Fourier Equation Research Articles

Heat conductive properties of set root canal sealers

Objective. The aim of this study was to examine the thermal conductivity of five different root canal sealers in vitro. Materials and methods. Sealapex, AH Plus, AH 26, Endomethasone and RoekoSeal root canal sealers were examined. These materials were prepared in accordance with the manufacturer's instructions and applied to standard molds. Three samples of each material were prepared. The samples were kept for 5 days under 37°C conditions. Measurements were taken using a heat conduction unit (P.A. Hilton Ltd. Stockbridge, Hants, UK). The thermal conductivity coefficient was calculated for each sample using the Fourier equation. Coefficients were analyzed statistically by the Kruskal–Wallis test. Results. Significant differences were found for thermal conductivity between some materials (p < 0.05). The conductivity coefficient of AH Plus was found to be higher than those of the other materials (p < 0.05). No statistically significant differences were found between AH 26 and RoekoSeal (p > 0.05) and the conductivity coefficients of these sealers were found to be lower than those of the other materials (p < 0.05). No statistically significant differences were found between Sealapex and Endomethasone (p > 0.05) and the conductivity coefficients of these sealers were found to be lower than that of AH Plus but higher than those of RoekoSeal and AH26 (p < 0.05). Conclusions. The results showed that root canal sealers functioned as thermal insulators and had different heat-conductive properties that depended on their composition.

Advantages and disadvantages associated with introducing an extra rarefied gas layer into a rotating microsystem filled with a liquid lubricant: First and second law analyses

In the present study, the effects of rarefaction, viscous dissipation, aspect ratio, and accommodation coefficients on transport characteristics associated with a two-phase flow within a rotating microsystem are investigated. It is assumed that centrifugal acceleration due to rotation separates the mixture into two immiscible gas and liquid layers. The incompressible Navier–Stokes–Fourier (NSF) equations in the cylindrical reference frame are employed, while the effects of velocity slip and temperature jump phenomena for the gaseous phase are taken into account. Considering external heating influence and applying the classical thermal boundary conditions of Uniform Heat Flux and Constant Wall Temperature, three different thermal cases are constructed. Expressions for the velocity and temperature distributions of these thermal cases as well as the average entropy generation rate and the Bejan number pertained to the gas and liquid layers are presented. In addition, the criteria for evaluating the impact of the gas layer on shear and temperature reductions at the shaft and also irreversibility rise in the medium are discussed.

Universality in heat conduction theory: weakly nonlocal thermodynamics

AbstractA linear irreversible thermodynamic framework of heat conduction in rigid conductors is introduced. The deviation from local equilibrium is characterized by a single internal variable and a current multiplier. A general constitutive evolution equation of the current density of the internal energy is derived by introducing a linear relationship between the thermodynamic forces and fluxes. The well‐known Fourier, Maxwell–Cattaneo–Vernotte, Guyer–Krumhansl, Jeffreys‐type, and Green–Naghdi‐type equations of heat conduction are obtained as special cases. The universal character of the approach is demonstrated by two examples. Solutions illustrating the properties of the equation with jump boundary conditions are given.

High resistive zero-crossing instant faults detection and location scheme based on wavelet analysis

This paper proposes an effective high resistive zero-crossing fault detection scheme based on transient signals obtained at single-end of the transmission line. The proposed scheme implements wavelet transform to provide time resolution for the high frequency components of the current fault transients. These signals are decomposed into their modal components: aerial and ground modes and, taking into account the fault inception angle; it implements one of both modes to detect faults in the power system. The proposed scheme also provides the distance to fault using the wavelet modulus maxima (WMM) technique. The performance of the proposed scheme has been evaluated in a 132kV transmission system modeled in PSCAD/EMTDC, and results have been presented considering total harmonic distortion of voltage (THDV) waveform and for varying ground resistivity, fault resistance and fault inception angle. The response time of the proposed algorithm has been compared with Fourier and differential equations algorithms and conventional wavelet transform scheme.

