Internal Energy Density Research Articles

Polylogarithmic Representation of Radiative and Thermodynamic Properties of Thermal Radiation in a Given Spectral Range: II. Real-Body Radiation

The general analytical expressions for the thermal radiative and thermodynamic properties of a real-body are obtained in a finite range of frequencies at different temperatures. The frequency dependence of the spectral emissivity is represented as a power series. The Stefan-Boltzmann law, total energy density, number density of photons, Helmholtz free energy density, internal energy density, enthalpy density, entropy density, heat capacity at constant volume, pressure, and total emissivity are expressed in terms of the polylogarithm functions. The general expressions for the thermal radiative and thermodynamic functions are applied for the study of thermal radiation of liquid and solid zirconium carbide. These functions are calculated using experimental data for the frequency dependence of the normal spectral emissivity in the visible-near infrared range at the melting (freezing) point. The gaps between the thermal radiative and thermodynamic functions of liquid and solid zirconium carbide are observed. The general analytical expressions obtained can easily be presented in wavenumber domain.

A reformulated flexoelectric theory for isotropic dielectrics

In flexoelectricity, a strain gradient can induce polarization and a polarization gradient can induce mechanical stress. In this paper, in order to identify the contributions of each strain gradient component, the flexoelectric theory is reformulated by splitting the strain gradient tensor into mutually independent parts. Two sets of orthogonal higher-order deformation metrics are inherited and perfected to reformulate the internal energy density for isotropic materials. The deviatoric stretch gradient and the symmetric part of the rotation gradient are proved to disappear in the coupling of strain gradient to polarization and, moreover, the independent higher-order constants associated with the coupling of strain gradient to strain gradient reduce from five to three. The constitutive relations are then reformulated in terms of the new deformation and electric field metrics, and the governing equations and boundary conditions are derived according to the variational principle of electric enthalpy. On the basis of the present simplified flexoelectric theory, a flexoelectric Bernoulli–Euler beam theory is specified. Solutions for a cantilever subjected to a force at the free end and a voltage cross the thickness are constructed and the size-dependent direct and inverse flexoelectric effects are captured.

Wave-number spectrum of dissipative drift waves and a transition scale.

We study the steady state spectrum of the Hasegawa-Wakatani (HW) equations that describe drift wave turbulence. Beyond a critical scale k_{c}, which appears as a balance between the nonlinear time and the parallel conduction time, the adiabatic electron response breaks down nonlinearly and an internal energy density spectrum of the form F(k_{⊥})∝k_{⊥}^{-3}, associated with the background gradient, is established. More generally a dual power law spectrum, approximately of the form F(k_{⊥})∝k_{⊥}^{-3}(k_{c}^{-2}+k_{⊥}^{-2}) is obtained, which captures this transition. Using dimensional analysis, an expression of the form k_{c}∝C/κ is derived for the transition scale, where C and κ are normalized parameters of the HW equations signifying the electron adiabaticity and the density gradient, respectively. The results are numerically confirmed using a shell model developed and used for the Hasegawa-Wakatani system.

Variations on an aethereal theme

We consider a class of Lorentz-violating theories of gravity involving a timelike unit vector field (the aether) coupled to a metric, two examples being Einstein-aether theory and Ho\v{r}ava gravity. The action always includes the Ricci scalar of the metric and the invariants quadratic in covariant derivatives of the aether, but the theories differ in how the aether is constructed from other fields, and whether those fields are varied in the action. Fields that are not varied define background structures breaking diffeomorphsim invariance, including threadings, folations, and clocks, which generally produce novel degrees of freedom arising from the violation of what would otherwise be initial value constraints. The principal aims of this paper are to survey the nature of the theories that arise, and to understand the consequences of breaking diffeomorphism invariance in this setting. In a companion paper [arXiv:1504.03305], we address some of the phenomenology of the "ponderable aether" case in which the presence of a background clock endows the aether with a variable internal energy density that behaves in some respects like dark matter.

A model for elastic flexoelectric materials including strain gradient effects

A constitutive model for elastic flexoelectric materials under small deformation based on second gradient continuum theory is developed, using a Toupin-like variational formulation to simultaneously obtain constitutive relations, balance equations and boundary conditions. The model includes three different electromechanical “stresses”: a higher-order stress, an extended local electric force and a generalized Cauchy stress tensor. The constitutive equations of the model are obtained by postulating an internal energy density function which depends on both the strain and its gradient as well as the polarization. Finally, as an application of the model, we derive the explicit analytical expressions of the polarization and displacement vector fields for the problem of the polarization induced over a thin spherical shell subjected to hydrostatic loading conditions.

