Fourier's Law Research Articles

Implementation of improved Fourier's law and Fick's law for rotational flow of nanofluid over an exponentially stretching sheet

PurposeThe purpose of the current article is to explore the rotational behavior on nanofluid flow over an exponentially stretching surface. Heat and mass flux are formulated upon Cattaneo–Christov theory.Design/methodology/approachEffect of thermophoretic, Brownian motion and thermally convective conditions is further retained. Novel boundary layer approximations are applied to transform the governing equations of continuity, momentum, energy and nanoparticle volume fraction. Convergent series solutions are obtained to manage the rotating flow with the aid of homotopy analysis method (HAM).FindingsDepending on the several dimensionless parameters including the local rotation parameter the Prandtl number Pr, the thermophoresis parameter, the Brownian motion parameter, the Lewis number Le, Biot number Bi, Deborah number in terms of heat flux relaxation parameter and Deborah number in terms of mass flux relaxation parameter with the dimensionless physical quantities are deliberated through graphs. Present results are also likened with the foregoing results in significance.Originality/valueNo such assumptions have been made for the development of analytical solution so far.

Double diffusion in stretched flow over a stretching cylinder with activation energy and entropy generation

Present study accords exploration of double diffusion in an axisymmetric flow of chemically reacting viscous fluid which impulsively starts flowing over a cylinder due to stretched surface. Non-Fourier law is employed to explain the heat flow whereas non-Fick's law of mass flux is employed to delve into the diffusion of mass. Activation energy is also taken into account. Transformed governing equations are solved using efficient mathematical technique called Homotopy Analysis Method (HAM). For the validity of the solution technique convergence table and h curves are plotted. Numerical solution has also been computed by using Shooting method for comparison with HAM. Entropy generation of the system has been investigated. Significant impact of pertinent flow parameters on thermal profile, concentration and velocity distribution are examined graphically. Velocity and thickness of momentum boundary layer increase by increasing curvature. Thermal profile is inversely related with Prandtl number and thermal relaxation time. Concentration profile trickles down by an increase in Schmidt number, mass relaxation parameter, Fitted rate constant and temperature ratio parameter. Most of the studies available in the literature focused on the double diffusion using generalized Fourier and Fick's laws neglecting activation energy and entropy generation which is an inevitable aspect in practical problems now a days. These features have been considered in the present article and it is the novelty of the present research.

Effect of modified Ohm's and Fourier's laws on magneto thermoviscoelastic waves with Green-Naghdi theory in a homogeneous isotropic hollow cylinder

This paper deals with the modified Ohm's law, including the temperature gradient and charge density effects, and the generalized Fourier's law, including the present current density impact, the problem of conveyance of thermal stresses and temperature in a generalized Magneto–Thermo-Viscoelastic Solid Cylinder of radius L. The formulation is applied to the generalized thermoelasticity dependent on the Green-Naghdi (G-N II) hypothesis. The Laplace change system is utilized to solve the problem. At last, the outcomes got are introduced graphically to show the impact of Magnetic Field and time and on the field variables.

Thermal Conducting Properties of Metal-fluoroplastic Material Determined by Tribological Method

The work is devoted to considering the possibility of using the Fourier law and the data of tribological tests for the approximate determination of the thermophysical characteristics of the sheet metal-fluoroplastic material on a steel substrate. The thermal conductivity coefficient of two different fluoroplastic materials was determined by tribological method using the temperature difference in the friction zone. It was shown that friction conditions change from viscoelastic to plastic

Thermodynamics of scalar-tensor gravity

Previously, the Einstein equation has been described as an equation of state, general relativity as the equilibrium state of gravity, and $f({\cal R})$ gravity as a non-equilibrium one. We apply Eckart's first order thermodynamics to the effective dissipative fluid describing scalar-tensor gravity. Surprisingly, we obtain simple expressions for the effective heat flux, "temperature of gravity", shear and bulk viscosity, and entropy density, plus a generalized Fourier law in a consistent Eckart thermodynamical picture. Well-defined notions of temperature and approach to equilibrium, missing in the current thermodynamics of spacetime scenarios, naturally emerge.