Rarefaction and dissipation effects on transport phenomena associated with an immiscible two-phase flow within a shaft-housing micro configuration

A parametric analytical study is performed to investigate the mechanism of fluid flow and heat transfer associated with an immiscible gas–liquid flow within a shaft-housing micro configuration. The incompressible Navier–Stokes–Fourier (NSF) equations in the cylindrical polar coordinate reference frame are employed, while simultaneous effects of viscous dissipation and rarefaction phenomenon are taken into account. Two different classical thermal boundary conditions, namely, the Uniform Heat Flux (UHF) and the Constant Wall Temperature (CWT) boundary conditions are considered. Throughout a combination of these boundary conditions three thermal case studies are constructed and then treated separately. Solutions for the velocity and temperature distributions for both liquid and gas phases of each thermal case are obtained. Employing these solutions, major quantities such as shear and temperature reductions at the shaft due to the gas layer and the Nusselt number pertained to the gas and liquid phases are evaluated. Effects of parameters such as the Knudsen and Brinkman numbers as well as the aspect ratio of the annuli on the aforementioned major variables are examined. The results indicate that introducing a gas layer into the micro domain always contributes to decreasing the shear stress at the shaft and consequently, reducing the torque required to initiate the flow, which is absolutely desirable. However, depending on the thermal case considered, this may be accompanied by serious thermal drawbacks like higher temperatures at the shaft surface. It is also found that increasing values of the Knudsen and Brinkman numbers may even deteriorate the situation.

Langmuir–Maxwell and Langmuir–Smoluchowski boundary conditions for thermal gas flow simulations in hypersonic aerodynamics

The simulation of nonequilibrium thermal gas flow is important for the aerothermodynamic design of re-entry and other high-altitude vehicles. In computational fluid dynamics, the accuracy of the solution to the Navier–Stokes–Fourier (N–S–F) equations depends on the accuracy of the surface boundary conditions. We propose new boundary conditions (called the Langmuir–Maxwell and the Langmuir–Smoluchowski conditions), for use with the N–S–F equations, which combine the Langmuir surface adsorption isotherm with the Maxwell/Smoluchowski slip/jump conditions in order to capture some of the physical processes involved in gas flow over a surface. These new conditions are validated for flat plate flow, circular cylinder in cross-flow, and the flow over a sharp wedge for Mach numbers ranging from 6 to 24, and for argon and nitrogen as the working gases. Our simulation results show that the new boundary conditions give better predictions for the surface pressures, compared with published experimental and DSMC data.

Abruptly autofocusing vortex beams

We generate abruptly autofocusing beams that produce vortices at the focus. We give explicit equations for the phase-only Fourier masks that generate these beams including explanations for controlling the focal distance and numerical aperture. We show experimental results for the focal distance, the vortex pattern and show that the diameter of the focused beam can be made smaller than the size of a comparable Airy beam from a lens. Finally we show how to move the focus spot in three dimensions by encoding additional optical elements onto the phase pattern.

Heat transfer and entropy generation analyses associated with mixed electrokinetically induced and pressure-driven power-law microflows

In the present study, the first and second law analyses associated with combined pressure-driven and electroosmotic flow of power-law liquids through a uniform microannulus are performed numerically. In the first law analysis, the Poisson–Boltzmann and incompressible Navier–Stokes–Fourier (NSF) equations in the cylindrical polar coordinate reference frame as the governing equations are numerically solved using a finite difference method. The obtained velocity and temperature distributions are then employed to evaluate the associated Nusselt number, while simultaneous effects of viscous dissipation and Joule heating are taken into consideration. In the second law analysis, the relevant general expression for the entropy generation process is derived and the contributions of thermal diffusion, fluid friction and Joule heating irreversibility to the total entropy generation in the micro domain are examined. Finally, the variations of the aforementioned major variables with influential parameters such as the aspect ratio of the annuli, flow behavior index, dimensionless Debye–Hückel parameter, forcing ratio, dimensionless viscous heating parameter, and dimensionless Joule heating parameter are investigated.

Thermo-Elastic Analysis of a Cracked Half-Plane Under a Thermal Shock Impact Using the Hyperbolic Heat Conduction Theory

The transient thermal stresses around a crack in a thermo-elastic half-plane are obtained under a thermal shock using the hyperbolic heat conduction theory. Fourier, Laplace transforms and singular integral equations are applied to solve the temperature and thermal stress fields consecutively. The integral equations are solved numerically and the asymptotic fields around the crack tip are obtained. Numerical results show that the hyperbolic heat conduction have significant influence on the dynamic temperature and stress field. It is suggested that to design materials and structures against fracture under thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.

The numerical solution of the transient heat conduction problem using the lattice Boltzmann method

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Global well-posedness for Navier–Stokes equations in critical Fourier–Herz spaces

We prove the global well-posedness for the 3D Navier–Stokes equations in critical Fourier–Herz spaces, by making use of the Fourier localization method and the Littlewood–Paley theory. The advantage of working in Fourier–Herz spaces lies in that they are more adapted than classical Besov spaces, for estimating the bilinear paraproduct of two distributions with the summation of their regularity indexes exactly zero. Our result is an improvement of a recent theorem by Lei and Lin (2011) [10].

