Velocity resolved - scalar modeled simulations of high Schmidt number turbulent transport

The objective of this thesis is to develop a framework to conduct velocity resolved - scalar modeled (VR-SM) simulations, which will enable accurate simulations at higher Reynolds and Schmidt (Sc) numbers than are currently feasible. The framework established will serve as a first step to enable future simulation studies for practical applications. To achieve this goal, in-depth analyses of the physical, numerical, and modeling aspects related to Sc>>1 are presented, specifically when modeling in the viscous-convective subrange. Transport characteristics are scrutinized by examining scalar-velocity Fourier mode interactions in Direct Numerical Simulation (DNS) datasets and suggest that scalar modes in the viscous-convective subrange do not directly affect large-scale transport for high Sc. Further observations confirm that discretization errors inherent in numerical schemes can be sufficiently large to wipe out any meaningful contribution from subfilter models. This provides strong incentive to develop more effective numerical schemes to support high Sc simulations. To lower numerical dissipation while maintaining physically and mathematically appropriate scalar bounds during the convection step, a novel method of enforcing bounds is formulated, specifically for use with cubic Hermite polynomials. Boundedness of the scalar being transported is effected by applying derivative limiting techniques, and physically plausible single sub-cell extrema are allowed to exist to help minimize numerical dissipation. The proposed bounding algorithm results in significant performance gain in DNS of turbulent mixing layers and of homogeneous isotropic turbulence. Next, the combined physical/mathematical behavior of the subfilter scalar-flux vector is analyzed in homogeneous isotropic turbulence, by examining vector orientation in the strain-rate eigenframe. The results indicate no discernible dependence on the modeled scalar field, and lead to the identification of the tensor-diffusivity model as a good representation of the subfilter flux. Velocity resolved - scalar modeled simulations of homogeneous isotropic turbulence are conducted to confirm the behavior theorized in these a priori analyses, and suggest that the tensor-diffusivity model is ideal for use in the viscous-convective subrange. Simulations of a turbulent mixing layer are also discussed, with the partial objective of analyzing Schmidt number dependence of a variety of scalar statistics. Large-scale statistics are confirmed to be relatively independent of the Schmidt number for Sc>>1, which is explained by the dominance of subfilter dissipation over resolved molecular dissipation in the simulations. Overall, the VR-SM framework presented is quite effective in predicting large-scale transport characteristics of high Schmidt number scalars, however, it is determined that prediction of subfilter quantities would entail additional modeling intended specifically for this purpose. The VR-SM simulations presented in this thesis provide us with the opportunity to overlap with experimental studies, while at the same time creating an assortment of baseline datasets for future validation of LES models, thereby satisfying the objectives outlined for this work.

Microstructure control and iodine doping of bismuth telluride

On the materials scale, thermoelectric efficiency is defined by the dimensionless figure of merit zT. This value is made up of three material components in the form zT = Tα²/ρκ, where α is the Seebeck coefficient, ρ is the electrical resistivity, and κ is the total thermal conductivity. Therefore, in order to improve zT would require the reduction of κ and ρ while increasing α. However due to the inter-relation of the electrical and thermal properties of materials, typical routes to thermoelectric enhancement come in one of two forms. The first is to isolate the electronic properties and increase α without negatively affecting ρ. Techniques like electron filtering, quantum confinement, and density of states distortions have been proposed to enhance the Seebeck coefficient in thermoelectric materials. However, it has been difficult to prove the efficacy of these techniques. More recently efforts to manipulate the band degeneracy in semiconductors has been explored as a means to enhance α.

The other route to thermoelectric enhancement is through minimizing the thermal conductivity, κ. More specifically, thermal conductivity can be broken into two parts, an electronic and lattice term, κ_e and κ_l respectively. From a functional materials standpoint, the reduction in lattice thermal conductivity should have a minimal effect on the electronic properties. Most routes incorporate techniques that focus on the reduction of the lattice thermal conductivity. The components that make up κ_l (κ_l = 1/3Cνl) are the heat capacity (C), phonon group velocity (ν), and phonon mean free path (l). Since the difficulty is extreme in altering the heat capacity and group velocity, the phonon mean free path is most often the source of reduction.