Polylogarithmic Representation of Radiative and Thermodynamic Properties of Thermal Radiation in a Given Spectral Range: I. Blackbody Radiation

There are several classes of materials and space objects for which the frequency dependence of the spectral emissivity is represented as a power series. Therefore, the study of the properties of thermal radiation for these real bodies is an important task for both fundamental science and industrial applications. The general analytical expressions for the thermal radiative and thermodynamic functions of a real body are obtained in a finite range of frequencies at different temperatures. The Stefan–Boltzmann law, total energy density, number density of photons, Helmholtz free energy density, internal energy density, enthalpy density, entropy density, heat capacity at constant volume, pressure, and total emissivity are expressed in terms of the polylogarithm functions. The obtained general expressions for the thermal radiative and thermodynamic functions are applied for the study of thermal radiation of liquid and solid zirconium carbide. These functions are calculated using experimental data for the frequency dependence of the normal spectral emissivity in the visible and near-infrared range at the melting (freezing) point. The gaps between the thermal radiative and thermodynamic functions of liquid and solid zirconium carbide are observed. The general analytical expressions obtained can easily be presented in the wavenumber domain.

A scheme to produce high density and high temperature plasma for opacity measurement

The opacity of shock-compressed material is of general scientific interest for astrophysical plasmas and for inertial confinement fusion research. A proposal is suggested to produce high temperature plasma with density around 1 g/cm−3. Two types of opacity target (the sandwich target and the foam enhanced sandwich target) are investigated numerically. The foam enhanced sandwich target has structure of foam–solid-sample-solid-foam. The foam will increase laser absorption efficiency and the ablating pressure. Hydrodynamic simulations confirm that the laser can be fully absorbed by the under-critical-density foam and a faster shock is produced inside the CH layer. High intensity lasers heat opacity target from both sides. The CH layers must be thick enough to keep the laser away from the sample. The laser-driven shocks move inward and collide at the center. Part of their kinetic energy is converted into internal energy and high density and high temperature local thermodynamic equilibrium sample plasma is produced. The plasma produced by laser heating the foam enhanced sandwich target has higher sample temperature than by laser heating the sandwich target. It may be useful for measuring the opacity of shock compressed material in laboratory.

The development of thermal lattice Boltzmann models in incompressible limit

In this paper, an incompressible two-dimensional (2-D) and three-dimensional (3-D) thermohydrodynamics for thelattice Boltzmann scheme are developed. The basic idea is to solve the velocity field and the temperature field usingtwo different distribution functions. A derivation of the lattice Boltzmann scheme from the continuous Boltzmannequation for 2-D is discussed in detail. By using the same procedure as in the derivation of the discretised densitydistribution function, we found that new lattice of four-velocity (2-D) and eight-velocity (3-D) models for internal energydensity distribution function can be developed where the viscous and compressive heating effects are negligible.These models are validated by the numerical simulation of the 2-D porous plate Couette flow problem where theanalytical solution exists and the natural convection flows in a cubic cavity.

Transient response of stiffened composite submersible hull to underwater explosion bubble

This paper deals with the transient response of a glass–epoxy composite submersible hull subjected to underwater explosive bubble. The boundary-element method (BEM) is used to simulate the physical process of the explosion bubble growth, contraction and collapse while the finite element method (FEM) is used to calculate the glass–epoxy composite structure response to the high pressure induced by the underwater explosion bubble. The coupled BEM–FEM is used to handle the interaction of the composite structures and the underwater explosion bubble. With the coupled code, the mutual effects of relative location between the bubble and the composite submersible hull are investigated; and also the transient responses (such as stresses and internal energy densities) of the composite submersible hull to the underwater explosive bubble can be predicted for different charge weights and charge distances. From the results obtained, some insights to the dynamical problem of the interaction between underwater explosion bubbles and composite structures are deduced.

Mixed convection in inclined lid driven cavity by Lattice Boltzmann Method and heat flux boundary condition

Laminar mixed convective heat transfer in two-dimensional rectangular inclined driven cavity is studied numerically by means of a double population thermal Lattice Boltzmann method. Through the top moving lid the heat flux enters the cavity whereas it leaves the system through the bottom wall; side walls are adiabatic. The counter-slip internal energy density boundary condition, able to simulate an imposed non zero heat flux at the wall, is applied, in order to demonstrate that it can be effectively used to simulate heat transfer phenomena also in case of moving walls. Results are analyzed over a range of the Richardson numbers and tilting angles of the enclosure, encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. As expected, heat transfer rate increases as increases the inclination angle, but this effect is significant for higher Richardson numbers, when buoyancy forces dominate the problem; for horizontal cavity, average Nusselt number decreases with the increase of Richardson number because of the stratified field configuration.

