Arbitrary Temperature Gradient Research Articles

Arbitrary temperature gradient in thermal-postbuckling behavior of polymer nanocomposites reinforced by boron nitride nanotubes

Polymer nanocomposites reinforced by boron nitride nanotubes (BNNTs) have great potential for use in a wide variety of thermal environments. A closed-form solution for nonlinear bending and thermal-postbuckling behavior of BNNT-reinforced nanocomposite beams under arbitrary temperature gradient in the thickness and length directions is presented for the first time. The uniform and functionally graded (FG) distributions of BNNTs through the thickness of macro-or micro-sized beams are considered. For constant temperature or constant heat flux conditions at two ends of the beam, thermal analysis is carried out to obtain temperature field ΔTx,z using the two-dimensional heat conduction equation and the assumption of arbitrary variation in the z direction. The governing equations of modified couple stress Timoshenko beam theory with von Kármán geometric nonlinearity are derived for BNNT-RC beams subjected to thermomechanical loads. Applying change of variable technique, the governing equations are exactly solved to develop closed-form expression for nonlinear transverse deflection. Finally, parametric study is performed to investigate the effects of length scale parameter, aspect ratio (L/h), distribution of BNNTs, and temperature gradient on the nonlinear bending and postbuckling equilibrium paths. It is shown that the thermal-buckling behavior of clamped and cantilever beams is completely different. Under thermomechanical loading, an initial snap-through buckling behavior is observed in the cantilever beam, while there exist both stable and unstable configurations in the static equilibrium path of clamped beam. Moreover, it is found that the assumption of parabolic variation of temperature through thickness is not compatible with real situations, since the temperature field is obtained as harmonic functions with very short wavelengths in the length direction.

Physics-informed deep learning for solving phonon Boltzmann transport equation with large temperature non-equilibrium

Phonon Boltzmann transport equation (BTE) is a key tool for modeling multiscale phonon transport, which is critical to the thermal management of miniaturized integrated circuits, but assumptions about the system temperatures (i.e., small temperature gradients) are usually made to ensure that it is computationally tractable. To include the effects of large temperature non-equilibrium, we demonstrate a data-free deep learning scheme, physics-informed neural network (PINN), for solving stationary, mode-resolved phonon BTE with arbitrary temperature gradients. This scheme uses the temperature-dependent phonon relaxation times and learns the solutions in parameterized spaces with both length scale and temperature gradient treated as input variables. Numerical experiments suggest that the proposed PINN can accurately predict phonon transport (from 1D to 3D) under arbitrary temperature gradients. Moreover, the proposed scheme shows great promise in simulating device-level phonon heat conduction efficiently and can be potentially used for thermal design.

Bandgap formation under temperature-induced quasi-periodicity in an acoustic duct with flexible walls

This paper is concerned with the acoustic bandgap formation in a duct with periodically flush-mounted flexible walls, subject to a stream-wise temperature variation. A numerical model, based on a piecewise treatment of the arbitrary temperature gradient, is proposed for the accurate prediction of sound propagation in a complex thermal environment. Effects of the system quasi-periodicity due to the temperature change on the bandgap formation, as well as possible mitigation measures through parameter tuning, are revealed. It is shown that, on the top of the resonance bandgap, Bragg reflection bandgap can still be created despite the system quasi-periodicity. In addition to an alteration to the bandgap central frequency due to the temperature-induced acoustic wavelength changes inside the duct, the bandwidths of the bandgaps are also adversely affected, especially when the temperature gradient is large. Numerical analyses show the possibility of tuning and customizing the bandgap formation through a proper selection of the lattice distance and the structural parameters. A combined tuning strategy would warrant a merging of the bandgaps, adjustment of their central frequencies as well as an enlargement of the bandwidth for a given temperature variation range through creating a favorable coupling between the Bragg reflections and the local resonances of the flexible walls.