Past routes to decreasing the phonon mean free path has been by alloying and grain size reduction. However, in these techniques the electron mobility is often negatively affected because in alloying any perturbation to the periodic potential can cause additional adverse carrier scattering. Grain size reduction has been another successful route to enhancing zT because of the significant difference in electron and phonon mean free paths. However, grain size reduction is erratic in anisotropic materials due to the orientation dependent transport properties. However, microstructure formation in both equilibrium and nonequilibrium processing routines can be used to effectively reduce the phonon mean free path as a route to enhance the figure of merit.

This work starts with a discussion of several different deliberate microstructure varieties. Control of the morphology and finally structure size and spacing is discussed at length. Since the material example used throughout this thesis is anisotropic a short primer on zone melting is presented as an effective route to growing homogeneous and oriented polycrystalline material. The resulting microstructure formation and control is presented specifically in the case of In₂Te₃-Bi₂Te₃ composites and the transport properties pertinent to thermoelectric materials is presented. Finally, the transport and discussion of iodine doped Bi₂Te₃ is presented as a re-evaluation of the literature data and what is known today.

Análisis del comportamiento de diferentes tecnologías de aerogeneradores ante ráfagas de viento y huecos de tensión

[ES]Ante el desarrollo que está experimentando la energía eólica en la actualidad, suponiendo un porcentaje cada vez más elevado de la totalidad de la potencia instalada, la adecuación de los requisitos técnicos que ha de cumplir esta tecnología para minimizar su impacto sobre la red eléctrica se ha convertido en una prioridad. Para aprovechar al máximo los beneficios medioambientales que presenta el aprovechamiento del viento como fuente de energía, se deben estudiar alternativas para mejorar su comportamiento ante ráfagas de viento y huecos de tensión. En este documento se incluye una breve explicación del funcionamiento de las tres diferentes tecnologías de aerogeneradores, así como los problemas que derivan de la aparición de huecos de tensión y el comportamiento que presentan ante variaciones en la velocidad de viento. Se comprueba, a su vez, el modo de operación y la eficacia de las posibles soluciones que existen en la actualidad para minimizar el impacto que los inconvenientes de la energía eólica tienen sobre la red y la calidad del suministro eléctrico mediante la realización de simulaciones.

Radar methods for the exploration of glaciers

The problem of finding the depths of glaciers and the current methods are discussed briefly. Radar methods are suggested as a possible improvement for, or adjunct to, seismic and gravity survey methods. The feasibility of propagating electromagnetic waves in ice and the maximum range to be expected are then investigated theoretically with the aid of experimental data on the dielectric properties of ice. It is found that the maximum expected range is great enough to measure the depth of many glaciers at the lower radar frequencies if there is not too much liquid water present. Greater ranges can be attained by going to lower frequencies.

The results are given of two expeditions in two different years to the Seward Glacier in the Yukon Territory. Experiments were conducted on a small valley glacier whose depth was determined by seismic sounding. Many echoes were received but their identification was uncertain. Using the best echoes, a profile was obtained each year, but they were not in exact agreement with each other. It could not be definitely established that echoes had been received from bedrock. Agreement with seismic methods for a considerable number of glaciers would have to be obtained before radar methods could be relied upon. The presence of liquid water in the ice is believed to be one of the greatest obstacles. Besides increasing the attenuation and possibly reflecting energy, it makes it impossible to predict the velocity of propagation. The equipment used was far from adequate for such purposes, so many of the difficulties could be attributed to this. Partly because of this, and the fact that there are glaciers with very little liquid water present, radar methods are believed to be worthy of further research for the exploration of glaciers.

