of a piezoelectric functionally graded (FG) rod due to a moveable axial heat source is considered in this paper, utilizing the dual-phase-lag (DPL) heat transfer model. It was supposed that the physical characteristics of the FG rod varied exponentially along the axis of the body. Both ends hold the rod, and there is no voltage across them. The Laplace transform and decoupling techniques were used to obtain the physical fields that have been analyzed. A range of heterogeneity, rotation, and heat source velocity measures were used to compare the results presented here and those in the previous literature.

An axisymmetric bioheat transfer analysis through a unified model

AbstractA unified model is developed for the analysis of heat transfer (radiation and non‐Fourier conduction) in an axisymmetric participating medium. The proposed model includes three different variants of hyperbolic–parabolic heat conduction models, that is, the single phase lag model, dual phase lag model, and the Fourier (no phase lag) model. The radiating‐conducting medium is radiatively absorbing, emitting, and isotropically scattering. Significance of all the above mentioned models on the heat transfer characteristics is investigated in a two‐dimensional axisymmetric geometry. The equation of transfer and the coupled non‐Fourier conduction‐radiation equation are solved via finite volume method. A fully implicit scheme is used to resolve the transient terms in the energy equation. For spatial resolution of radiation information, the STEP scheme is applied. Tri‐diagonal‐matrix‐algorithm is used to solve the resulting set of linear discrete equations. Effects of two important influencing parameters: the scattering albedo and the radiation‐ conduction parameter are studied on the temporal evolution of temperature field in the radiatively participating medium. The non‐Fourier effect of heat transport captured well with the proposed unified model. A good agreement can be found between the proposed model predictions and those available in the literature. It is also found that when the phase lag of the temperature gradient and the heat flux are the same, it reduces to conventional Fourier conduction‐radiation and the wave behavior diminishes. However, the reduction to this Fourier model fails in the presence of constant blood perfusion and metabolic heat generation.

Simulation of wave propagation in thermoporoelastic media with dual-phase-lag heat conduction

The theory of wave propagation in non-isothermal porous rocks has been introduced in geophysics in recent years by combining the single-phase-lag (SPL) Lord-Shulman (LS) model of heat conduction with Biot’s poroelasticity theory. However, the theory of SPL thermoporoelasticity is inadequate in describing the lagging behavior of thermal relaxation for wave dissipation due to fluid and heat flow effects. We address this problem by incorporating a dual-phase-lags (DPL) model of heat conduction into thermoporoelasticity, utilizing analytical solutions and numerical simulations. The DPL model involves two lagging times: the (macroscopic) heat-flux lagging time τq from the LS model and an additionally introduced lagging time τT of temperature gradient that characterizes the fluid phase. A plane-wave analysis predicts four propagation waves, namely, fast P, slow P, thermal (T), and shear (S). We calculate wavefield snapshots by using a finite-difference solver for the DPL thermoelastic equations and provide further insight into the physics of two lagging times for porous rocks. The simulations show that the DPL model induces higher thermal attenuation and larger velocity dispersion compared to the SPL model, especially at high frequencies. The influence of fluids is crucial for wave propagation within thermoporoelastic media.

Modelling of thin metal film heating using the dual-phase lag equation with temperature-dependent parameters

The paper presents a mathematical description and numerical algorithm simulating the thermal processes occurring in the metal microdomain subjected to the ultrashort laser pulse. The model of these processes is based on an equation with two delay times (the dual-phase lag equation – DPLE) supplemented by the appropriate boundary and initial conditions. The energy equation is formulated in the version assuming the variability of thermophysical parameters with temperature (volumetric specific heat and thermal conductivity are temperature-dependent). Taking into account the geometric properties of the laser beam, the problem is treated as an axisymmetric task, while the thermal impact of the laser is taken into account by the introducing into DPLE a component related to the internal heat sources in the domain considered. At the stage of numerical modeling the implicit scheme of the finite difference method is used. In the program simulating the heating/cooling processes occurring in the metal microdomain, the possibility of the melting and resolidification effects are also considered. In the final part of the work, the numerous examples of numerical computations and the resulting conclusions are presented. It turned out, among others, that especially at high temperatures, the consideration of parameters variability (gold, nickel) causes that the results of numerical simulations differ visibly from classical solutions for the constant values of parameters.

Photothermal Response for the Thermoelastic Bending Effect Considering Dissipating Effects by Means of Fractional Dual-Phase-Lag Theory

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey’s equation, an extension of the Generalized Cattaneo Equations (GCEs). Obtaining the temperature distribution by incorporating the effects of fractional differential operators enables us to determine the TE effects in solid samples accurately. This study contributes to understanding the mechanisms that contribute to the PA signal and highlights the importance of considering fractional differential operators in the analysis of thermoelastic bending. As a result, we can determine the PA signal’s TE component. Our findings demonstrate that the fractional differential operators lead to a wide range of behaviors, including dissipative effects related to anomalous diffusion.

