Dual-phase-lag in the balance: Sufficiency bounds for the class of Jeffreys' equations to furnish physical solutions. (September 2020)

• Record Type: Journal Article
• Authors: Awad, Emad
• Is Part Of: International journal of heat and mass transfer. Volume 158(2020)
• Publication Date: 2020-09
• Subjects: Bernstein functions -- Cattaneo equation -- Dual-phase-lag -- Green's functions -- Guyer-Krumhansl-type heat conduction -- Jeffreys-type heat equation -- Parabolic two-step model -- Riemann-Liouville fractional derivative; Heat -- Transmission -- Periodicals; Mass transfer -- Periodicals; Chaleur -- Transmission -- Périodiques; Transfert de masse -- Périodiques; Electronic journals; 621.4022
• DOI: 10.1016/j.ijheatmasstransfer.2020.119742
• Languages: English
• ISSNs: 0017-9310

Abstract

Highlights: The non-negativity of solutions of the parabolic dual-phase-lag, parabolic two-step and Guyer-Krumhansl models is studied in the higher dimensions. Bernstein functions approach and numerical techniques are used. A fractional DPL model has been proposed to handle the negativity of the DPL solutions in the higher dimensions. Abstract: Recent studies (see Rukolaine 2014, 2017) have deduced solutions of the parabolic and hyperbolic dual-phase-lag (DPL) models in the three-dimensional space which record negative (unphysical) values at specific choice of the characteristic parameter Z, consequently, rejected them as appropriate models of heat and mass transfer. The present work re-sheds light on the non-negativity of solutions of the parabolic DPL when different values of the parameter Z are considered. It is found that when Z > 1, the DPL model provides nonnegative solutions, while the case Z < 1 may provide negative solutions. These results have stimulated examining the non-negativity of the microscopic pictures of the DPL; the parabolic two-step (PTS) and the Guyer-Krumhansl (GK) models. This examination enhances what anticipated from the macroscopic DPL model. A fractional Jeffreys-type constitutive law is phenomenologically proposed to address this shortcoming in the DPL model when Z < 1. However, it may be useful in data fitting when Z > 1. The Bernstein functions technique is employed for the theoretical proofs of the non-negativity. Numerical schemes are adopted to verify the theoretical predictions. Two temporal integral transformations which map the DPL temperature to the electron and lattice temperatures are derived. Otherwise, it is shown that the internal thermal energies of both the parabolic DPL and PTS models are exactly equivalent.