A report on increasing the thrust of a turbojet engine by combustion of fuel in the tail pipe

The purpose of this work was to develop a means of increasing the thrust of a turbojet engine by burning kerosene in the tail pipe.

A combustion system was developed which gave the following results:
(l) Maximum thrust increase using a G.E. I-14 engine was 64 per cent over straight tail pipe thrust corresponding to 42 per cent increase over the normal engine thrust. This increase was accomplished at an engine rpm of 12,000.
(2) Increase of maximum thrust obtained was 51 per cent over the straight tail pipe thrust corresponding to 23 per cent over the normal engine thrust. This increase was accomplished at an engine rpm of l6,000.
(3) For the thrust increases mentioned in (1) and (2) above, increases of Specific Fuel Consumption were 66 per cent and 76 per cent respectively over normal engine SFC.

Phase effects on group velocity propagation in a V-type system with spontaneously generated coherence

The effects of the relative phase between two laser beams on the propagation of a weak electromagnetic pulse are investigated in a V-type system with spontaneously generated coherence (SGC). Due to the relative phase, the subluminal and superluminal group velocity can be unified. Meanwhile, SGC can be regarded as a knob to manipulate light propagation between subluminal and superluminal.

Upper level and phase controlling group velocity in V-type system

With the external field coupling the two upper levels, we investigate the light pulse propagation properties with weak probe field in a V-type system. Due to the external upper level (UL) coupling field, the dispersion of the system has been influenced by the relative phase. It is shown that the UL field and the relative phase can be regarded as switches to manipulate light propagation between subluminal and superluminal. (c) 2007 Elsevier B.V. All rights reserved.

Controlling the group velocity in a five-level K-type atomic system

The properties of a five-level K-type system are investigated. With the controlling fields, the properties of the dispersion and absorption of the system are changed greatly. The system can produce anomalous dispersion regions with absorption and normal dispersion regions with absorption or transparency. Furthermore, the group velocity can be varied from subluminal to superluminal by varying the intensity of the controlling field and the probe detunings in principle. (C) 2008 Elsevier B.V. All rights reserved.

Ultra-slow atomic beam generation by velocity selective resonance

We describe a method to generate an ultra-slow atomic beam by velocity selective resonance (VSR). A VSR experiment on a metastable helium beam in a magnetic field is presented and the results show that the transverse velocity of the defected beam can be cooled and precisely controlled to less than the recoil velocity, depending on the magnitude of the magnetic field. We extend this idea to a cold atomic cloud to produce an ultra-slow Rb-87 beam that can be used as a source of an atomic fountain clock or a space clock.

I. The Superstition Hills, California, earthquakes of 24 November 1987. II. Three-dimensional velocity structure of southern California.

Part 1 of this thesis is about the 24 November, 1987, Superstition Hills earthquakes. The Superstition Hills earthquakes occurred in the western Imperial Valley in southern California. The earthquakes took place on a conjugate fault system consisting of the northwest-striking right-lateral Superstition Hills fault and a previously unknown Elmore Ranch fault, a northeast-striking left-lateral structure defined by surface rupture and a lineation of hypocenters. The earthquake sequence consisted of foreshocks, the M_s 6.2 first main shock, and aftershocks on the Elmore Ranch fault followed by the M_s 6.6 second main shock and aftershocks on the Superstition Hills fault. There was dramatic surface rupture along the Superstition Hills fault in three segments: the northern segment, the southern segment, and the Wienert fault.

In Chapter 2, M_L≥4.0 earthquakes from 1945 to 1971 that have Caltech catalog locations near the 1987 sequence are relocated. It is found that none of the relocated earthquakes occur on the southern segment of the Superstition Hills fault and many occur at the intersection of the Superstition Hills and Elmore Ranch faults. Also, some other northeast-striking faults may have been active during that time.

