A comparative numerical study of a semi-infinite heat conductor subject to double strip heating under non-Fourier models

Abstract

The present work considers a two-dimensional (2D) heat conduction problem in the semi-infinite domain based on the classical Fourier model and other non-Fourier models, e.g., the Maxwell–Cattaneo–Vernotte (MCV) equation, parabolic, hyperbolic, and modified hyperbolic dual-phase-lag (DPL) equations. Using the integral transform technique, Laplace, and Fourier transforms, we provide a solution of the problem (Green’s function) in Laplace domain. The thermal double-strip problem, allowing the wave interference within the heat conductor, is considered. A numerical technique, based on the Durbin series for inverting Laplace transform and the trapezoidal rule for calculating an integral form of the solution in the double-strip case, is adopted to recover the solution in the physical domain. Finally, discussions for different non-Fourier heat transfer situations are presented. We compare among the speeds of hyperbolic heat transfer models and shed light on the concepts of flux-precedence and temperature-gradient-precedence, hallmarks of the lagging response idea. Otherwise, we emphasize the existence of a relationship between the waves speed and the time instant of interference onset, underlying the five employed heat transfer models.

On the plane strain problem of dynamical flexoelectric effect in dielectrics subject to Sinc-shaped surface loading

“…The method of Laplace inversion has been successfully employed to verify the non-negativity of Jeffreys equation in the higher dimensions [55], and in approximating the behavior of stresses in Cauchy problem [81]. The numerical inversion of Fourier transform was used to describe the thermomechanical behaviors in micropolar thermoelasticity [59] and has been employed in describing the finite speed propagation in Awad [82]. Numerical experiments have been carried out at a fixed time instant, t = 0 . 1 .…”

Anomalous Thermally Induced Deformation in Kelvin–Voigt Plate with Ultrafast Double-Strip Surface Heating

“…Before the dimensionless time instant t = 0.09, the heat flux is directed from the upper surface to the lower surface which precedes and stimulates the temperature gradient. In the case of a semi-infinite conductor [50], the temperature peak diffuses with time progress, leaving the boundary surface. Here, in the finite domain setting, because of the presence of the reflecting boundary condition on the lower boundary (36) and the acute heat flux precedence, τ θ τ q , the heat flux vector reverses its direction between the dimensionless instants t = 0.08 and t = 0.09, so that the upper surface always has the highest temperature though the overall dissipation of heat (i.e., the decrease in temperature value with time).…”

“…Furthermore, we assume that there are no body forces or heat sources. The upper surface of the plate is assumed to be traction-free and subjected to ultrafast double-strip heating [50], which decays exponentially with time, while the lower surface is assumed to be traction-free and thermally insulated; see Figure 1.…”

“…Furthermore, we assume that there are no body forces or heat sources. The upper surface of the plate is assumed to be traction-free and subjected to ultrafast double-strip heating [50], From the above settings, all the considered functions should depend on x, y, and t. Also, the displacement vector is given in the form u = u, v, 0 , u = u(x, y, t), v = v(x, y, t). (10) Hence, the governing equations for Kelvin-Voigt thermoelastic plate on the domain (x, y, t) ∈ (−∞, ∞) × [0, l] × (0, ∞) consist of the following:…”

“…In the present article, we consider a Kelvin-Voigt nanoscale two-dimensional plate with infinite extension subject to an ultrafast surface heating on its upper surface taking the double-strip shape [50,51]. The delayed strain response underlying the Kelvin-Voigt assumption, the nanoscale thickness of the plate, and the ultrafast heating method can be considered as reasoning causes for adopting non-Fourier heat transfer mechanisms such as accelerating, retarding, and crossover from super-to sub-thermal conduction.…”