Beyond Navier–Stokes

The Navier–Stokes–Fourier (NSF) equations are shown to be strictly applicable only to incompressible flows, namely those involving fluids whose density is uniform throughout. Linear irreversible thermodynamic principles are used to derive an amended set of mass, momentum, and energy equations applicable to all fluids, compressible and incompressible, liquids as well as gases. These reduce to those of NSF in the incompressible limit. A modification of Fourier’s heat-conduction constitutive law is also required as a consequence of the mass/volume velocity difference, wherein the heat flux is now defined in relation to the Second- rather than First-law of thermodynamics; that is, the heat flux is expressed in terms of entropy transport rather than energy transport, with the distinction between the two fluxes vanishing for incompressible flows. When solved subject to a no-slip boundary condition imposed on the fluid’s volume velocity rather than on its mass velocity, the amended NSF equations are noted to furnish results in accord with a variety of experimental data for isothermal and nonisothermal, compressible and incompressible, gas and liquid flows. The difference between the fluid’s volume and mass velocities, namely the diffuse flux of volume, is shown to constitute the hydrodynamic-level manifestation of the fluid’s biased Brownian motion, with the bias arising from the inhomogeneity in mass distribution arising in compressible fluids from temperature or pressure gradients. Previously, continuum hydrodynamics, as embodied in the Navier–Stokes–Fourier equations, has overlooked the fundamental contribution to fluid mechanics emanating from biased Brownian motion.

Simulation of Characteristic of a Thermoelectric Power Sensor Based on MEMS Technology

A thermal equivalent model based on Fourier equation is introduced to optimize a thermoelectric microwave power sensor in this paper. The different distance between the load resistor and thermopile is considered carefully to obtain good matching characteristic and high sensitivity characteristic. The sensitivity of the thermoelectric power sensor has been chosen as a goal function with the distance, the length of thermopile and the number of thermopiles as independent variables. The simulation results show that the sensitivity of the power sensor decreases with the distance, but increases with the length and the number of thermopiles. Then the input matching characteristic of this thermoelectric power sensor with different distance are simulated with HFSS. The power sensor has a good performance on sensitivity characteristic and input match characteristic with an optimization distance is about 10μm.

Nonstationary heat conduction in one-dimensional models with substrate potential

The paper investigates nonstationary heat conduction in one-dimensional models with substrate potential. To establish universal characteristic properties of the process, we explore three different models: Frenkel-Kontorova (FK), phi4+ (φ(4)+), and phi4- (φ(4)-). Direct numeric simulations reveal in all these models a crossover from oscillatory decay of short-wave perturbations of the temperature field to smooth diffusive decay of the long-wave perturbations. Such behavior is inconsistent with the parabolic Fourier equation of heat conduction and clearly demonstrates the necessity for hyperbolic corrections in the phenomenological description of the heat conduction process. The crossover wavelength decreases with an increase in the average temperature. The decay patterns of the temperature field almost do not depend on the amplitude of the perturbations, so the use of linear evolution equations for the temperature field is justified. In all models investigated, the relaxation of thermal perturbations is exponential, contrary to a linear chain, where it follows a power law. The most popular lowest-order hyperbolic generalization of the Fourier law, known as the Cattaneo-Vernotte or telegraph equation, is also not valid for the description of the observed behavior of the models with the substrate potential, since the characteristic relaxation time in an oscillatory regime strongly depends on the excitation wavelength. For some of the models, this dependence seems to obey a simple scaling law.

Investigation of flow and heat transfer characteristics of rarefied gaseous slip flow in nonplanar micro-Couette configuration

In the present study the mechanism of transport phenomena in a low-speed rarefied gas flow confined between a shaft and its concentric housing, namely, the cylindrical Couette problem, is investigated analytically in the slip-flow regime. The incompressible Navier–Stokes–Fourier (NSF) equations in the cylindrical polar coordinate reference frame are employed while the simultaneous effects of viscous dissipation and rarefaction phenomenon are taken into account. Two different thermal boundary conditions are considered: the Uniform Heat Flux (UHF) and the Constant Wall Temperature (CWT). Solutions for the velocity and temperature distributions, the Nusselt number and the wall heat flux are obtained for different values of the Knudsen and Brinkman numbers and curvature parameter.

