Energy Flux Vector Research Articles

Mechanical Work at a Boundary and its Associated Exergy Transfer

This paper is a didactic discussion of mechanical work at a boundary from the standpoint of exergy analysis. It is shown that the use of the material velocity vector and the mechanical stress tensor of continuum mechanics allows the magnitude and direction of the energy flux vector at a point to be determined unambiguously. Hence, the amount and direction of the energy transfer at a boundary can be established. This methodology is then extended to the calculation of the associated exergy flux vector and the exergy transfer at the boundary. It is noted that the exergy transfer is not always equal to the energy transfer as mechanical work. In general, the exergy transfer also involves a term consisting of the product of the environmental pressure and a volume displacement or transfer.

The range of existence of reflected homogeneous and surface waves in elastic orthotropic materials

Reflection of elastic waves from the free surface of orthotropic incompressible materials is examined under plane strain conditions in a material symmetry plane. Use of the physically allowable energy flux vectors in conjunction with the propagation condition and the generalized Snell’s law, yields formulas for the range of existence of the different types of reflected waves with respect to the two elastic parameters: the material orientation of the free surface and the angle of incidence. It is the consideration of the energy flux vectors that, in general, introduces the second elastic parameter and enlarges the range for the existence of either two homogeneous shear plane waves or one homogeneous and one supersonic surface wave. In the special case of transverse isotropy, the above formalas yield a homogeneous shear wave and a surface wave which diminishes when the angle of incidence is 45°, the wave-number vector coincides with the energy flux vector, and there enters only one elastic parameter.

Computation of Sea-Wave Direction of Propagation of Random Waves

In the last 20 years, the application of the random wave theory in maritime engineering has been given the utmost importance, especially for evaluating design wave conditions, loads on structures, and sand littoral drift. The aim of the paper is to present a general expression for the evaluation of the local energy flux vector of random waves. This expression can be used for computing a representative direction of propagation of the wave spectrum in the presence of any scattering source. The effectiveness of the presented expression is shown by a numerical experiment. Moreover the expression is compared with the relationship that is normally used in field observations to represent the mean wave direction of a two-dimensional spectrum. The results of the comparison, carried out with a measured directional wave spectrum, are discussed.

FURTHER PROPERTIES OF FINITEAMPLITUDE PLANE WAVES IN DEFORMED MOONEY-RIVLIN MATERIALS

In a previous paper, Currie and Hayes (J. Inst. Maths Applics 5 (1969) 140-161) showed that two linearly polarized finite-amplitude plane shear waves may propagate with constant speeds along any direction in a Mooney-Rivlin material which is maintained in a state of arbitrary static finite homogeneous deformation. Later, Boulanger and Hayes (Q. Jl Mech. appl. Math 45 (1992) 575-593), using special directions called 'acoustic axes', obtained explicit expressions for the polarization directions and the phase speeds of these waves in terms of the propagation direction.Here further properties of these waves are obtained. An energy-flux velocity vector is introduced and a duality between slowness and energy-flux velocity is exhibited. The slowness and ray surfaces are obtained and their singular points and singular tangent planes are studied. As in crystal optics and crystal acoustics, it follows that cones of internal and external conical refraction may be introduced.Also, it is shown that for a given direction of the energy-flux velocity ('ray direction'), two plane shear waves may propagate. Introducing new special directions called 'ray axes', explicit expressions are obtained for the various properties of these waves in terms of the ray direction. The duality between the results for a given propagation direction and for a given ray direction is clearly brought out.Finally, it is shown that a finite-amplitude plane shear wave is uniquely determined by its polarization direction, provided this direction is not along a principal axis of the basic static homogeneous deformation. Indeed all the properties of such a wave may be expressed in terms of its polarization direction. © 1995 Oxford University Press.