Lattice Boltzmann method with heat flux boundary condition applied to mixed convection in inclined lid driven cavity

Mixed convection in a inclined cavity has not been investigated by LBM in case of an imposed non zero heat flux. This type of boundary condition, representing very usual situations in physical world, is not simple to model in lattice Boltzmann schemes. In effect, the only boundary condition able to simulate an imposed temperature and an imposed heat flux at a boundary has been presented by D’Orazio et al. in previous works, where the boundary was at rest. In this work, laminar mixed convective heat transfer in two-dimensional rectangular inclined driven cavity is studied numerically by means of a double population thermal Lattice Boltzmann method. The counter-slip internal energy density boundary condition, able to simulate an imposed heat flux at the wall, is applied. Through the top moving lid the heat flux enters the cavity and it leaves the system through the bottom wall; side walls are adiabatic. Results are analyzed over a range of the Richardson numbers and tilting angles of the enclosure, encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. The results show that, as expected, heat transfer rate increases as increases the inclination angle, but this effect is significant for higher Richardson numbers, when buoyancy forces dominate the problem; for horizontal cavity, average Nusselt number decreases with the increase of Richardson number because of the stratified field configuration. This study shows that the counter-slip internal energy density boundary condition can be effectively used to simulate heat transfer phenomena also in case of moving walls and it makes the Lattice Boltzmann Method able to simulate a wide class of cooling process where a given thermal power must be removed.

Non-linear behaviours in complex fluid dynamics via non-differentiability. Separation control of the solid components from heterogeneous mixtures

In the framework of the Scale Relativity Theory, non-linear effects in complex fluids and, implicitly, the separation control of the solid components from heterogeneous mixtures are analysed. Assuming that the movements of the complex fluid entities occur on continuous but non-differentiable curves, the specific momentum and the local energy density conservation law equations are obtained. For potential movements at a fractal scale, as well as for non-potential ones at a differentiable scale, the non-differentiable hydrodynamic model is established. In such context, for different curve motions at various resolution scales, the non-differentiable hydrodynamic model is reduced to either a quantum hydrodynamic model, a standard hydrodynamic model, a random walk model, or a Stokes model. By numerical simulations using the non-differentiable hydrodynamic equations and the internal energy density conservation law with adequate initial and boundary conditions, some non-linear effects are obtained. Moreover, eliminating the time between the viscosity stress tension type and temperature field, for various given positions, thermal hysteresis type effects can be obtained. For each of the models mentioned above, the separation control processes are discussed.

Topology Optimization for Polymeric Foam Shock-Absorbing Structure Using Hybrid Cellular Automata

Foam shock-absorbing structures such as cushioned packages are often utilized to protect various products from mechanical shock and vibration during transportation. The goal of packaging design engineers is to design a cushioned package structure that improves the shock-absorbing performance and minimizes the volume of the package. Some optimization techniques, combined with computational simulation, provide engineers with a way to design an optimal structure. In this paper, we propose a modified topology optimization method suitable for a polymeric foam shock-absorbing structure under dynamic drop loads in multiple directions. Our approach uses a heuristic topology optimization method, known as the Hybrid Cellular Automata (HCA). The HCA algorithm uniformly distributes internal energy density and controls the relative density of Cellular Automata (CAs) making up the design space. This allows the algorithm to maintain or increase the performance of shock absorption and to decrease the amount of material. In particular, this paper presents a modified Solid IsotropicMaterial with Penalization (SIMP) model for foam materials, which parameterizes the design region and interpolates the material properties. We attempt to optimize a simple bottom-cushioned package for a refrigerator by using the proposed foam SIMP model with commercial software: LS-DYNA for drop dynamic simulation and LS-OPT/Topology for the HCA algorithm. Drop simulation and topology optimization are performed considering multiple drop-directions. As a result, our method removes elements that are not related to the shock-absorption performance and provide an optimal cushioning structure using foam material.

Sound propagation in a model ocean with internal waves: experiments and simulations

We conduct laboratory experiments and numerical simulations in a laboratory tank where sound waves and internal gravity waves are simultaneously generated and detected. The tank contains a density-stratified fluid with internal gravity waves produced by a wave generator. Acoustic sources and receivers are arranged such that the acoustic track crosses the internal wave beam. The internal wave energy density, energy flux, and total power are determined from particle imaging velocimetry measurements. The internal wave field is also computed in direct numerical simulations of the Navier-Stokes equations, using a parallelized finite-volume based solver code. The fluctuations in sound speed and intensity are determined as a function of the acoustic track location and the internal wave amplitude, frequency, phase, and modulation frequency. This research is designed to achieve a better understanding of sound propagation and scattering by internal waves in the oceans. [This research was supported by ONR MURI Grant N000141110701 (WHOI). L.Z. acknowledges the support of the 2013-14 ASA F. V. Hunt Postdoctoral Research Fellowship.]