Chapter 3 discusses the Superstition Hills earthquake sequence using data from the Caltech-U.S.G.S. southern California seismic array. The earthquakes are relocated and their distribution correlated to the type and arrangement of the basement rocks. The larger earthquakes occur only where continental crystalline basement rocks are present. The northern segment of the Superstition Hills fault has more aftershocks than the southern segment.

An inversion of long period teleseismic data of the second mainshock of the 1987 sequence, along the Superstition Hills fault, is done in Chapter 4. Most of the long period seismic energy seen teleseismically is radiated from the southern segment of the Superstition Hills fault. The fault dip is near vertical along the northern segment of the fault and steeply southwest dipping along the southern segment of the fault.

Chapter 5 is a field study of slip and afterslip measurements made along the Superstition Hills fault following the second mainshock. Slip and afterslip measurements were started only two hours after the earthquake. In some locations, afterslip more than doubled the coseismic slip. The northern and southern segments of the Superstition Hills fault differ in the proportion of coseismic and postseismic slip to the total slip.

The northern segment of the Superstition Hills fault had more aftershocks, more historic earthquakes, released less teleseismic energy, and had a smaller proportion of afterslip to total slip than the southern segment. The boundary between the two segments lies at a step in the basement that separates a deeper metasedimentary basement to the south from a shallower crystalline basement to the north.

Part 2 of the thesis deals with the three-dimensional velocity structure of southern California. In Chapter 7, an a priori three-dimensional crustal velocity model is constructed by partitioning southern California into geologic provinces, with each province having a consistent one-dimensional velocity structure. The one-dimensional velocity structures of each region were then assembled into a three-dimensional model. The three-dimension model was calibrated by forward modeling of explosion travel times.

In Chapter 8, the three-dimensional velocity model is used to locate earthquakes. For about 1000 earthquakes relocated in the Los Angeles basin, the three-dimensional model has a variance of the the travel time residuals 47 per cent less than the catalog locations found using a standard one-dimensional velocity model. Other than the 1987 Whittier earthquake sequence, little correspondence is seen between these earthquake locations and elements of a recent structural cross section of the Los Angeles basin. The Whittier sequence involved rupture of a north dipping thrust fault bounded on at least one side by a strike-slip fault. The 1988 Pasadena earthquake was deep left-lateral event on the Raymond fault. The 1989 Montebello earthquake was a thrust event on a structure similar to that on which the Whittier earthquake occurred. The 1989 Malibu earthquake was a thrust or oblique slip event adjacent to the 1979 Malibu earthquake.

At least two of the largest recent thrust earthquakes (San Fernando and Whittier) in the Los Angeles basin have had the extent of their thrust plane ruptures limited by strike-slip faults. This suggests that the buried thrust faults underlying the Los Angeles basin are segmented by strike-slip faults.

Earthquake and explosion travel times are inverted for the three-dimensional velocity structure of southern California in Chapter 9. The inversion reduced the variance of the travel time residuals by 47 per cent compared to the starting model, a reparameterized version of the forward model of Chapter 7. The Los Angeles basin is well resolved, with seismically slow sediments atop a crust of granitic velocities. Moho depth is between 26 and 32 km.

The Equation of State and Petrogenesis of Komatiite

(1) Equation of State of Komatiite

The equation of state (EOS) of a molten komatiite (27 wt% MgO) was detennined in the 5 to 36 GPa pressure range via shock wave compression from 1550°C and 0 bar. Shock wave velocity, U_S, and particle velocity, U_P, in km/s follow the linear relationship U_S = 3.13(±0.03) + 1.47(±0.03) U_P. Based on a calculated density at 1550°C, 0 bar of 2.745±0.005 glee, this U_S-U_P relationship gives the isentropic bulk modulus K_S = 27.0 ± 0.6 GPa, and its first and second isentropic pressure derivatives, K'_S = 4.9 ± 0.1 and K"_S = -0.109 ± 0.003 GPa^-1.