Stages in the energetics of baroclinic systems

The results from several idealized and case studies are drawn together to form a comprehensive picture of “downstream baroclinic evolution” using local energetics. This new viewpoint offers a complementary alternative to the more conventional descriptions of cyclone development. These additional insights are made possible largely because the local energetics approach permits one to define an energy flux vector which accurately describes the direction of energy dispersion and quantifies the role of neighboring systems in local development. In this view, the development of a system's energetics is divided into three stages. In Stage 1, a pre-existing disturbance well upstream of an incipient trough loses energy via ageostrophic geopotential fluxes directed downstream through the intervening ridge, generating a new energy center there. In Stage 2, this new energy center grows vigorously, at first due to the convergence of these fluxes, and later by baroclinic conversion as well. As the center matures, it begins to export energy via geopotential fluxes to the eastern side of the trough, initiating yet another energy center. In Stage 3, this new energy center continues to grow while that on the western side of the trough decays due to a dwinding supply of energy via fluxes from the older upstream system and also as a consequence of its own export of energy downstream. As the eastern energy center matures, it exports energy further downstream, and the sequence begins anew. The USA “Blizzard of'93” is used as a new case study to test the limits to which this conceptual sequence might apply, as well as to augment the current limited set of case studies. It is shown that, despite the extraordinary magnitude of the event, the evolution of the trough associated with the Blizzard fits the conceptual picture of downstream baroclinic evolution quite well, with geopotential fluxes playing a critical role in three respects. First, fluxes from an old, decaying system in the Pacific were convergent over the west coast of North America, creating a kinetic energy center there and modifying the jet, resulting in a large extension of the overall kinetic energy center well into Mexico. Second, energy fluxes from this extension of the northwesterly flow were strongly convergent east of the trough, producing explosive growth of kinetic energy over the northwestern Gulf of Mexico, with baroclinic conversion following shortly thereafter. Lastly, the kinetic energy generated by the vigorous baroclinic conversion in the cold advection on the west side of the trough was very effectively transferred to the energy center on the east side of the trough via geopotential fluxes. DOI: 10.1034/j.1600-0870.1995.00108.x

Fundamental Nonlinear Theory for Micropolar Electrically Conducting Fluids. Conservation Law, Nonequilibrium Thermodynamics, Peltier Effect.

In the present paper, the conservation laws for micropolar electrically conducting fluids with many small fine particles are discussed on the basis of a nonequilibrium thermodynamics relation which leads easily to precise expressions of the energy flux vector and the dissipation function of micropolar fluids. The thermodynamic pressure defined here is based on the local conservation laws of mass, momentum and entropy, which satisfies the mutually complementary relationship with the sum of kinetic energy and internal energy. As a result we obtain Euler's thermodynamic relation as a homogeneous function in the first order of entropy, density and pressure. The above discussion takes into account the three parts of nonpolar fluid, polar fluid and micropolar electrically conducting fluid.

Energy flux vector field in an ensonified elastodynamic sphere

Recent interest has been shown in the energy flux density inside an ensonified elastodynamic scatterer. A spherical model was chosen as the simplest and most general geometry to analyze. To properly model the energy flow in an elastodynamic medium, it is necessary to employ the elastodynamic equivalent of the Poynting vector. Plots of the vector field of the elastodynamic intensity are presented with an eye toward the classification of the type of resonance behavior displayed. Analogies with other resonance phenomena in related geometries suggest the possibility of fruitful comparison with resonance modes in shells and plates. [Work in this area supported by ONR/NRL and by ASEE/ONT Summer Fellowship Program.]

Finite-amplitude plane waves in deformed Hadamard elastic materials

SUMMARY It has been shown previously (John 1966; Currie & Hayes 1969) that Hadamard compressible elastic materials are the only ones for which three linearly polarized finite-amplitude plane waves, one longitudinal and two transverse, may propagate in any direction when the material is maintained in a state of arbitrary static finite homogeneous deformation. Here we first obtain simple explicit expressions for the three wave speeds and a simple characterization of the polarization directions of the transverse waves. Also, although the theory is non-linear, it is seen that the three waves may be superposed: they do not interact. Special directions, called acoustic axes, are introduced. These are the only directions along which circularly polarized transverse waves may propagate. They are determined only by the basic static deformation of the material. Results are expressed in terms of the angles that the propagation direction makes with the acoustic axes. Then, the energy flux and energy density of the waves are considered. Relations between the projection on to the propagation direction of the energy flux vector and the energy density are derived for each individual wave. Finally, periodic transverse waves are considered.

On the propagation of pure plane waves in anisotropic media

In this letter it is shown that, for plane longitudinal waves propagating in generally anisotropic media, any direction in which the polarization vector is collinear with the propagation direction is also a direction in which the energy flux vector is collinear with the propagation direction. For transverse waves, the condition that the polarization vector is perpendicular to the propagation direction is shown to imply that the energy flux vector is collinear with the propagation direction only if the anisotropic media satisfies certain implicit symmetry conditions. Possible applications of these results to the determination of the elastic moduli of anisotropic media are discussed.

