scholarly journals First and second order dual phase lag equation. Numerical solution using the explicit and implicit schemes of the finite difference method

2018 ◽  
Vol 240 ◽  
pp. 05018 ◽  
Author(s):  
Ewa Majchrzak ◽  
Bohdan Mochnacki
Keyword(s):  
Thin Metal Film ◽  
Phase Lag ◽  
Dual Phase ◽  
Mathematical Form ◽  
Fourier Law ◽  
Delay Times ◽  
Left And Right ◽  
Explicit And Implicit Schemes ◽  
Theoretical Considerations ◽  
Generalized Fourier Law

In the paper the different variants of the dual phase lag equation (DPLE) are considered. As one knows, the mathematical form of DPLE results from the generalization of the Fourier law in which two delay times are introduced, namely the relaxation time τq and the thermalization one τT. Depending on the order of development of the left and right hand sides of the generalized Fourier law into the Taylor series one can obtain the different forms of the DPLE. It is also possible to consider the others forms of equation discussed resulting from the introduction of the new variable or variables (substitution). In the paper a thin metal film subjected to a laser pulse is considered (the 1D problem). Theoretical considerations are illustrated by the examples of numerical computations. The discussion of the results obtained is also presented.

scholarly journals Second-Order Dual Phase Lag Equation. Modeling of Melting and Resolidification of Thin Metal Film Subjected to a Laser Pulse

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 999 ◽  
Author(s):  
Ewa Majchrzak ◽  
Bohdan Mochnacki
Keyword(s):  
Laser Pulse ◽  
Latent Heat ◽  
Metal Film ◽  
Thin Metal Film ◽  
Second Order ◽  
Phase Changes ◽  
Thin Metal ◽  
Phase Lag ◽  
Dual Phase ◽  
Numerical Computations

The process of partial melting and resolidification of a thin metal film subjected to a high-power laser beam is considered. The mathematical model of the process is based on the second-order dual phase lag equation (DPLE). Until now, this equation has not been used for the modeling of phase changes associated with heating and cooling of thin metal films and the considerations regarding this issue are the most important part of the article. In the basic energy equation, the internal heat sources associated with the laser action and the evolution of phase change latent heat are taken into account. Thermal processes in the domain of pure metal (chromium) are analyzed and it is assumed that the evolution of latent heat occurs at a certain interval of temperature to which the solidification point was conventionally extended. This approach allows one to introduce the continuous function corresponding to the volumetric fraction of solid or liquid state at the neighborhood of the point considered, which significantly simplifies the phase changes modeling. At the stage of numerical computations, the authorial program based on the implicit scheme of the finite difference method (FDM) was used. In the final part of the paper, the examples of numerical computations (including the results of simulations for different laser intensities and different characteristic times of laser pulse) are presented and the conclusions are formulated.

scholarly journals Dual Phase Lag Model of Melting Process in Domain of Metal Film Subjected to an External Heat Flux

2016 ◽  
Vol 16 (4) ◽  
pp. 85-90 ◽  
Author(s):  
B. Mochnacki ◽  
M. Ciesielski
Keyword(s):  
Mushy Zone ◽  
Metal Film ◽  
Control Volume ◽  
Extreme Temperature ◽  
Internal Heat ◽  
Melting Process ◽  
Thin Metal Film ◽  
Heating Process ◽  
Phase Lag ◽  
Dual Phase

Abstract Heating process in the domain of thin metal film subjected to a strong laser pulse are discussed. The mathematical model of the process considered is based on the dual-phase-lag equation (DPLE) which results from the generalized form of the Fourier law. This approach is, first of all, used in the case of micro-scale heat transfer problems (the extremely short duration, extreme temperature gradients and very small geometrical dimensions of the domain considered). The external heating (a laser action) is substituted by the introduction of internal heat source to the DPLE. To model the melting process in domain of pure metal (chromium) the approach basing on the artificial mushy zone introduction is used and the main goal of investigation is the verification of influence of the artificial mushy zone ‘width’ on the results of melting modeling. At the stage of numerical modeling the author’s version of the Control Volume Method is used. In the final part of the paper the examples of computations and conclusions are presented.

NONEQUILIBRIUM DUAL-PHASE-LAG HEAT TRANSPORT THROUGH BIOLOGICAL TISSUES

2015 ◽  
Vol 18 (1) ◽  
pp. 57-69 ◽  
Author(s):  
Hossein Askarizadeh ◽  
Hossein Ahmadikia
Keyword(s):  
Heat Transport ◽  
Biological Tissues ◽  
Phase Lag ◽  
Dual Phase

scholarly journals The exact analytical solution of the dual-phase-lag two-temperature bioheat transfer of a skin tissue subjected to constant heat flux

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Hamdy M. Youssef ◽  
Najat A. Alghamdi
Keyword(s):  
Heat Flux ◽  
Analytical Solution ◽  
Exact Analytical Solution ◽  
Constant Heat Flux ◽  
Bioheat Transfer ◽  
Skin Tissue ◽  
Phase Lag ◽  
Dual Phase ◽  
Constant Heat ◽  
Two Temperature

Abstract This work is dealing with the temperature reaction and response of skin tissue due to constant surface heat flux. The exact analytical solution has been obtained for the two-temperature dual-phase-lag (TTDPL) of bioheat transfer. We assumed that the skin tissue is subjected to a constant heat flux on the bounding plane of the skin surface. The separation of variables for the governing equations as a finite domain is employed. The transition temperature responses have been obtained and discussed. The results represent that the dual-phase-lag time parameter, heat flux value, and two-temperature parameter have significant effects on the dynamical and conductive temperature increment of the skin tissue. The Two-temperature dual-phase-lag (TTDPL) bioheat transfer model is a successful model to describe the behavior of the thermal wave through the skin tissue.

The initial stress effect on a thermoelastic micro-elongated solid under the dual-phase-lag model

2021 ◽  
Vol 127 (9) ◽  
Author(s):  
Mohamed I. A. Othman ◽  
Sarhan Y. Atwa ◽  
Ebtesam E. M. Eraki ◽  
Mohamed F. Ismail
Keyword(s):  
Initial Stress ◽  
Stress Effect ◽  
Phase Lag ◽  
Dual Phase ◽  
Lag Model
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