Simulations of Transport in Hard Particle Systems

Hard particle systems are among the most successful, inspiring and prolific models of physics. They contain the essential ingredients to understand a large class of complex phenomena, from phase transitions to glassy dynamics, jamming, or the physics of liquid crystals and granular materials, to mention just a few. As we discuss in this paper, their study also provides crucial insights on the problem of transport out of equilibrium. A main tool in this endeavour are computer simulations of hard particles. Here we review some of our work in this direction, focusing on the hard disks fluid as a model system. In this quest we will address, using extensive numerical simulations, some of the key open problems in the physics of transport, ranging from local equilibrium and Fourier’s law to the transition to convective flow in the presence of gravity, the efficiency of boundary dissipation, or the universality of anomalous transport in low dimensions. In particular, we probe numerically the macroscopic local equilibrium hypothesis, which allows to measure the fluid’s equation of state in nonequilibrium simulations, uncovering along the way subtle nonlocal corrections to local equilibrium and a remarkable bulk-boundary decoupling phenomenon in fluids out of equilibrium. We further show that the the hydrodynamic profiles that a system develops when driven out of equilibrium by an arbitrary temperature gradient obey universal scaling laws, a result that allows the determination of transport coefficients with unprecedented precission and proves that Fourier’s law remains valid in highly nonlinear regimes. Switching on a gravity field against the temperature gradient, we investigate numerically the transition to convective flow. We uncover a surprising two-step transition scenario with two different critical thresholds for the hot bath temperature, a first one where convection kicks but gravity hinders heat transport, and a second critical temperature where a percolation transition of streamlines connecting the hot and cold baths triggers efficient convective heat transport. We also address numerically the efficiency of boundary heat baths to dissipate the energy provided by a bulk driving mechanism. As a bonus track, we depart from the hard disks model to study anomalous transport in a related hard-particle system, the 1d diatomic hard-point gas. We show unambiguously that the universality conjectured for anomalous transport in 1d breaks down for this model, calling into question recent theoretical predictions and offering a new perspective on anomalous transport in low dimensions. Our results show how carefully-crafted numerical simulations of simple hard particle systems can lead to unexpected discoveries in the physics of transport, paving the way to further advances in nonequilibrium physics.

Turbulent thermal diffusion in strongly stratified turbulence: Theory and experiments

Turbulent thermal diffusion is a combined effect of the temperature stratified turbulence and inertia of small particles. It causes the appearance of a non-diffusive turbulent flux of particles in the direction of the turbulent heat flux. This non-diffusive turbulent flux of particles is proportional to the product of the mean particle number density and the effective velocity of inertial particles. The theory of this effect has been previously developed only for small temperature gradients and small Stokes numbers (Phys. Rev. Lett. {\bf 76}, 224, 1996). In this study a generalized theory of turbulent thermal diffusion for arbitrary temperature gradients and Stokes numbers has been developed. The laboratory experiments in the oscillating grid turbulence and in the multi-fan produced turbulence have been performed to validate the theory of turbulent thermal diffusion in strongly stratified turbulent flows. It has been shown that the ratio of the effective velocity of inertial particles to the characteristic vertical turbulent velocity for large Reynolds numbers is less than 1. The effective velocity of inertial particles as well as the effective coefficient of turbulent thermal diffusion increase with Stokes numbers reaching the maximum at small Stokes numbers and decreases for larger Stokes numbers. The effective coefficient of turbulent thermal diffusion also decreases with the mean temperature gradient. It has been demonstrated that the developed theory is in a good agreement with the results of the laboratory experiments.

Behavior of plane waves propagating through a temperature-inhomogeneous region

Description and analysis of acoustic waves in ducts with a region containing temperature-inhomogeneous fluid represent a significant problem of scientific and practical interest. This interest is induced by the need of understanding how temperature fields affect acoustic processes which would lead to a more efficient design and control of systems involving thermoacoustic interactions. Most of the works addressing these problems limit themselves to the assumption of weak temperature profile gradients or to temperature profiles which do not connect neighboring temperature-homogeneous regions smoothly. In our work we investigate the behavior of plane acoustic waves that enter a region with an arbitrary temperature gradient. A polynomial character of the used temperature profile ensures smooth connection with constant-temperature regions. The one-dimensional wave equation for ducts with an axial mean temperature gradient is solved analytically. The derived solutions based on Heun functions extend the class of published exact analytical solutions of model wave equations taking into account the medium temperature gradient. Due to the property that our proposed polynomial temperature function has derivatives equal to zero at points which are connected with the surrounding temperature-homogeneous regions we can form more complex smooth temperature profiles for which it is possible to use the transfer matrix method.