Lattice Boltzmann Simulation of Two- and Three-Dimensional Incompressible Thermal Flows

This work is concerned with the application of the thermal lattice Boltzmann method (TLBM) to compute incompressible two- and three-dimensional flows in cavities. Two convection test cases, namely, the laminar flow in a differentially heated square cavity and a cubic cavity, are numerically analyzed through TLBM. The internal energy density distribution function approach with two three-dimensional particle velocity models, namely, the 15-velocity and the 19-velocity, and a two-dimensional model, namely, the nine-velocity, have been used in the present work. Computations are carried out for laminar flows in a differentially heated square cavity and a cubical cavity (Rayleigh numbers = 103 to 105). The boundary conditions used are stable and of good accuracy. To lend credibility to the thermal lattice Boltzmann model square cavity results, they are further compared with those obtained from a finite-difference-based code developed for this purpose.

The particle content of low-power radio galaxies in groups and clusters

The synchrotron-radiating particles and magnetic fields in low-power radio galaxies (including most nearby cluster-centre sources), if at equipartition, can provide only a small fraction of the total internal energy density of the radio lobes or plumes, which is now well constrained via X-ray observations of their external environments. We consider the constraints on models for the dominant energy contribution in low-power radio-galaxy lobes obtained from a detailed comparison of how the internal equipartition pressure and external pressure measured from X-ray observations evolve with distance for two radio galaxies, 3C 31 and Hydra A. We rule out relativistic-lepton dominance of the radio lobes, and conclude that models in which magnetic field or relativistic protons/ions carried up the jet dominate lobe energetics are unlikely. Finally, we argue that entrainment of material from the jet surroundings can provide the necessary pressure, and construct a simple self-consistent model of the evolution of the entrainment rate required for pressure balance along the 100-kpc scale plumes of 3C 31. Such a model requires that the entrained material is heated to temperatures substantially above that of the surrounding intra-group or cluster medium, and that the temperature of the thermal component of the jet increases with distance, though remaining sub-relativistic.

Numerical Simulation to Study Riemann Problem for Euler Equations in Case of Two Supersonic Flows Collide

In this work we run numerical simulation of gasdynamic to study Riemann problem for the case of collision between two supersonic flows. We analyzed the profiles of density, velocity, pressure, furthermore we inspect for the late time the Mach number which is represent the shock strengths, the specific internal energy density, the total energy density, and the entropy related quantity. Our results reveal that the collision between two supersonic flows leads to the formation of two shocks separated by a contact discontinuity, as well as the numerical results for Euler equations and the exact solution are close to each other.

On the effective stress for unsaturated soils with residual water

In the present work an expression describing the rate of internal energy density for unsaturated soils is derived. The main difference here, with respect to preceding works, lies in the fact that the free and residual water are treated as two different phases from the soil. The latter presents intrinsic velocity equal to that of the solid phase, and is able to exchange mass with the free water phase. Under these conditions, a Bishop-type effective stress is identified as the energy-conjugated variable with the solid deformation rate, and presents a Bishop parameter equal to the effective degree of saturation. Furthermore, the negative rate of effective degree of saturation is energy-conjugated with the suction times the effective porosity. Other contributors to the internal energy come from the constituent compressibilities, the relative flow and the mass exchange between the free and residual water.

On spatial evolution of the solution of a non-standard problem in the bending theory of elastic plates

In this paper we consider the bending theory of Mindlin-type elastic plates. We study the spatial evolution of the solution of a non-standard problem. This problem is characterized by some constraint conditions in which the displacement and velocity at a given time are proportional with their respective initial values. The study is organized in two parts: in the first part we consider the materials with internal energy density positive definite and in the end we study the strongly elliptic materials. Using numerical analysis for some materials, we compare the different Saint-Venant’s type decay rates obtained.

Internal energy density of the critical three-state Potts model on the kagome lattice

The internal energy density of the Potts model on a semi-infinite strip with a width L has been conjectured to have no finite-size corrections at the critical point K = K c . By factorizing the transfer matrix for the kagome lattice with larger widths, we have found that this conjecture is not correct in that the internal energy density varies slightly with L at the critical point. From this size dependence of the internal energy density, we obtain an upper bound as K c < 1.0565615, which is close to a recent estimate K JS = 1.0565600(7) by Jacobsen and Scullard [J. Phys. A 45, 494003 (2012)]. We also obtain a lower bound as K c > 1.0560 by calculating the correlation length along the strips.