The calculated liquidus compression curve agrees within error with the static compression results of Agee and Walker [1988a] to 6 GPa. We detennine that olivine (FO₉₄) will be neutrally buoyant in komatiitic melt of the composition we studied near 8.2 GPa. Clinopyroxene would also be neutrally buoyant near this pressure. Liquidus garnet-majorite may be less dense than this komatiitic liquid in the 20-24 GPa interval, however pyropic-garnet and perovskite phases are denser than this komatiitic liquid in their respective liquidus pressure intervals to 36 GPa. Liquidus perovskite may be neutrally buoyant near 70 GPa.

At 40 GPa, the density of shock-compressed molten komatiite would be approximately equal to the calculated density of an equivalent mixture of dense solid oxide components. This observation supports the model of Rigden et al. [1989] for compressibilities of liquid oxide components. Using their theoretical EOS for liquid forsterite and fayalite, we calculate the densities of a spectrum of melts from basaltic through peridotitic that are related to the experimentally studied komatiitic liquid by addition or subtraction of olivine. At low pressure, olivine fractionation lowers the density of basic magmas, but above 14 GPa this trend is reversed. All of these basic to ultrabasic liquids are predicted to have similar densities at 14 GPa, and this density is approximately equal to the bulk (PREM) mantle. This suggests that melts derived from a peridotitic mantle may be inhibited from ascending from depths greater than 400 km.

The EOS of ultrabasic magmas was used to model adiabatic melting in a peridotitic mantle. If komatiites are formed by >15% partial melting of a peridotitic mantle, then komatiites generated by adiabatic melting come from source regions in the lower transition zone (≈500-670 km) or the lower mantle (>670 km). The great depth of incipient melting implied by this model, and the melt density constraint mentioned above, suggest that komatiitic volcanism may be gravitationally hindered. Although komatiitic magmas are thought to separate from their coexisting crystals at a temperature =200°C greater than that for modern MORBs, their ultimate sources are predicted to be diapirs that, if adiabatically decompressed from initially solid mantle, were more than 700°C hotter than the sources of MORBs and derived from great depth.

We considered the evolution of an initially molten mantle, i.e., a magma ocean. Our model considers the thermal structure of the magma ocean, density constraints on crystal segregation, and approximate phase relationships for a nominally chondritic mantle. Crystallization will begin at the core-mantle boundary. Perovskite buoyancy at > 70 GPa may lead to a compositionally stratified lower mantle with iron-enriched mangesiowiistite content increasing with depth. The upper mantle may be depleted in perovskite components. Olivine neutral buoyancy may lead to the formation of a dunite septum in the upper mantle, partitioning the ocean into upper and lower reservoirs, but this septum must be permeable.

(2) Viscosity Measurement with Shock Waves

We have examined in detail the analytical method for measuring shear viscosity from the decay of perturbations on a corrugated shock front The relevance of initial conditions, finite shock amplitude, bulk viscosity, and the sensitivity of the measurements to the shock boundary conditions are discussed. The validity of the viscous perturbation approach is examined by numerically solving the second-order Navier-Stokes equations. These numerical experiments indicate that shock instabilities may occur even when the Kontorovich-D'yakov stability criteria are satisfied. The experimental results for water at 15 GPa are discussed, and it is suggested that the large effective viscosity determined by this method may reflect the existence of ice VII on the Rayleigh path of the Hugoniot This interpretation reconciles the experimental results with estimates and measurements obtained by other means, and is consistent with the relationship of the Hugoniot with the phase diagram for water. Sound waves are generated at 4.8 MHz at in the water experiments at 15 GPa. The existence of anelastic absorption modes near this frequency would also lead to large effective viscosity estimates.