A thorough study on thermal mass flux of rarefied flow through micro/nanochannels

Most of the previous works on mass flux due to thermal creep focused on thermal creep due to wall temperature gradient. In this work, we show that even without wall temperature gradient there exists thermal creep mass flux if one applies heat flux over the wall. This heat flux could induce a temperature gradient inside the flow and subsequently leads to a temperature gradient across the channel. Additionally, we deduce that rarefied gas under thermal non-equilibrium conditions is governed by three separate mechanisms, i.e., thermal creep effect and viscosity and density changes. There is a critical Knudsen number, around unity, where the mass flux remains invariant against the heating/cooling wall heat fluxes. Some analytical expressions for all sorts of thermal mass flux, including that due to the heat flux and that due to the wall temperature gradient are derived. Previous works on mass flux due to the thermal creep only considered linearized flow while this work considers arbitrary temperature gradient effects.

Method to determine cutoff frequencies for acoustic waves propagating in nonisothermal media

A method to determine cutoff frequencies for linear acoustic waves propagating in nonisothermal media is introduced. The developed method is based on wave variable transformations that lead to Klein-Gordon equations, and the oscillation theorem is applied to obtain the turning point frequencies. Physical arguments are used to justify the choice of the largest turning point frequency as the cutoff frequency. The method is used to derive the cutoff frequencies in nonisothermal media modeled by exponential and power law temperature gradients, for which the cutoffs cannot be obtained based on known analytical solutions. An interesting result is that the acoustic cutoff frequencies calculated by the method are local quantities that vary in the media, and that their specific values at a given height determine the frequency that acoustic waves must have in order to be propagating at this height. To extend this physical interpretation of the acoustic cutoff frequency to nonisothermal media of arbitrary temperature gradients, a generalized version of the method applicable to these media is also presented.

Calculating Seebeck Coefficients for Arbitrary Temperature Gradients

ABSTRACTA novel computational scheme has been used to predict the electric potentials generated by arbitrary temperature gradients in semiconductor materials. Written in object-oriented code, the Discrete State Simulation (DSS) is a coupled cellular automata simulator that builds upon the objects and rules of quantum mechanics. The DSS represents global non-equilibrium processes as patterns that emerge through an ensemble of scattering events that are localized at vibronic nodes. By tracking the energy-momentum-position coordinates of the individual particles that define the vibronic state at a node, the DSS undercuts equilibrium concepts such as temperature. Consequently, the DSS can represent physical systems that are described by more than one temperature or that contain physical features that defy definitions of temperature. Using modified bootstrap sampling algorithms, the DSS depicted (1) shifts in distribution functions induced by external fields and temperature gradients, (2) field-dependent transitions from linear mobility to non-linear mobility, (3) saturation velocities, (4) non-exponential decay functions generated by multiple phonon scattering modes, and (5) charge separations and electric potentials generated by temperature gradients. Ensemble averages were sensitive to the structure of dispersion relations, to the energy of the system, and to quantum coupling strengths. Seebeck coefficients were sensitive to the features of the electronic and the vibrational band structures, and their associated coupling coefficients.

Heat Transport by Countercurrent Blood Vessels in the Presence of an Arbitrary Temperature Gradient

This paper presents a three-dimensional analysis of the temperature field around a pair of countercurrent arteries and veins embedded in an infinite tissue that has an arbitrary temperature gradient along the axes of the vessels. Asymptotic methods are used to show that such vessels are thermally similar to a highly conductive fiber in the same tissue. Expressions are developed for the effective radius and thermal conductivity of the fiber so that it conducts heat at the same rate that the artery and vein together convect heat and so that its local temperature equals the mean temperature of the vessels. This result allows vascular tissue to be viewed as a composite of conductive materials with highly conductive fibers replacing the convective effects of the vasculature. By characterizing the size and thermal conductivity of these fibers, well-established methods from the study of composites may be applied to determine when an effective conductive model is appropriate for the tissue and vasculature as a whole.

Generalized transport coefficients in a gas with large shear rate

We get a solution of the Bhatnagar-Gross-Krook (BGK) model kinetic equation by means of a perturbative expansion of a temperature gradient to study the transport properties in a gas with large shear rate. The irreversible fluxes are evaluated exactly to first order in the expansion for Maxwell molecules. The transport coefficients obtained are highly nonlinear functions of the shear rate. This dependence on shear rate is analysed and compared with previous results for several transport coefficients. Finally, we have found a solution for a simple model of constant collision frequency for which a large shear rate coexists with an arbitrary temperature gradient.