Axisymmetric half-space problem in thermoelastic diffusion under phase lags and hyperbolic two temperature

The motive of this work is to figure out the impact of hyperbolic two temperature on the forced vibrations of thermodiffusive axisymmetric half-space in the framework of fractional thermoelastic diffusion with three-phase lags. The governing equations are simplified by introducing potential functions and solved by applying the Laplace and Hankel transform techniques. The boundary surface is restricted by axisymmetric thermal, mechanical, and mass concentration loads. The expressions for field quantities in closed form are obtained analytically. To convert the field quantities into physical space, a mathematical inversion process is employed. The numerical calculation is performed for copper material. The numerically computed results are presented graphically to portray the different physical effects. The results are compared with thermoelastic diffusion with single-phase lag and dual-phase lag.

Refined Dual-Phase-Lag Theory for the 1D Behavior of Skin Tissue under Ramp-Type Heating

In this article, a mathematical analysis of thermoelastic skin tissue is presented based on a refined dual-phase-lag (DPL) thermal conduction theory that considers accounting for the effect of multiple time derivatives. The thin skin tissue is regarded as having mechanically clamped surfaces that are one-dimensional. Additionally, the skin tissue undergoes ramp-type heating on its outer surface, whereas its inner surface keeps the assessed temperature from vanishing. Some of the previous generalized thermoelasticity theories were obtained from the proposed model. The distributions of temperature, displacement, dilatation, and stress are attained by applying the Laplace transform and its numerical reversal approaches. The outcomes are explicitly illustrated to examine the significant influences on the distributions of the field variables. The refined DPL bioheat conduction model in this study predicts temperature, and the findings revealed that the model is located among the existing generalized thermoelastic theories. These findings offer a more thorough understanding of how skin tissue behaves when exposed to a particular boundary condition temperature distribution.

Modeling the viscosity and rotation on a generalized micropolar thermoelasticity with two-temperature under five theories

ABSTRACT The current study establishes the generalized micropolar thermo-visco-elasticity model in an isotropic elastic material with two temperatures. At a uniform angular velocity, the entire elastic medium is rotated. The description is implemented under five generalized thermoelasticity theories: Lord-Shulman (L-S) with one relaxation time, Green-Lindsay (G-L) with two relaxation times, Green-Naghdi theory (G-N II) without energy dissipation, and Tzou theory with dual phase lags (DPL), much like the coupled theory. We used the normal mode analysis to obtain the specific articulations for the considering factors. Additionally, some specific cases are discussed with regard to the problem. Numerical findings are collected and displayed graphically for the variables under consideration. Different theories (L-S, G-L, G-N II, and DPL), rotation, viscosity, and two temperatures were used to predict the results.

The effect of memory and stiffness on energy ratios at the interface of distinct media

PurposeThis paper aims to study the energy ratios of plane waves on an imperfect interface of elastic half-space (EHS) and orthotropic piezothermoelastic half-space (OPHS).Design/methodology/approachThe dual-phase lag (DPL) theory with memory-dependent derivatives is employed to study the variation of energy ratios at the imperfect interface.FindingsA plane longitudinal wave (P) or transversal wave (SV) propagates through EHS and strikes at the interface. As a result, two waves are reflected, and four waves are transmitted, as shown in Figure 2. The amplitude ratios are determined by imperfect boundaries having normal stiffness and transverse stiffness. The variation of energy ratios is computed numerically for a particular model of graphite (EHS)/cadmium selenide (OPHS) and depicted graphically against the angle of incidence to consider the effect of stiffness parameters, memory and kernel functions.Research limitations/implicationsThe energy distribution of incident P or SV waves among various reflected and transmitted waves, as well as the interaction of waves for imperfect interface (IIF), normal stiffness interface (NSIF), transverse stiffness interface (TSIF), and welded contact interface (WCIF), are important factors to consider when studying seismic wave behavior.Practical implicationsThe present model may be used in various disciplines, such as high-energy particle physics, earthquake engineering, nuclear fusion, aeronautics, soil dynamics and other areas where memory-dependent derivative and phase delays are significant.Originality/valueIn a variety of technical and geophysical scenarios, wave propagation in an elastic/piezothermoelastic medium with varying magnetic fields, initial stress, temperature, porosity, etc., gives important information regarding the presence of new and modified waves.