The Space-Time Coupled Fractional Cattaneo-Friedrich Maxwell Model with Caputo Derivatives.

In the current article, we have thoroughly investigated the collective impact of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid (Cattaneo–Friedrich Maxwell-CFM model) over a permeable surface embedded in a porous medium. Here we have utilized the Caputo time-fractional derivatives and mechanical laws (generalized shear stress constitutive equation and generalized Fourier’s and Fick’s laws) are being used to fractionalize the presented model. The effects of radiative heat flux, Ohmic dissipation, and internal absorption are presented through generalized Fourier’s law while Fick’s law or mass transfer equation offers the effects of first order chemically reactive species. The finite element method and finite difference method are being utilized to numerically solve the nonlinear coupled differential equations. It is established, through compression of numerical and analytical solutions, that the presented model is convergent. Further, error analysis of the subject model is also carried out. Moreover, for better illustration of results, we have also offered a graphical and tabular presentation of impacts of the parameters of interest on velocity, temperature, concentration profile, local skin friction coefficient, and heat and mass transfer. It is evident from the obtained results that velocity near and away from the surface increases with the enhancement of fractional derivative parameter whereas an opposite trend is observed in the case of temperature. Furthermore, it is noticed that temperature shows a decreasing behavior for the value {Lambda }_{theta }<2 and {Lambda }_{phi }<2, on the other hand entirely opposite trend is witnessed for {Lambda }_{theta }ge 3 and {Lambda }_{phi }ge 3. From an engineering perspective, we have acquired comprehensive outcomes such that the heat transfer offers an increasing trend in the case of TR and thermal fractional parameter beta_{1} . Additionally, the chemical reaction parameter and Sc significantly contribute towards the mass transfer rate. Since, in literature, one cannot refer to such results with non-integer Caputo fractional derivatives thus the results obtained through the current assessment hold significance for future research avenues. Moreover, the numerical inferences of the subject study may contribute to an advanced thermal processing method in the food industry to swiftly increase the temperature for cooking or sterilization drives.

In-plane interfacial phonon transport through multi-layer thin films by theoretical analyses and Monte Carlo simulations

This work studies the in-plane phonon transport through double-layer, superlattice, and four-layer thin films by theoretical analyses and Monte Carlo simulations. The spectral diffuse mismatch model and the adiabatic diffuse scattering are taken as the interface and surface boundary conditions respectively. The results indicate that the number of thin-film layers has little impact on the heat flux distribution within each layer and there is a heat flux discontinuity at the interface. Indentical in-plane thermal conductivities are obtained for double-layer, superlattice, and four-layer thin films with a uniform thickness of each layer. This means that the number of thin-film layers has negligible impacts on the in-plane thermal conductivity, which can provide the first explicit explanation to the observation in a recent experiment. Finally, the minimum in-plane thermal conductivity is found in thin films when varying thickness ratios. The extreme points are also analytically obtained. This law is completely different from that for the cross-plane thermal conductivity or that predicted by the Fourier's law of heat conduction for in-plane heat transport which leads to a monotonic relation between the thermal conductivity and the thickness ratio.

Derivation of Heat Conductivity from Temperature and Heat Flux Measurements in Soil

