Modeling and analysis of hysteretic structural behavior

For damaging response, the force-displacement relationship of a structure is highly nonlinear and history-dependent. For satisfactory analysis of such behavior, it is important to be able to characterize and to model the phenomenon of hysteresis accurately. A number of models have been proposed for response studies of hysteretic structures, some of which are examined in detail in this thesis. There are two popular classes of models used in the analysis of curvilinear hysteretic systems. The first is of the distributed element or assemblage type, which models the physical behavior of the system by using well-known building blocks. The second class of models is of the differential equation type, which is based on the introduction of an extra variable to describe the history dependence of the system.

Owing to their mathematical simplicity, the latter models have been used extensively for various applications in structural dynamics, most notably in the estimation of the response statistics of hysteretic systems subjected to stochastic excitation. But the fundamental characteristics of these models are still not clearly understood. A response analysis of systems using both the Distributed Element model and the differential equation model when subjected to a variety of quasi-static and dynamic loading conditions leads to the following conclusion: Caution must be exercised when employing the models belonging to the second class in structural response studies as they can produce misleading results.

The Massing's hypothesis, originally proposed for steady-state loading, can be extended to general transient loading as well, leading to considerable simplification in the analysis of the Distributed Element models. A simple, nonparametric identification technique is also outlined, by means of which an optimal model representation involving one additional state variable is determined for hysteretic systems.

Extracting material response from simple mechanical tests on hardening-softening-hardening viscoplastic solids

Compliant foams are usually characterized by a wide range of desirable mechanical properties. These properties include viscoelasticity at different temperatures, energy absorption, recoverability under cyclic loading, impact resistance, and thermal, electrical, acoustic and radiation-resistance. Some foams contain nano-sized features and are used in small-scale devices. This implies that the characteristic dimensions of foams span multiple length scales, rendering modeling their mechanical properties difficult. Continuum mechanics-based models capture some salient experimental features like the linear elastic regime, followed by non-linear plateau stress regime. However, they lack mesostructural physical details. This makes them incapable of accurately predicting local peaks in stress and strain distributions, which significantly affect the deformation paths. Atomistic methods are capable of capturing the physical origins of deformation at smaller scales, but suffer from impractical computational intensity. Capturing deformation at the so-called meso-scale, which is capable of describing the phenomenon at a continuum level, but with some physical insights, requires developing new theoretical approaches.

A fundamental question that motivates the modeling of foams is ‘how to extract the intrinsic material response from simple mechanical test data, such as stress vs. strain response?’ A 3D model was developed to simulate the mechanical response of foam-type materials. The novelty of this model includes unique features such as the hardening-softening-hardening material response, strain rate-dependence, and plastically compressible solids with plastic non-normality. Suggestive links from atomistic simulations of foams were borrowed to formulate a physically informed hardening material input function. Motivated by a model that qualitatively captured the response of foam-type vertically aligned carbon nanotube (VACNT) pillars under uniaxial compression [2011,“Analysis of Uniaxial Compression of Vertically Aligned Carbon Nanotubes,” J. Mech.Phys. Solids, 59, pp. 2227–2237, Erratum 60, 1753–1756 (2012)], the property space exploration was advanced to three types of simple mechanical tests: 1) uniaxial compression, 2) uniaxial tension, and 3) nanoindentation with a conical and a flat-punch tip. The simulations attempt to explain some of the salient features in experimental data, like
1) The initial linear elastic response.
2) One or more nonlinear instabilities, yielding, and hardening.

The model-inherent relationships between the material properties and the overall stress-strain behavior were validated against the available experimental data. The material properties include the gradient in stiffness along the height, plastic and elastic compressibility, and hardening. Each of these tests was evaluated in terms of their efficiency in extracting material properties. The uniaxial simulation results proved to be a combination of structural and material influences. Out of all deformation paths, flat-punch indentation proved to be superior since it is the most sensitive in capturing the material properties.

Pulsed neutron experiments in graphite

Detailed pulsed neutron measurements have been performed in graphite assemblies ranging in size from 30.48 cm x 38.10 cm x 38.10 cm to 91.44 cm x 66.67 cm x 66.67 cm. Results of the measurement have been compared to a modeled theoretical computation.