Bio-thermal response and thermal damage in biological tissues with non-equilibrium effect and temperature-dependent properties induced by pulse-laser irradiation

Comprehension of thermal behavior underlying the living biological tissues helps successful applications of current heat therapies. The present work is to explore the heat transport properties of irradiated tissue during tis thermal treatment, in which the local thermal non-equilibrium effect as well as temperature-dependent properties arose from complicated anatomical structure, is considered. Based on the generalized dual-phase lag (GDPL) model, a non-linear governing equation of tissue temperature with variable thermal physical properties is proposed. The effective procedure constructed on an explicit finite difference scheme is then developed to predict numerically the thermal response and thermal damage irradiated by a pulse laser as a therapeutic heat source. The parametric study on variable thermal physical parameters including the phase lag times, heat conductivity, specific heat capacity and blood perfusion rate has been performed to evaluate their influence on temperature distribution in time and space. On this basis, the thermal damage with different laser variables such as laser intensity and exposure time are further analyzed.

Thermodynamic behavior of rectangular nanoplate under moving laser pulse based on nonlocal dual-phase-lag model

Recently, the moving laser pulse is commonly employed in laser-assisted manufacturing for nanostructures synthesis and modification, however, the involved thermodynamic coupling process is still unclear. In the present work, a nonlocal thermo-elastic model is formulated to reveal the thermodynamic response of rectangular nanoplate subjected to a moving laser pulse, in which the heat conduction at the nanoscale is captured via the nonlocal dual-phase-lag (DPL) model. The temperature distribution in the nanoplate is obtained by Green's function method, which is inserted into the governing equation of the nonlocal nanoplate to obtain its dynamic response. The proposed theoretical model is validated by comparing with the previous experimental study and a good agreement is achieved. This study demonstrates that the size-scale effect plays a significant role in the dynamic response of nanoplate under a moving laser pulse, i.e., the nonlocal heat parameter can reduce the deflection, while the nonlocal structural parameter can enlarge the deflection amplitude of the nanoplate. Furthermore, the influences of size-scale parameters, the moving speed and pulse duration of the laser spot on the temperature profile and dynamic response of the rectangular nanoplate are also investigated in detail. The estimation method presented in this work can be used to guide the laser-assisted manufacturing process for nanostructures.

Fractional Dual-Phase-Lag Model for Nonlinear Viscoelastic Soft Tissues

The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) bioheat model and Biot’s model. The more complex shapes of nonlinear viscoelastic soft tissues can be handled by the boundary element method, which also avoids the need for the interior domain to be discretized. The fractional dual-phase-lag bioheat equation was solved using the general boundary element method (GBEM) based on the local radial basis function collocation method (LRBFCM). The poroelastic fields are then calculated using the convolution quadrature boundary element method (CQBEM) The numerical findings show that our proposed numerical model is valid, efficient, and accurate.

Temperature profile and thermal piston component of photoacoustic response calculated by the fractional dual-phase-lag heat conduction theory

We present the temperature distribution predictions for photothermal systems by considering an extension of dual-phase lag. It is an extension of the GCE-II and GCE-III models with a fractional dual-phase lag from kinetic relaxation time. Solving the one-dimensional problem considering a planar and periodic excitation, we obtained the temperature distribution and the Photoacoustic (PA) signal for the transmission setup. We also analyze the effects of fractional order derivatives and kinetic relaxation time. It is shown that the derived models have promising results that could be used to explain the experimentally observed behavior of PA signals measured on thin films with an inhomogeneous internal structure.

Analytical and numerical analysis of the dual-pulse lag heat transfer in a three-dimensional tissue subjected to a moving multi-point laser beam

An extensive algorithm based on both analytical and numerical solution methodologies is proposed to obtain transient temperature distributions in a three-dimensional living tissue subjected to a moving single-point and multi-point laser beam by considering metabolic heat generation and blood perfusion rate. Here, the dual-phase lag/Pennes equation is analytically solved by using the method of Fourier series and the Laplace transform. The ability to model single-point or multi-point laser beams as an arbitrary function of place and time is a significant advantage of the proposed analytical approach, which can be used to solve similar heat transfer problems in other living tissues. Besides, the related heat conduction problem is numerically solved based on the finite element method. The effects of laser beam transitional speed, laser power, and the number of laser points on the temperature distribution within the skin tissue are investigated. Moreover, the temperature distribution predicted by the dual-phase lag model is compared with that of the Pennes model under different working conditions. For the studied cases, it is observed that the maximum tissue temperature decreased about 63% by an increase of 6mm/s in the speed of the laser beam. An increase in the laser power from 0.8W/cm3 to 1.2W/cm3 results in a 28 °C increase in the maximum temperature of the skin tissue. It is observed that the maximum temperature predicted by the dual-phase lag model is always lower than that of the Pennes model and the temperature variations over time are sharper, while their results are entirely consistent over the simulation time. The obtained numerical results indicated that the dual-phase lag model is preferred in heating processes occurring at short intervals. Among the investigated parameters, the laser beam speed has the most considerable effect on the difference between the results of the Pennes and the dual-phase lag models.

A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.