High tip angle approximation based on a modified Bloch–Riccati equation

When designing a radio-frequency pulse to produce a desired dependence of magnetization on frequency or position, the small flip angle approximation is often used as a first step, and a Fourier relation between pulse and transverse magnetization is then invoked. However, common intuition often leads to linear scaling of the resulting pulse so as to produce a larger flip angle than the approximation warrants--with surprisingly good results. Starting from a modified version of the Bloch-Riccati equation, a differential equation in the flip angle itself, rather than in magnetization, is derived. As this equation has a substantial linear component that is an instance of Fourier's equation, the intuitive approach is seen to be justified. Examples of the accuracy of this higher tip angle approximation are given for both constant- and variable-phase pulses.

An example illustrating the incompleteness of the Navier–Stokes–Fourier equations for thermally compressible fluids

This paper illustrates, by example, the incompleteness of the Navier–Stokes–Fourier (NSF) equations for the case of thermally compressible fluids, namely fluids possessing a nonzero coefficient of thermal expansion. The work is a follow-up to a recent publication that offered elementary arguments quantifying that incompleteness but did not provide an explicit physical example thereof. The present example was chosen strictly for the simplicity of the calculations required to bring it to fruition, rather than for its importance in applications. The example analyzes steady-state, one-dimensional (albeit nonunidirectional) heat conduction through a quiescent fluid bounded by concentric spheres maintained at different temperatures. This example is counterpart to the classic NSF case of steady-state, one-dimensional (but now unidirectional) heat conduction through a quiescent fluid bounded by flat plates maintained at different temperatures. The contrasting results obtained for the two cases illustrates effects arising from the proposed amendments to the traditional NSF equations. For the case of gases the amended results indicate the possibility of their differing significantly from classical results based on the NSF equations when the gas is rarefied. For liquids, however, physically realizable values of the relevant parameters governing the amended equations are such that no sensible deviations from classical NSF behavior are observed. The difference owes to the relative incompressibility of liquids compared with gases. The smallness of the effect for liquids is, however, noted to be atypical of the amended consequences arising in circumstances where the temperature varies along, rather than purely normal to solid surfaces, as in the present concentric-sphere example. In that case Maxwell thermal creep effects would create more profound effects than in the present example, whether for liquids or gases.

Analytical solution of the parabolic and hyperbolic heat transfer equations with constant and transient heat flux conditions on skin tissue

In this article, the parabolic (Pennes bioheat equation) and hyperbolic (thermal wave) bioheat transfer models for constant, periodic and pulse train heat flux boundary conditions are solved analytically by applying the Laplace transform method for skin as a semi-infinite and finite domain. The bioheat transfer analysis with transient heat flux on skin tissue has only been studied by Pennes equation for a semi-infinite domain. For modeling heat transfer in short duration of an initial transient, or when the propagation speed of the thermal wave is finite, there are major differences between the results of parabolic and hyperbolic heat transfer equations. The non-Fourier bioheat transfer equation describes the thermal behavior in the biological tissues better than Fourier equation. The outcome of transient heat flux condition shows that by penetrating into the depths beneath the skin subjected to heat, the amplitude of temperature response decreases significantly. The blood perfusion rate can be predicted using the phase shift between the surface temperature and transient surface heat flux. The thermal damage of the skin is studied by applying both the parabolic and hyperbolic bioheat transfer equations.

Natural convection in horizontal annuli: evaluation of the error for two approximations

We consider convection inside annuli, driven by a uniform temperature gap on the boundaries and gravitation as outer force. It takes place for any Rayleigh number while steady convective motions are observed only for small ones (but any Prandtl number and gap width). We provide estimates for the relative error of two popular approximations to the full Navier–Stokes–Fourier equations. For this we propose a new method. In particular we have to derive a lower bound for the norm of the velocity and the temperature both for steady nonlinear coupled and decoupled approximations in two space dimensions.

Principle of Virtual Power for thermomechanics of fluids and solids with dissipation

A fundamental variational principle, called the Principle of Virtual Power, is proposed in this paper to govern the irreversible thermomechanical processes for a material body. The principle and the corresponding variational equation of virtual velocity are formulated by integrating the classical field theories of continuum mechanics and thermodynamics. From the variational equation, we establish simultaneously the balance equations of linear momentum, of angular momentum, and of virtual internal power by applying the procedure of relaxation in Lagrangian mechanics. By assuming different types of system functions for internal power consisting of internal energy and newly introduced working dissipation, we are able to derive the constitutive equations for gas, liquid, or solid and the Fourier equation of heat conduction from the balance of virtual internal power. The combination of the law of conservation of mass, the first law of thermodynamics, and the proposed principle forms a complete set of fundamental principles for the thermomechanics of deformable bodies.