The stability of buoyancy-driven rolls aligned with a shear flow when the temperature gradient is nonlinear

The buoyancy driven instability of rolls aligned with a shear flow of a Boussinesq fluid is considered. It is supposed that the temperature varies not only with height but also in the direction of the flow thus allowing the vertical temperature profile to be nonlinear. Conditions are derived for the neglect of the horizontal temperature gradient in the perturbation equations. These are (i) that the vertical temperature gradient varies sufficiently slowly in the direction of the flow and (ii) that Pr h/d≫1, where Pr is the Prandtl number and h and d are the length scales for the vertical variation of velocity and temperature, respectively. Under these conditions, stability is governed by the equation for Bénard convection. The principle of exchange of stabilities is proved for an arbitrary temperature gradient profile, with isothermal, free-surface boundary conditions, and the equation is solved numerically and by an asymptotic method for a model thermal boundary layer. The analysis is then applied to an experimental study of spoke patterns, observed in the growth of electronic materials. This work also indicates that a previous investigation of the stability of stratified shear flow has a restricted range of validity.

Thermally driven motion of strongly heated fluids

A formal, unified method is used to derive two sets of approximate equations governing the thermally driven motion of strongly heated fluids. Account is taken of the temperature dependence of the fluid properties: The limits considered correspond to: (a) a strongly heated, shallow fluid and (b) a strongly heated, deep liquid with a small coefficient of volume expansion. Two additional limits that are considered correspond to: (c) a weakly heated, deep fluid; and (d) a weakly heated, shallow fluid. For cases (a), (c) and (d), use is made of two basic expansion parameters: the nondimensionalized depth ϵ of the fluid and the nondimensionalized strength δ′ of the heat source distribution. Case (a) corresponds to ϵ ⪡ 1; the resulting equations govern the temperature, density and pressure to lowest order. The lowest order pressure is uniform but time dependent. Case (c) corresponds to δ′ ⪡ 1. The unperturbed state for this case is taken to be steady and quiescent; this restricts it to the so-called adiabatic state. Case (d) is obtained from the results of case (c) by also taking ϵ ⪡ 1. The resulting equations constitute the Boussinesq approximation, and are valid independent of the relative magnitude of δ′ and ϵ. An arbitrary temperature gradient of order δ′ can be added to the unperturbed state of a weakly heated fluid. For the case of a weakly heated, shallow ideal gas maintained at constant volume and heated throughout the volume, the equations account for the effect of perturbation pressure on perturbation density and on compression heating. Case (b) yields an energy equation that is significantly different from that for case (d). When applied to a liquid under normal conditions, case (b) leads to the Boussinesq equations with temperature-dependent fluid properties. The sets of equations obtained all are based on a characteristic velocity that is vanishingly small compared with the speed of sound. As a result, the equations do not contain acoustic waves, and their numerical solution is no more difficult than that of incompressible flow equations.

Creep Buckling of Thin-Walled Circular Cylindrical Shells Subject to Radial Pressure and Thermal Gradients

A method of calculating the creep deflections and predicting the creep collapse of a thin-walled circular cylindrical shell, subject to uniform external radial pressure and arbitrary temperature gradients, is shown. This method may also be applied to investigate the behavior of a shell subject to time-dependent temperature gradients and deformations due to other inelastic causes. The set of simultaneous differential equations of equilibrium in terms of creep, thermal, and other inelastic strains is presented. Applications of this method to long cylindrical shells are illustrated by two numerical examples.

An exact general solution for the temperature distribution and the composite radiation convection heat exchange along a constant cross-sectional area fin

An exact general solution is obtained for the nonlinear differential equation governing the one-dimensional steady-state heat exchange by composite radiation and convection along constant area fins with uniform temperature at the base and an arbitrary temperature gradient at the other end. The fin can dissipate or receive energy. The fin is assumed to have constant thermal properties, and the radiant interaction between the fin and the base surfaces is neglected.

Neutron transport in differentially heated media

Calculations have been made of the average energy of neutrons diffusing in an infinite, non-absorbing moderator possessing an arbitrary temperature gradient. The results are based upon the heavy gas model of neutron scattering, but the exact neutron transport equation is employed; thus it has been possible to assess the accuracy of diffusion theory for this type of problem.