The paradox of fourier heat equation: A theoretical refutation

Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

An examination is made of some aspects of the modeling of heat conduction or diffusion process described by Fourier equations on analog computers. A scheme is given for reproducing a sum of exponential functions with the aid of a single amplifier and a group of switching circuits.

Aim. The investigation into the applicability of classical macroscopic approximations to obtain the nonequilibrium local distribution function inside the structure of a strong shock wave. Methodology. This paper examines the capability of various macroscopic models (the Navier – Stokes – Fourier equations, the Burnett equations, and the original and regularized 13-moment Grad equations) to approximate a nonequilibrium molecular velocity distribution function. Results. The locally reconstructed distribution functions obtained from the flow macro-parameters for the considered models are compared with each other and with a benchmark solution at different locations within the structure of a planar shock wave. The benchmark solution is provided by the Direct Simulation Monte Carlo (DSMC) method, which supplies the flow macro-parameters required for the reconstruction of the distribution function. Research implications. All considered models predict the distribution function rather poorly in the supersonic part of the shock-wave structure, where strong oscillations and nonphysical negative values are observed.

It is well known that the general radiation hydrodynamics models include two mainly coupled parts: one is the macroscopic fluid part, which is governed by the compressible Navier--Stokes--Fourier equations; another is the radiation field part, which is described by the transport equation of photons. Under the two physical approximations, “gray” approximation and P1 approximation, one can derive the so-called Navier--Stokes--Fourier--P1 approximation radiation hydrodynamics model from the general one. In this paper, we study the nonrelativistic limit problem for the Navier--Stokes--Fourier--P1 approximation model due to the fact that the speed of light is much larger than the speed of the macroscopic fluid. Our results give a rigorous derivation of the widely used macroscopic model in radiation hydrodynamics.

A kinetic model and corresponding high-order macroscopic model for the accurate description of rarefied polyatomic gas flows are introduced. The different energy exchange processes are accounted for with a two term collision model. The proposed kinetic model, which is an extension of the S-model, predicts correct relaxation of higher moments and delivers the accurate Prandtl ($Pr$) number. Also, the model has a proven linear H-theorem. The order of magnitude method is applied to the primary moment equations to acquire the optimized moment definitions and the final scaled set of Grad’s 36 moment equations for polyatomic gases. At the first order, a modification of the Navier–Stokes–Fourier (NSF) equations is obtained. At third order of accuracy, a set of 19 regularized partial differential equations (R19) is obtained. Furthermore, the terms associated with the internal degrees of freedom yield various intermediate orders of accuracy, a total of 13 different orders. Thereafter, boundary conditions for the proposed macroscopic model are introduced. The unsteady heat conduction of a gas at rest is studied numerically and analytically as an example of a boundary value problem. The results for different gases are given and effects of Knudsen numbers, degrees of freedom, accommodation coefficients and temperature-dependent properties are investigated. For some cases, the higher-order effects are very dominant and the widely used first-order set of the NSF equations fails to accurately capture the gas behaviour and should be replaced by the proposed higher-order set of equations.

A certain number of techniques have been designed in the kinetic theory of gases to derive macroscopic time evolution equations from the Boltzmann equation. Most of these methods require the single-particle distribution function to be parameterized by a set of distinguished fields, such as the hydrodynamic ones: the number (or mass) density, momentum vector, and temperature. This is a plausible assumption as long as the microscopic dynamics enjoys a vast separation of time scales and local thermodynamic equilibrium exists. Moreover, the derivation of hydrodynamics from kinetic theory is often concerned with the hydrodynamic limit of the Boltzmann equation. Loosely speaking, one is interested, typically, in the scaling of the Boltzmann equation with respect to some reference macroscopic length and time scales, which are expected to largely dominate the intrinsic kinetic scales. Nonetheless, it makes sense to consider the extension of the hydrodynamic description beyond the standard domain, considering reference scales comparable with the kinetic ones. This is the subject dealt with by generalized hydrodynamics. There are several delicate aspects hindering this line of investigation. A first, natural, objection points to the fact that below a certain length scale, the notion itself of “local equilibrium,” which is brought about by a sufficiently large number of particle collisions, is questionable. Moreover, from the technical side, one typically deals, in this context, with perturbative methods, such as Hilbert’s procedure or the Chapman–Enskog (CE) technique, which, at a certain order of truncation, may give rise to artificial instabilities. In particular, the CE method introduces an expansion of the distribution function in terms of a parameter, the Knudsen number, defined as the ratio of the mean free path to a representative macroscopic length. For small values of the Knudsen number, the CE method recovers the standard Navier–Stokes–Fourier (NSF) equations of hydrodynamics. In more refined approximations, referred to as the Burnett and super-Burnett hydrodynamics, the hydrodynamic modes become polynomials of higher order in the wave vector. In such an extension, the resulting hydrodynamic equations may become unstable and violate the H-theorem, as first shown by Bobylev for a particular case of Maxwell molecules. This indicates that the CE theory cannot be immediately trusted away from the hydrodynamic limit. Thus, while the mathematical framework concerning the hydrodynamic limit of the Boltzmann equation is well established, there is no consolidated counterpart addressing the short-wavelength domain.