Display of the flow of energy in molecules

AbstractIn this article, we develop a method to graphically display the flow of energy within molecules. An energy continuity equation is derived leading to a molecular energy flux vector field. Computation of the flux calls for the intramolecular potential, any external interactions, and the phase space trajectories of the molecular motion. The flux provides a means to display energy flow in still frames and as a tool to visualize hitherto undiscovered dynamic pathways in molecules. Examples are presented that show energy flow in three molecular systems and illustrate the point that depiction of energy flux patterns has increasing utility and meaning as one moves to larger molecules. Simple extensions to this work would also allow visualization of the flux of such quantities as linear and angular momentum. © 1994 by John Wiley & Sons, Inc.

Fundamental Nonlinear Theory for Micropolar Electrically Conducting Fluids. 3rd Report. Conservation Law, Nonequilibrium Thermodynamics, Peltier Effect.

In the present paper, the conservation laws for micropolar electrically conducting fluids with many small fine particles are discussed on the basis of a nonequilibrium thermodynamics relation which leads easily to precise expressions of the energy flux vector and the dissipation function of micropolar fluids. The thermodynamic pressure defined here is based on the local conservation laws of mass, momentum and entropy, which satisfies the mutually complemental relation with the sum of kinetic energy and internal energy. As a result we obtain Euler's thermodynamic relation as a homogeneous function in the first order of entropy, density and pressure. The above discussion takes into account the three parts of nonpolar fluid, polar fluid and micropolar electrically conducting fluid.

Aspects of energy propagation in highly anisotropic elastic solids

Anomalies in the theory of wave propagation in constrained materials may be reconciled with the standard theory of wave propagation in unconstrained materials by relaxing the constraint slightly and then taking the limit as the constraint is obeyed exactly. In this paper the same method is employed in an attempt to reconcile anomalies in the propagation of energy in a constrained material with the known propagation propertics for unconstrained materials. On relaxing the constraint in a singly constrained material, it is found that the energetics associated with two of the three propagating waves tend to the appropriate known forms for the corresponding constrained material in the limit where the constraint holds exactly. The third wave has no counterpart in the constrained theory and it is conjectured that both the total energy density and the energy flux vector tend to zero as the constrained limit is approached. This conjecture is shown to be true for two simple boundary value problems involving incompressible, and inextensible, elastic half-spaces.

Phase and group velocity considerations for dynamic modulus measurement in anisotropic media

In anisotropic media, the direction of energy propagation does not necessarily coincide with the wave normal, i.e. the energy flux vector does not coincide with the wave normal. Since, experimentally, one measures group velocity not phase velocity, one must therefore be careful in interpreting ultrasonic wave speed measurements in anisotropic media. This is of particular importance in elastic property reconstruction where acoustic velocity measurements are used as the basis for determining anisotropic material properties. In this work, the consequences of energy flux deviation from the wave normal are considered for typical experimental geometries. Particular attention is devoted to developing appropriate relationships between the phase velocity and ultrasonic transit time measurements, as these relations are most useful for elastic property reconstruction. In all the cases considered, it is shown that the phase velocity can be directly calculated from appropriate time delay measurements.

ENERGY-FLUX VECTOR OF SURFACE THERMOELASTIC WAVES

The energy-flux vector connected with the kinetic and internal energy of surface thermoelastic waves is considered. It is shown that this vector is not parallel to the surface of the half-space, as it is in classical elasticity. Its direction at a given point varies with the distance of this point from the surface. Its magnitude decays to 0 when this distance tends to infinity and decays exponentially to 0 in the direction of movement of the straight tine in which the planes of constant phase of the superposed waves intersect the surface of the half-space. Some numerical results are presented when the half space is made of copper and thermally insulated.

Energy flux densities of electromagnetic waves

AbstractIn time‐harmonic electromagnetic fields, the Poynting vector, the cross product of electric field with the complex conjugate of the magnetic field, is well known as an energy flux density vector. Here, a new type of vector is proposed in which the cross products of electric and magnetic fields with their corresponding complex conjugates are taken, and its relationship with the Poynting vector is established as a pair of Max‐wellian‐type equations. Moreover, the usefulness of this vector is demonstrated by the examples of optical theorem, reciprocity theorem and decomposition of radiated power into circularly polarized fields, where it supplements the Poynting vector. This vector, like the Poynting vector, can also be considered as the energy flux density vector of the electromagnetic wave.

