On the stability of the exact solutions of the dual-phase lagging model of heat conduction

The dual-phase lagging (DPL) model has been considered as one of the most promising theoretical approaches to generalize the classical Fourier law for heat conduction involving short time and space scales. Its applicability, potential, equivalences, and possible drawbacks have been discussed in the current literature. In this study, the implications of solving the exact DPL model of heat conduction in a three-dimensional bounded domain solution are explored. Based on the principle of causality, it is shown that the temperature gradient must be always the cause and the heat flux must be the effect in the process of heat transfer under the dual-phase model. This fact establishes explicitly that the single- and DPL models with different physical origins are mathematically equivalent. In addition, taking into account the properties of the Lambert W function and by requiring that the temperature remains stable, in such a way that it does not go to infinity when the time increases, it is shown that the DPL model in its exact form cannot provide a general description of the heat conduction phenomena

Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study

[EN] Parabolic heat equation based on Fourier's theory (FHE), and hyperbolic heat equation (HHE), has been used to mathematically model the temperature distributions of biological tissue during thermal ablation. However, both equations have certain theoretical limitations. The FHE assumes an infinite thermal energy propagation speed, whereas the HHE might possibly be in breach of the second law of thermodynamics. The relativistic heat equation (RHE) is a hyperbolic-like equation, whose theoretical model is based on the theory of relativity and which was designed to overcome these theoretical impediments. In this study, the three heat equations for modelling of thermal ablation of biological tissues (FHE, HHE and RHE) were solved analytically and the temperature distributions compared. We found that RHE temperature values were always lower than those of the FHE, while the HHE values were higher than the FHE, except for the early stages of heating and at points away from the electrode. Although both HHE and RHE are mathematically hyperbolic, peaks were only found in the HHE temperature profiles. The three solutions converged for infinite time or infinite distance from the electrode. The percentage differences between the FHE and the other equations were larger for higher values of thermal relaxation time in HHE.This work received financial support from the Spanish Government (Ministerio de Ciencia e Innovacion, Ref. TEC2011-27133-C02-01).López Molina, JA.; Rivera Ortun, MJ.; Berjano, E. (2014). Fourier, hyperbolic and relativistic heat transfer equations: a comparative analytical study. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 470:1-16. https://doi.org/10.1098/rspa.2014.0547S11647

On the stability in phase-lag heat conduction with two temperatures

We investigate the well-posedness and the stability of the solutions for several Taylor approximations of the phase-lag two-temperature equations.We give conditions on the parameters which guarantee the existence and uniqueness of solutions as well as the stability and the instability of the solutions for each approximationPeer ReviewedPostprint (author's final draft

Heat transfer flow of Maxwell hybrid nanofluids due to pressure gradient into rectangular region

In this work, infuence of hybrid nanofuids (Cu and Al2O3) on MHD Maxwell fuid due to pressure gradient are discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions are converted into dimensionless form. Fractional model for Maxwell fuid is established by Caputo time fractional diferential operator. The dimensionless expression for concentration, temperature and velocity are found using Laplace transform. As a result, it is found that fuid properties show dual behavior for small and large time and by increasing volumetric fraction temperature increases and velocity decreases respectively. Further, we compared the Maxwell, Casson and Newtonian fuids and found that Newtonian fuid has greater velocity due to less viscosity. Draw the graphs of temperature and velocity by Mathcad software and discuss the behavior of fow parameters and the efect of fractional parameters

Toward nanofluids of ultra-high thermal conductivity

The assessment of proposed origins for thermal conductivity enhancement in nanofluids signifies the importance of particle morphology and coupled transport in determining nanofluid heat conduction and thermal conductivity. The success of developing nanofluids of superior conductivity depends thus very much on our understanding and manipulation of the morphology and the coupled transport. Nanofluids with conductivity of upper Hashin-Shtrikman (H-S) bound can be obtained by manipulating particles into an interconnected configuration that disperses the base fluid and thus significantly enhancing the particle-fluid interfacial energy transport. Nanofluids with conductivity higher than the upper H-S bound could also be developed by manipulating the coupled transport among various transport processes, and thus the nature of heat conduction in nanofluids. While the direct contributions of ordered liquid layer and particle Brownian motion to the nanofluid conductivity are negligible, their indirect effects can be significant via their influence on the particle morphology and/or the coupled transport