The general inverse problem formulation for a heat conductance equation is adopted for the types of measurement routinely carried out in the soil active layer. The problem solution delivers a constant thermal diffusivity coefficient a0 (in general, different from true value a) and respective heat conductivity λ0 for the layer, located between two temperature sensors and equipped with a temperature or heat flux sensor in the middle. We estimated the error of solution corresponding to systematic shifts in sensor readings and mislocation of sensors in the soil column. This estimation was carried out by a series of numerical experiments using boundary conditions from observations on Mukhrino wetland (Western Siberia, Russia), performed in summer, 2019. Numerical results were corroborated by analytical estimates of inverse problem solution sensitivity derived from classical Fourier law. The main finding states that heat conductivity error due to systematic shifts in temperature measurements become negligible when using long temperature series, whereas the relative error of a is approximately twice the relative error of sensor depth. The error a0−a induced by heat flux plate displacement from expected depth is 3–5 times less than the same displacement of thermometers, which makes the requirements for heat flux installation less rigid. However, the relative errors of heat flux observation typical for modern sensors (±15%) cause the uncertainty of a above 15% in absolute value. Comparison of the inverse problem solution to a estimated from in situ moss sampling on Mukhrino wetland proves the feasibility of the method and corroborates the conclusions of the error sensitivity study.

Decay of quasi-static porous-thermo-elastic waves

We study the behavior in time of the solutions to several systems of equations for porous-thermo-elastic problems when one of the variables is considered to be quasi-static or, in other words, whose second time derivative can be neglected. We analyze three different situations using the classical Fourier law and also the type II or type III Green–Naghdi heat conduction models.

A heat conduction equation for heterogeneous media and its connections to various known equations

We derive the governing equation of the macroscopically average temperature field in a heterogeneous medium. Although the constituents of the heterogeneous medium obey Fourier's law, the average temperature field is governed by a spatiotemporally nonlocal integro-differential formulation. Then, we find that the average fluctuation of the temperature in the matrix region of the heterogeneous medium plays a key role in determining the various degenerate forms of the governing equation. Depending on the value of this average fluctuation, the governing equation can degenerate into the well-known equations in the literature, including the Jeffreys-type equation, Nunziato equation, Gurtin and Pipkin equation, peridynamic formulation, and dual-phase-lag (DPL) equation. These connections establish the relations between the coefficients in these equations with the properties of the constituents of a heterogeneous medium, and provide guidelines for their applications to the prediction of heat conduction in heterogeneous media.

Heat conduction in a three-dimensional momentum-conserving fluid.

Size dependence of energy transport and the effects of reduced dimensionality on transport coefficients are of key importance for understanding nonequilibrium properties of matter on the nanoscale. Here, we perform nonequilibrium and equilibrium simulations of heat conduction in a three-dimensional (3D) fluid with the multiparticle collision dynamics, interacting with two thermal walls. We find that the bulk 3D momentum-conserving fluid has a finite nondiverging thermal conductivity. However, for large aspect ratios of the simulation box, a crossover from 3D to one-dimensional (1D) abnormal behavior of the thermal conductivity occurs. In this case, we demonstrate a transition from normal to abnormal transport by a suitable decomposition of the energy current. These results not only provide a direct verification of Fourier's law, but also further confirm the validity of existing theories for 3D fluids. Moreover, they indicate that abnormal heat transport persists also for almost 1D fluids over a large range of sizes.

Transient thermoelastic response of a size‐dependent nanobeam under the fractional order thermoelasticity

AbstractWith the miniaturization of structures, such as MEMS/NEMS, the size‐dependent effect has become an issue and attracted much attention. The well‐known theories describing the size‐dependent effect mainly include the nonlocal elasticity theory, the strain gradient theory and the modified coupled stress theory. Based on these theories, a number of works have been conducted to explore the size‐dependent behaviors of structures or devices in micro/nano‐scale, among them, majorities are on elastic performances, while, minorities are on thermoelastic performances. It is inevitable for structures suffering changeable temperature, as a consequence, thermal‐induced stress and deformation occur in structures and they are worth being fully concerned. For thermoelastic behaviors limited to small scale problems, the classical Fourier's heat conduction law may fail, meanwhile, new models, for example, fractional order heat conduction model, have been developed to modify Fourier's law. In present paper, the transient thermoelastic response of a nanobeam subjected to a ramp heating is investigated by combining the nonlocal elasticity theory and the fractional order heat conduction model. The governing equations are formulated and then solved by Laplace transform and its numerical inversion. The non‐dimensional temperature, displacement, stress, and deflection in the nanobeam are obtained and illustrated graphically. In calculation, the effects of the ramp‐heating time parameter, the nonlocal parameter and the fractional order parameter on the considered physical quantities are examined and discussed in detail.