A century ago, Edgar Buckingham presented data and a theoretical conceptualization of soil moisture movement. His work constitutes a milestone in the history of soil physics and more generally, of movement of multiple fluid phases in porous media. Starting from first principles, Buckingham formulated a conceptual model to make rational sense of long‐term observations of evaporation from soil columns. Central to his model were the notion of a capillary potential, soil moisture retention curve, and potential‐dependent hydraulic conductivity. Buckingham recognized that whereas heat capacity and thermal conductivity were independent of temperature in Fourier's heat equation, in the case of soil moisture, the slope of the soil‐moisture retention curve (analogous to specific heat) and capillary conductivity were both strong functions of capillary potential. Noting that available solutions of Fourier's linear differential equation did not apply to moisture movement in soils, Buckingham was skeptical that the nonlinear problem could be solved mathematically. This is perhaps why he did not present a partial differential equation for soil‐moisture movement. Such an equation would be gvien in 1931 by Richards. Despite considerable efforts, analytical solutions to Richards' equation can be obtained only under simplifying assumptions. While these solutions give valuable insights into patterns of soil moisture movement, they cannot adequately address problems of the natural soil environment. Although Buckingham's model remains the only workable physical‐mathematical conceptualization for studying moisture movement in soils, his own skepticism of its ability to reliably describe moisture movement in soils is still valid. More profound, his skepticism captures the limitations inherent in precisely describing the behavior of earth systems. This paper examines Buckingham's central ideas in light of developments in groundwater hydrology and soil mechanics and reflects on the limits of our ability to quantitatively understand moisture movement in unsaturated soils.

In the present study, a rarefied gas flow behavior is investigated numerically using two different approaches. A rarefied gas flow is considered within a double-sided lid-driven cavity which the upper and lower walls are moving in the same direction. As a macroscopic description, the set equations of Navier-Stockes and Fourier (NSF) are resolved numerically. The Monte Carlo simulation (DSMC) method is used as a kinetic based approach to capture the non-equilibrium effects. The flow properties are determined using both approaches as a function of the Knudsen number which estimates the gas flow rarefaction degree.

Non-Fourier heat conduction with radiation in an absorbing, emitting, and isotropically scattering medium

The non-stationary heat conduction in an infinite solid medium internally bounded by an infinitely long cylindrical surface is considered. A uniform and time- dependent temperature is prescribed on the boundary surface. An analytical solution of the hyperbolic heat conduction equation is obtained. The solution describes the wave nature of the temperature field in the geometry under consideration. A detailed analysis of the cases in which the temperature imposed on the boundary surface behaves as a square pulse or as an exponentially decaying pulse is provided. The evolution of the temperature field in the case of hyperbolic heat conduction is compared with that obtained by solving Fourier's equation.

In order to design new materials at atomic-length scales, there is a need to connect the fractal nature of fracture surfaces at the atomic scale using quantum mechanics modeling with that of the experimental data of fracture surfaces at macroscopic-length scales. We use a semi-empirical quantum mechanics simulation of fracture in amorphous silica to calculate a parameter identified as a critical characteristic length, a0, which has been experimentally derived from the fractal nature of fracture for many materials that fail in a brittle matter. To our knowledge, there are no known simulation models other than our related research that use the fractal parameter a0 to describe the fractal fracture of the fracture surface using quantum mechanical simulations. We provide evidence that a0 can be calculated at both the atomic and macroscopic scale, making it a fundamental property of the structure and one of the elements of fractal fracture. We use a continuous random network model and reaction coordinate method to simulate fracture. We propose that fracture in amorphous silica occurs due to bond reconfiguration resulting in increased strain volume at the crack tip. We hypothesize two specific configurations leading to fracture from a four-fold ring reconfiguration to three-fold ring or (newly observed) five-fold ring configurations resulting in a change in volume. Finally, we define a reconfiguration fracture energy at the atomic level, which is approximately the value of the experimental fracture surface energy.