(3) Equation of State of Molybdenum at 1400°C

Shock compression data to 96 GPa for pure molybdenum, initially heated to 1400°C, are presented. Finite strain analysis of the data gives a bulk modulus at 1400°C, K'_S. of 244±2 GPa and its pressure derivative, K'_OS of 4. A fit of shock velocity to particle velocity gives the coefficients of U_S = C_O+S U_P to be C_O = 4.77±0.06 km/s and S = 1.43±0.05. From the zero pressure sound speed, C_O, a bulk modulus of 232±6 GPa is calculated that is consistent with extrapolation of ultrasonic elasticity measurements. The temperature derivative of the bulk modulus at zero pressure, θK_OSθT|_P, is approximately -0.012 GPa/K. A thermodynamic model is used to show that the thermodynamic Grüneisen parameter is proportional to the density and independent of temperature. The Mie-Grüneisen equation of state adequately describes the high temperature behavior of molybdenum under the present range of shock loading conditions.

Studies on the role of histones in the structure and function of chromatin

Studies on the dissociation of histones from chromatin by increasing concentrations of sodium deoxycholate (DOC) have shown that histrone II is removed at lowest concentrations of DOC, while slightly higher concentrations remove histones III and IV. Still higher concentrations remove histone I.

The complete separation of chromatin and ¹⁴C-DOC by sucrose sedimentation indicated that the binding of DOC to chromatin is readily and completely reversible.

The dissociation of histones from chromatin by increasing concentrations of related cholanic acids and some of their conjugated derivatives were studied. The results suggested that the driving force for the interaction between the cholanic acid anion and histones is the lowering of the activity coefficient of the cholanic acid anion which occurs when it is partially removed from solution by interaction with hydrophobic regions of the positively charged histones.

The role of histones in the structure of chromatin has been studied by comparing the effects of selective removal of histones from chromatin by increasing concentrations of DOC with those caused by NaCl (removes histone I at lowest concentrations, while higher concentrations remove histones II, III, and IV). Properties studied included thermal denaturation, sedimentation velocity, flow dichroism, relaxation times of molecules oriented in a flow field, and the irreversible disruption of a 130 S, cross-linked component of sheared chromatin. The data indicated that none of the structural or chemical parameters with which these properties are correlated show a dependence on the presence of one particular histone fraction.

The template activity (ability to prime a 0.2 M KC1 DNA-dependent RNA synthesis system catalyzed by E. coli RNA polymerase) increases from that of native chromatin (approximately 25 per cent of that pure DNA) to that of pure DNA in a fashion which shows a nearly linear relationship to the amount of histone coverage of the template. The precipitability of partially dehistonized chromatin samples in 0.15 M NaCl shows a large dependence on the presence of histone I.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

Three different categories of flow problems of a fluid containing small particles are being considered here. They are: (i) a fluid containing small, non-reacting particles (Parts I and II); (ii) a fluid containing reacting particles (Parts III and IV); and (iii) a fluid containing particles of two distinct sizes with collisions between two groups of particles (Part V).

Part I

A numerical solution is obtained for a fluid containing small particles flowing over an infinite disc rotating at a constant angular velocity. It is a boundary layer type flow, and the boundary layer thickness for the mixture is estimated. For large Reynolds number, the solution suggests the boundary layer approximation of a fluid-particle mixture by assuming W = W_p. The error introduced is consistent with the Prandtl’s boundary layer approximation. Outside the boundary layer, the flow field has to satisfy the “inviscid equation” in which the viscous stress terms are absent while the drag force between the particle cloud and the fluid is still important. Increase of particle concentration reduces the boundary layer thickness and the amount of mixture being transported outwardly is reduced. A new parameter, β = 1/Ω τ_v, is introduced which is also proportional to μ. The secondary flow of the particle cloud depends very much on β. For small values of β, the particle cloud velocity attains its maximum value on the surface of the disc, and for infinitely large values of β, both the radial and axial particle velocity components vanish on the surface of the disc.

