Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating
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Analytical and numerical analysis of time fractional dual-phase-lag heat conduction during short-pulse laser heating Xiaoping Wang1 · Huanying Xu1 · Haitao Qi1 Received: 16 May 2019 / Accepted: 23 December 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this study, we analytically and numerically investigate the non-Fourier heat conduction behavior within a finite medium based on the time fractional dual-phase-lag model. Firstly, the time fractional dual-phase-lag model and the corresponding fractional heat conduction equation for short-pulse laser heating is built. Laplace and Fourier cosine transforms are performed to derive the semi-analytical expression of temperature distribution in the Laplace domain. Then, by the L1 approximation for the Caputo derivative, the finite difference algorithm is developed for the short-pulse laser heating problem. The solvability, stability, and convergence of this algorithm are also examined. Meanwhile, the efficiency and accuracy of this method have been verified by using three numerical examples. Finally, based on numerical analysis, we study the non-Fourier heat conduction behavior and discuss the effect of variability of parameters, such as fractional parameter and the ratio between the relaxation and retardation times, on the temperature distribution graphically. We believe that this analysis, besides benefiting the laser heating applications, will also provide a deep theoretical insight to interpret the anomalous heat transport process. Keywords Fractional calculus · Dual-phase-lag model · Short-pulse laser heating · Finite difference method
1 Introduction With the development of laser technology, laser heating has become a very important aspect of mechanical processing, biomedicine, modern materials science, and
Haitao Qi
[email protected] 1
School of Mathematics and Statistics, Shandong University, Weihai, 264209, People’s Republic of China
Numerical Algorithms
so on [1, 2]. In dealing with pulse laser heating problems, several heat conduction models have been usually employed in the literature, such as Fourier’s law for heat conduction [1, 3], the Cattaneo-Vernotte thermal wave model [2, 3], and the dualphase-lag (DPL) model proposed by Tzou [4]. Fourier’s law is a classical empirical law, which implies the unphysical infinite propagation speed of thermal disturbance. And it fails to describe the temperature distribution in the cases of ultrafast heating or low-temperature conditions [5]. Hence, the Cattaneo-Vernotte thermal wave model has been proposed as a non-Fourier model, which leads to a hyperbolic-type partial differential equation and predicts a finite propagation speed of heat [6]. Furthermore, these two models are both capable of modeling some macroscopic behaviors of heat conduction, but which cannot be used to describe heat transport in different microsystems. Thus, the DPL model is proposed to fill the gap from micro- to macroscopic theories by introducing the relaxation time and the retardation time [4]. And,
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