A consistent algorithm for quasigeostrophic energy and energy flux analysis

The energy flux vector is familiar to geophysical fluid dynamics, but only in the abstract. Equations for computing the energy balance and the energy flux vector (S) for a quasigeostrophic numerical ocean model are provided in both analytic and finite difference form. The multilayer model includes wind, lateral and bottom friction, and bathymetry. The finite difference approximation to the analytic energy balance is consistent with the finite difference approximation to the quasigeostrophic potential vorticity equation. An application of the algorithms to a simple steady ocean gyre illustrates their use. This choice of the flux vector (from an infinity of possibilities) has several advantages: it is computationally convenient; it is capable of field measurement; it has nice mathematical properties in the limit of small amplitude motion; and it facilitates intuition about the dynamics.

Propagation of low frequency acoustic waves in periodic 2D-lattices of tubes

The problem of an acoustic, 2D-lattice of tubes is investigated. A “ladder” lattice is analyzed as a superposition of two 1D-lattices, symmetrical and antisymmetrical, respectively. Then infinite, complete 2D-lattices are analyzed by using the Green function method adapted from solid state physics. In the far field produced by a point source, the energy flux vector is found to have a radial direction, but not the wavevector, except on the main axes. In the horizontal or vertical directions, one finds from a recurrence relation a superposition of two waves (one is evanescent, and the other one is propagating or evanescent). Then a numerical computation of a finite lattice allows one to analyze the reflections on the boundaries. Finally, the agreement between experimental results on a small lattice and numerical results is found to be satisfactory.

Ricci collineation vectors in fluid space-times

The properties of fluid space-times that admit a Ricci collineation vector (RCV) parallel to the fluid unit four-velocity vector ua are briefly reviewed. These properties are expressed in terms of the kinematic quantities of the timelike congruence generated by ua. The cubic equation derived by Oliver and Davis [Ann. Inst. Henri Poincaré 30, 339 (1979)] for the equation of state p=p(μ) of a perfect fluid space-time that admits an RCV, which does not degenerate to a Killing vector, is solved for physically realistic fluids. Necessary and sufficient conditions for a fluid space-time to admit a spacelike RCV parallel to a unit vector na orthogonal to ua are derived in terms of the expansion, shear, and rotation of the spacelike congruence generated by na. Perfect fluid space-times are studied in detail and analogues of the results for timelike RCVs parallel to ua are obtained. Properties of imperfect fluid space-times for which the energy flux vector qa vanishes and na is a spacelike eigenvector of the anisotropic stress tensor πab are derived. Fluid space-times with anisotropic pressure are discussed as a special case of imperfect fluid space-times for which na is an eigenvector of πab.

Reflection and transmission of elastic waves from a fluid-saturated porous solid boundary

Based on the classical work of Biot [J. Acoust. Soc. Am. 28, 168 (1956)] which predicts that three different kinds of bulk waves may propagate in the fluid-saturated porous solid, the wave equation is solved to determine the energy reflection and transmission coefficients of plane elastic waves at oblique incidence on an interface between a fluid and a fluid-saturated porous solid. For this purpose, the necessary formalism of the energy equation, the Poynting energy flux vector, and the sound intensity of elastic waves in fluid-saturated porous media are presented. Two general cases of mode conversion have been investigated: (1) The initial wave is incident from the fluid to the interface and generates three transmitted bulk waves in the fluid-saturated porous solid, and (2) the initial wave is incident from the fluid-saturated porous solid to the interface and generates three reflected bulk waves in the same medium. Furthermore, the transmission of sound through a fluid-saturated porous solid plate immersed in fluid is calculated. Measurement of sound transmission through a porous plate was carried out using experimental techniques suggested by Plona [Appl. Phys. Lett. 36, 259 (1980)]. Good correlation between measured and calculated values of the angular behavior of the transmission coefficients for fast, shear, and slow waves was obtained after adjusting the theory for experimentally obtained attenuation.

Three-dimensional nonequilibrium interface conditions for electron transport at band edge discontinuities

The author addresses the problem of boundary conditions for electron transport at interfaces of different material species from a fundamental microscopic point of view in order to derive self-consistent boundary conditions for various transport models and to help select a suitable model for the consideration of interface effects. A three-dimensional nonequilibrium interface condition for the Boltzmann equation at heterojunctions is set up, where the heterojunction is modeled by a band-edge discontinuity. From this interface condition, respective conditions for the moments of the distribution function are derived. Application to a selected transport model results in conditions stating continuity of quasi-Fermi level, electron temperature, and the normal component of the particle current density, and yields relations between the tangential components of the particle- and energy-flux vectors on both sides of the interface. These interface conditions enable numerical simulations of hot electron transport to be carried out at abrupt interfaces of different material species.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>