Thermal conduction in a harmonic chain coupled to two cavity-optomechanical systems

We propose a model including a one-dimensional harmonic chain of oscillators whose two ends coupled two cavity-optomechanical systems for studying one-dimensional thermal conductivity in statistical physics. In this model, the cavity-optomechanical systems function as two laser-engineerable thermal reservoirs. When the effective temperatures of the two reservoirs are not equal, a heat flux through the chain can be generated, and moreover the classical and quantum features of the heat flux have been studied. We further show that the heat flux does not obey the Fourier's law in this model. The thermal switching phenomenon could be induced by controlling the on-site vibrational frequency. Finally, we propose to apply the correlation between two mechanical oscillators to measure the heat flux through the chain.

On the Cauchy problem of the standard linear solid model with Fourier heat conduction

In this paper, we consider the standard linear solid model in $$\mathbb {R}^N$$ coupled with the Fourier law of heat conduction. First, we give the appropriate functional setting to prove the well-posedness of this model under certain assumptions on the parameters (that is, $$0<\tau \le \beta $$ ). Second, using the energy method in the Fourier space, we obtain the optimal decay rate of a norm related to the solution. In particular, we prove that when $$0<\tau <\beta $$ the decay rate is the same as in the Cauchy problem without heat conduction (see Pellicer and Said-Houari in Appl Math Optim 80: 447–478, 2019), and that it does not exhibit the well-known regularity loss phenomenon which is present in some of these models. When $$\tau =\beta >0$$ (that is, when the only dissipation comes through the heat conduction), we still have asymptotic stability, but with a slower decay rate. We also prove the optimality of the previous decay rate for the solution itself by using the eigenvalues expansion method. Finally, we complete the results in Pellicer and Said-Houari (Appl Math Optim 80: 447–478, 2019) by showing how the condition $$0<\tau < \beta $$ is not only sufficient but also necessary for the asymptotic stability of the problem without heat conduction.

Research and Discussion on Welding Mechanism of Reflow Furnace

The flexible grasp of automatic welding technology is very important for the production of electronic products such as integrated circuit board, and the most critical step is to accurately control the temperature of the welding area. This paper will establish the mechanism model of the influence of the temperature of each temperature zone of the reflow furnace and the speed of the conveyor belt on the temperature of the welding area through the relevant thermodynamic theory, so as to help the relevant electronic product manufacturers to accurately control the temperature of the welding area. The mechanism model of temperature change in the center of welding area is established, and the heat transfer process is divided into three steps. Firstly, according to the temperature field of the air in the furnace, the air temperature corresponding to the gap in the small temperature zone is determined, and then the heat convection process between the air temperature in the furnace and the surface of the welding area is described by using Newton’s cooling law. Finally, Fourier law is used to study the heat conduction process in the welding area.

Negative-temperature Fourier transport in one-dimensional systems

We investigate nonequilibrium steady states in a class of one-dimensional diffusive systems that can attain negative absolute temperatures. The cases of a paramagnetic spin system, a Hamiltonian rotator chain and a one-dimensional discrete linear Schrödinger equation are considered. Suitable models of reservoirs are implemented to impose given, possibly negative, temperatures at the chain ends. We show that a phenomenological description in terms of a Fourier law can consistently describe unusual transport regimes where the temperature profiles are entirely or partially in the negative-temperature region. Negative-temperature Fourier transport is observed both for deterministic and stochastic dynamics and it can be generalized to coupled transport when two or more thermodynamic currents flow through the system.

