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In this work, a new model for porothermoelastic waves under a fractional time derivative and two time delays is utilized to study temperature increments, stress and the displacement components of the solid and fluid phases in porothermoelastic media. The governing equations are presented under Lord–Shulman theory with thermal relaxation time. The finite element method has been adopted to solve these equations due to the complex formulations of this problem. The effects of fractional parameter and porosity in porothermoelastic media have been studied. The numerical outcomes for the temperatures, the stresses and the displacement of the fluid and the solid are presented graphically. These results will allow future studies to gain a detailed insight into non-simple porothermoelasticity with various phases.

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.

The Pennes bioheat equation is the most widely used model for describing heat transfer in living tissue during thermal exposure. It is derived from the classical Fourier law of heat conduction and assumes energy exchange between blood vessels and surrounding tissues. The literature presents various numerical methods for solving the bioheat equation, with exact solutions developed for different boundary conditions and geometries. However, analytical models based on this framework are rarely reported. This study aims to develop an analytical three-dimensional model using MATHEMATICA software, with subsequent mathematical validation performed through COMSOL simulations, to characterize heat transfer in biological tissues induced by laser irradiation under various therapeutic conditions. The objective is to refine the conventional bioheat equation by introducing three key improvements: (a) incorporating a non-Fourier framework for the Pennes equation, thereby accounting for the relaxation time in thermal response; (b) integrating Dirac functions and the telegraph equation into the bioheat model to simulate localized point heating of diseased tissue; and (c) deriving a closed-form analytical solution for the Pennes equation in both its classical (Fourier-based) and improved (non-Fourier-based) formulations. This paper investigates the nuanced relationship between the relaxation time parameter in the telegraph equation and the thermal relaxation time employed in the bioheat transfer equation. Considering all these aspects, the optimal thermal relaxation time determined for these simulations was 1.16 s, while the investigated thermal exposure time ranged from 0.01 s to 120 s. This study introduces a generalized version of the model, providing a more realistic representation of heat exchange between biological tissue and blood flow by accounting for non-uniform temperature distribution. It is important to note that a reasonable agreement was observed between the two modeling approaches: analytical (MATHEMATICA) and numerical (COMSOL) simulations. As a result, this research paves the way for advancements in laser-based medical treatments and thermal therapies, ultimately contributing to more optimized therapeutic outcomes.

This report examines the heat and mass transfer in three-dimensional second grade non-Newtonian fluid in the presence of a variable magnetic field. Heat transfer is presented with the involvement of thermal relaxation time and variable thermal conductivity. The generalized theory for mass flux with variable mass diffusion coefficient is considered in the transport of species. The conservation laws are modeled in simplified form via boundary layer theory which results as a system of coupled non-linear partial differential equations. Group similarity analysis is engaged for the conversion of derived conservation laws in the form of highly non-linear ordinary differential equations. The solution is obtained vial optimal homotopy procedure (OHP). The convergence of the scheme is shown through error analysis. The obtained solution is displayed through graphs and tables for different influential parameters.

A novel nonlocal model with one thermal relaxation time is presented to investigate the propagation of waves in a thermoelastic semi-infinite medium. We used Eringen’s theory of the nonlocal continuum to develop these models. Analytical solutions in all physical quantities are provided by using Laplace transforms and eigenvalue techniques. All physical quantities are presented as symmetric and asymmetric tensors. The temperature, the displacement, and the stress variations in semi-infinite materials have been calculated. The effects of nonlocal parameters, ramp type heating, and the thermal relaxation times on the wave propagation distribution of physical fields for mediums are graphically displayed and analyzed.

This research addresses the phenomena of thermoelastic damping (TED) and frequency shift (FS) of a thin flexible piezoelectro-magneto-thermoelastic (PEMT) composite beam. Its motion is constrained by two linear flexible springs attached to both ends. The novelty behind the proposed study is to mimic the uncertainties during the fabrication of the beam. Therefore, the equation of motion was derived utilizing the linear Euler–Bernoulli theory accounting for the flexible boundary conditions. The beam’s eigenvalues, mode shapes, and the effects of the thermal relaxation time (t1), the dimensions of the beam, the linear spring coefficients (KL0 and KLL), and the critical thickness (CT) on both TED and FS of the PEMT beam were investigated numerically employing the Newton–Raphson method. The results show that the peak value of thermoelastic damping (Qpeak−1) and the frequency shift (Ω) of the beam increase as t1 escalates. Another observation was made for the primary fundamental mode, where an increase in the spring coefficient KLL leads to a further increase in Ω. On the other hand, the opposite trend is noted for the higher modes. Indeed, the results show the possibility of using the proposed design in a variety of applications that involve damping dissipation.

In this paper, the thermoelastic behavior of a rod made of an isotropic material under the action of a moving heat source was investigated using a new theory of thermoelasticity related to fractional-order time with two relaxation times. A mathematical model of the one-dimensional thermoelasticity problem was established based on the new thermoelasticity theory. We considered the symmetry of the material, and the fractional-order thermoelasticity control equation was given. Subsequently, the control equations were solved and analyzed using the Laplace transform and its inverse transform. This study examined the effects of fractional-order parameters, time, two thermal relaxation times, and the speed of movement of the heat source on the displacement, temperature, and stress distribution patterns in the rod.

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a q uasi-longitudinal P rimary ( qP ) wave and a q uasi-transverse S econdary ( qS ) wave; and (2) a mainly thermal ( qT ) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.

Investigation of nanofluid flow, mass and heat transfer, and impact of thermal relaxation time parameter against temperature in porous spinning rotating stretching disks was done. The behavior of heat and mass transfer, velocity, temperature and particles concentration volume fraction profiles against other parameters are investigated. A decrease in heat transfer in the lower disk and upper disk was observed as Eckert number, porosity parameter and Brownian diffusion parameter increased in value though with an increase in temperature ratio and thermal relaxation time parameter. The mass transfer rate at both lower and upper disk increased as Lewis number, Eckert number alongside porosity parameter increase in values, but decreases in value as Brownian diffusion parameter and temperature ratio increase in values. Increasing thermal relaxation time parameter caused decrease in temperature. The physical meaning is that because longer time is needed for transporting heat to nearby particles, the upper disk stretching parameter increases whenever radial velocity profile towards the lower disk is increased, which was exhibited by a negative sign towards the lower disk and positive sign in the neighbourhood of the upper disk. This implies that the fluid inside the two disks continues to flow in opposite directions of upward and downward directions. The partial differential equations which are non-linear are, transformed to non-linear coupled ordinary differential equations by app-lying Van Karman transformations which are then solved using MATLAP bvp4c with shooting technique.

The impact of Cattaneo–Christov heat flux on cylindrical surfaces using Carbon Nanotube (CNT) ternary Hybrid Nanofluids with convective boundary conditions is investigated in this work, especially within the context of solar-powered ships. A numerical simulation is performed to assess the thermal characteristics and effectiveness of CNT ternary Hybrid Nanofluids over traditional fluids. The thermal relaxation effects and the heat flux are attained using the Cattaneo–Christov heat flux model which leads to a better prediction of heat transfer processes in the nanofluids. By assessing the existing behaviors and energy transferal characteristics of CNT ternary hybrid nanofluids, the findings perfectly show that the nanoparticle impacts improving the thermal conductivity and heat transfer efficiency. This is useful for optimization of cooling systems of ships driven by solar energy. This work is beneficial to the efforts put towards designing and optimal thermal management strategies for solar-powered ships using nanofluid and novel heat transfer.