Part II

The “inviscid” equation for a gas-particle mixture is linearized to describe the flow over a wavy wall. Corresponding to the Prandtl-Glauert equation for pure gas, a fourth order partial differential equation in terms of the velocity potential ϕ is obtained for the mixture. The solution is obtained for the flow over a periodic wavy wall. For equilibrium flows where λ_v and λ_T approach zero and frozen flows in which λ_v and λ_T become infinitely large, the flow problem is basically similar to that obtained by Ackeret for a pure gas. For finite values of λ_v and λ_T, all quantities except v are not in phase with the wavy wall. Thus the drag coefficient C_D is present even in the subsonic case, and similarly, all quantities decay exponentially for supersonic flows. The phase shift and the attenuation factor increase for increasing particle concentration.

Part III

Using the boundary layer approximation, the initial development of the combustion zone between the laminar mixing of two parallel streams of oxidizing agent and small, solid, combustible particles suspended in an inert gas is investigated. For the special case when the two streams are moving at the same speed, a Green’s function exists for the differential equations describing first order gas temperature and oxidizer concentration. Solutions in terms of error functions and exponential integrals are obtained. Reactions occur within a relatively thin region of the order of λ_D. Thus, it seems advantageous in the general study of two-dimensional laminar flame problems to introduce a chemical boundary layer of thickness λ_D within which reactions take place. Outside this chemical boundary layer, the flow field corresponds to the ordinary fluid dynamics without chemical reaction.

Part IV

The shock wave structure in a condensing medium of small liquid droplets suspended in a homogeneous gas-vapor mixture consists of the conventional compressive wave followed by a relaxation region in which the particle cloud and gas mixture attain momentum and thermal equilibrium. Immediately following the compressive wave, the partial pressure corresponding to the vapor concentration in the gas mixture is higher than the vapor pressure of the liquid droplets and condensation sets in. Farther downstream of the shock, evaporation appears when the particle temperature is raised by the hot surrounding gas mixture. The thickness of the condensation region depends very much on the latent heat. For relatively high latent heat, the condensation zone is small compared with Ʌ_D.

For solid particles suspended initially in an inert gas, the relaxation zone immediately following the compression wave consists of a region where the particle temperature is first being raised to its melting point. When the particles are totally melted as the particle temperature is further increased, evaporation of the particles also plays a role.

The equilibrium condition downstream of the shock can be calculated and is independent of the model of the particle-gas mixture interaction.

Part V

For a gas containing particles of two distinct sizes and satisfying certain conditions, momentum transfer due to collisions between the two groups of particles can be taken into consideration using the classical elastic spherical ball model. Both in the relatively simple problem of normal shock wave and the perturbation solutions for the nozzle flow, the transfer of momentum due to collisions which decreases the velocity difference between the two groups of particles is clearly demonstrated. The difference in temperature as compared with the collisionless case is quite negligible.

Relativistic velocity - potential hydrodynamics and stellar stability

The equations of relativistic, perfect-fluid hydrodynamics are cast in Eulerian form using six scalar "velocity-potential" fields, each of which has an equation of evolution. These equations determine the motion of the fluid through the equation

U_ʋ=µ^-1 (ø,_ʋ + αβ,_ʋ + ƟS,_ʋ).

Einstein's equations and the velocity-potential hydrodynamical equations follow from a variational principle whose action is

I = (R + 16π p) (-g)^1/2 d⁴x,

where R is the scalar curvature of spacetime and p is the pressure of the fluid. These equations are also cast into Hamiltonian form, with Hamiltonian density –T₀⁰ (-g^oo)^-1/2.

The second variation of the action is used as the Lagrangian governing the evolution of small perturbations of differentially rotating stellar models. In Newtonian gravity this leads to linear dynamical stability criteria already known. In general relativity it leads to a new sufficient condition for the stability of such models against arbitrary perturbations.

By introducing three scalar fields defined by

ρ ᵴ = ∇λ + ∇x(xi + ∇xɣi)

(where ᵴ is the vector displacement of the perturbed fluid element, ρ is the mass-density, and i, is an arbitrary vector), the Newtonian stability criteria are greatly simplified for the purpose of practical applications. The relativistic stability criterion is not yet in a form that permits practical calculations, but ways to place it in such a form are discussed.