The nonlinear thermo-hyperelasticity wave propagation analysis of near-incompressible functionally graded medium under mechanical and thermal loadings

This article presents a non-Fourier thermo-hyperelastic model to investigate thermal and stress wave propagation phenomenon in a near-incompressible functionally graded medium (FGM) for various thermal and mechanical boundary conditions. A strain energy function is chosen to modify FGM’s hyperelastic equations considering the coupling effects of mechanical and thermal terms. By switching the strain tensor's invariants, equations are developed to estimate a near-incompressible model for rubber. The rubber is characterized by a gradual variation in the longitudinal direction. Therefore, the material properties of rubber mainly depend on coordinates in through an exponential function. The nonlinear governing equations are derived from the large displacement approach using Finite Strain Theory. To find an acceptable solution of nonlinear time-dependent thermo-hyperelastic equations, Newmark's time integration process and a nonlinear Hermitian finite element algorithm are employed. The final system’s responses to different boundary conditions such as input surface traction, heat flux and variable material properties are described schematically, and their influence on the wave propagation is calculated. It is shown that the amplitude of oscillation in a functionally graded hyperelastic medium is less than that of a medium with constant properties. The results also show that the wave travels through the medium faster than the FGM. Moreover, the modified Fourier law of heat conduction is applied and the impact of enhanced heat conduction model on the thermo-hyperelastic responses is discussed.

3D Bio-convective nanofluid Bödewadt slip flow comprising gyrotactic microorganisms over a stretched stationary disk with modified Fourier law

The renowned Bödewadt flow problem is considered here for the case of Bio-convective nanofluid flow encompassing gyrotactic microorganisms over a stationary disk. The flow is initiated owing to the rotation of water-based nanofluid which is at a large distance from an immovable disk. The analysis is performed with modified Fourier law in a Darcy-Forchheimer spongy media accompanied by slip and zero mass flux conditions at the surface. The subject Bödewadt nanofluid flow is considered with gyrotactic microorganisms and Cattaneo-Christov (C-C) heat flux effects for the first time. Conformist Von Kármán relations are affianced to attain a similar governing system of differential equations. An efficient MATLAB software-based numerical scheme bvp4c is employed to address the envisaged novel mathematical model. An outstanding synchronization is achieved when a comparative statement is formulated in a limiting case. Outcomes of the proclaimed parameters versus involved distributions are discussed with logical reasoning. It is comprehended that the microorganism density boundary layer is prominent for higher values of the Peclet number. Furthermore, it is remarked that the fluid temperature is diminished for significant values of the thermal relaxation time.

Mix convection non- boundary layer flow of unsteady MHD oblique stagnation point flow of nanofluid

This article demonstrates the analysis of hydrodynamics in the existence of a magnetic field. Heat transfer is analyzed for the unsteady flow of non-Newtonian fluid. The thermal conductivity is enhanced by introducing nanoparticles in the non-Newtonian fluid. Magnetohydrodynamics is introduced to control the turbulence in the flow. The dimensional equations satisfying the stated assumptions of the fluid flow and the motion of nanoparticles are presented. Fourier law is incorporated in the evaluation of heat flux. The analysis is tested in the fixed frame of reference. The mathematical coupled equations of motion are solved numerically. It is concluded that the Casson parameter leads to enhance velocity profile whereas the temperature profile is declined.

Derivation of a hydrodynamic heat equation from the phonon Boltzmann equation for general semiconductors

We present a formalism to solve the phonon Boltzmann transport equation (BTE) for finite Knudsen numbers that supplies a hydrodynamic heat transport equation similar to the Navier-Stokes equation for general semiconductors. This generalization of Fourier's law applies in general cases, from systems dominated by momentum-preserving normal collisions, as is well known, to kinetic materials dominated by resistive collisions, where it captures nonlocal effects. The key feature of our framework is that the macrostate is described in terms of the heat flux and its first derivatives. We obtain explicit expressions for the nonequilibrium phonon distribution and for the geometry-independent macroscopic parameters as a function of phonon properties that can be calculated from first principles. Ab initio model predictions are found to agree with a wide range of experiments in silicon. In contrast to approaches directly based on the BTE, the hydrodynamic equation can be solved in arbitrary geometries, thus providing a powerful tool for nanoscale heat modeling at a low computational cost.