We investigate the macroscopic quasi-static description of a deformable porous medium with a double porosity constituted by pores and fractures. For this purpose, we use an homogenization technique which gives the macroscopic modelling from the description at the pore and fracture levels. It appears that the macroscopic description is sensitive to the ratios between the different scales,l/l′ andl′/l″, wherel, l′ l″ are characteristic lengths of the pores, the fractures and the macroscopic medium, respectively. In the first paper we investigate the casel′/l″=(l/l′)2, which exhibits a coupling between the flows through the pores and the fractures. The macroscopic description is shown to depend on a single pressure field and exhibits a broken symmetry. Other situations will be examined in a subsequent paper. Large spectra of pore and fracture sizes are also evoked.

In this study the thermal field is presented for pulse laser processing of nanoscale Au films. Fourier law is inadequate for describing the heat conduction in nanoscale process due to the boundary scattering and the finite relaxation time of heat carriers. In the regime where the particle description of electrons and phonons is valid, the Boltzmann equation is the most accurate option to model heat transfer in such problems. However, solving the Boltzmann equation is generally difficult due to involving three spatial, three momentums and one time. Dual-phase-lag (DPL) model is averaged over the momentum space and thus involves only spatial coordinates plus time, as in the Fourier equation. Therefore this paper utilizes the dual-phase-lag (DPL) model with scattering boundary condition to study the temperature field for laser processing of nanometer-sized thin films instead of Boltzmann equation. The results obtained from the dual-phase-lag heat conduction model, hyperbolic and parabolic heat conduction equations were compared with the available experimental data to validate the compatibility of the thermal models for analyzing the heat transfer in nanoscale thin film irradiated by laser. The temperature history at different locations of the thin film and the effects of boundary phonon scattering on the normalized temperature were also discussed.

The propagation and transport of electrons in crystals is a fundamental process pertaining to the functioning of most electronic devices. Microscopic theories describe this phenomenon as being based on the motion of Bloch wave packets. These wave packets are superpositions of individual Bloch states with the group velocity determined by the dispersion of the electronic band structure near the central wavevector in momentum space. This concept has been verified experimentally in artificial superlattices by the observation of Bloch oscillations--periodic oscillations of electrons in real and momentum space. Here we present a direct observation of electron wave packet motion in a real-space and real-time experiment, on length and time scales shorter than the Bloch oscillation amplitude and period. We show that attosecond metrology (1 as = 10(-18) seconds) now enables quantitative insight into weakly disturbed electron wave packet propagation on the atomic length scale without being hampered by scattering effects, which inevitably occur over macroscopic propagation length scales. We use sub-femtosecond (less than 10(-15) seconds) extreme-ultraviolet light pulses to launch photoelectron wave packets inside a tungsten crystal that is covered by magnesium films of varied, well-defined thicknesses of a few ångströms. Probing the moment of arrival of the wave packets at the surface with attosecond precision reveals free-electron-like, ballistic propagation behaviour inside the magnesium adlayer--constituting the semi-classical limit of Bloch wave packet motion. Real-time access to electron transport through atomic layers and interfaces promises unprecedented insight into phenomena that may enable the scaling of electronic and photonic circuits to atomic dimensions. In addition, this experiment allows us to determine the penetration depth of electrical fields at optical frequencies at solid interfaces on the atomic scale.

To date, all radiofrequency heating (RFH) theoretical models have employed Fourier's heat transfer equation (FHTE), which assumes infinite thermal energy propagation speed. Although this equation is probably suitable for modeling most RFH techniques, it may not be so for surgical procedures in which very short heating times are employed. In such cases, a non-Fourier model should be considered by using the hyperbolic heat transfer equation (HHTE). Our aim was to compare the temperature profiles obtained from the FHTE and HHTE for RFH modeling. We built a one-dimensional theoretical model based on a spherical electrode totally embedded and in close contact with biological tissue of infinite dimensions. We solved the electrical–thermal coupled problem analytically by including the power source in both equations. A comparison of the analytical solutions from the HHTE and FHTE showed that (1) for short times and locations close to the electrode surface, the HHTE produced temperatures higher than the FHTE, however, this trend became negligible for longer times, when both equations produced similar temperature profiles (HHTE always being higher than FHTE); (2) for points distant from the electrode surface and for very short times, the HHTE temperature was lower than the FHTE, however, after a delay time, this tendency inverted and the HHTE temperature increased to the maximum; (3) from a mathematical point of view, the HHTE solution showed cuspidal-type singularities, which were materialized as a temperature peak traveling through the medium at a finite speed. This peak rose at the electrode surface, and clearly reflected the wave nature of the thermal problem; (4) the differences between the FHTE and HHTE temperature profiles were smaller for the lower values of thermal relaxation time and locations further from the electrode surface.