In the first set of experiments, we measured the effective decay constant of the neutron population in ten graphite stacks as a function of time after the source burst. We found the decay to be non-exponential in the six smallest assemblies, while in three larger assemblies the decay was exponential over a significant portion of the total measuring interval. The decay in the largest stack was exponential over the entire ten millisecond measuring interval. The non-exponential decay mode occurred when the effective decay constant exceeded 1600 sec^( -1).

In a second set of experiments, we measured the spatial dependence of the neutron population in four graphite stacks as a function of time after the source pulse. By doing an harmonic analysis of the spatial shape of the neutron distribution, we were able to compute the effective decay constants of the first two spatial modes. In addition, we were able to compute the time dependent effective wave number of neutron distribution in the stacks.

Finally, we used a Laplace transform technique and a simple modeled scattering kernel to solve a diffusion equation for the time and energy dependence of the neutron distribution in the graphite stacks. Comparison of these theoretical results with the results of the first set of experiments indicated that more exact theoretical analysis would be required to adequately describe the experiments.

The implications of our experimental results for the theory of pulsed neutron experiments in polycrystalline media are discussed in the last chapter.

Band structure and transport properties of simple cubic tellurium-base alloys

The electrical transport properties and lattice spacings of simple cubic Te-Au, Te-Au-Fe, and Te-Au-Mn alloys, prepared by rapid quenching from the liquid state, hove been measured and correlated with a proposed bond structure. The variations of superconducting transition temperature, absolute thermoelectric power, and lattice spacing with Te concentration all showed related anomalies in the binary Te-Au alloys. The unusual behavior of these properties has been interpreted by using nearly free electron theory to predict the effect of the second Brillouin zone boundary on the area of the Fermi surface, and the electronic density of states. The behavior of the superconducting transition temperature and the lattice parameter as Fe and Mn ore added further supports the proposed interpretation as well as providing information on the existence of localized magnetic states in the ternary alloys. In addition, it was found that a very distinct bond structure effect on the transition temperatures of the Te-Au-Fe alloys could be identified.

Theory of s-d exchange scattering in dilute magnetic alloys

The problem of s-d exchange scattering of conduction electrons off localized magnetic moments in dilute magnetic alloys is considered employing formal methods of quantum field theoretical scattering. It is shown that such a treatment not only allows for the first time, the inclusion of multiparticle intermediate states in single particle scattering equations but also results in extremely simple and straight forward mathematical analysis. These equations are proved to be exact in the thermodynamic limit. A self-consistent integral equation for electron self energy is derived and approximately solved. The ground state and physical parameters of dilute magnetic alloys are discussed in terms of the theoretical results. Within the approximation of single particle intermediate states our results reduce to earlier versions. The following additional features are found as a consequence of the inclusion of multiparticle intermediate states;

(i) A non analytic binding energy is pre sent for both, antiferromagnetic (J < o) and ferromagnetic (J > o) couplings of the electron plus impurity system.

(ii) The correct behavior of the energy difference of the conduction electron plus impurity system and the free electron system is found which is free of unphysical singularities present in earlier versions of the theories.

(iii) The ground state of the conduction electron plus impurity system is shown to be a many-body condensate state for J < o and J > o, both. However, a distinction is made between the usual terminology of "Singlet" and "Triplet" ground states and nature of our ground state.

(iv) It is shown that a long range ordering, leading to an ordering of the magnetic moments can result from a contact interaction such as the s-d exchange interaction.

(v) The explicit dependence of the excess specific heat of the Kondo systems is obtained and found to be linear in temperatures as T→ o and T ℓnT for 0.3 T_K ≤ T ≤ 0.6 T_K. A rise in (ΔC/T) for temperatures in the region 0 < T ≤ 0.1 T_K is predicted. These results are found to be in excellent agreement with experiments.

(vi) The existence of a critical temperature for Ferromagnetic coupling (J > o) is shown. On the basis of this the apparent contradiction of the simultaneous existence of giant moments and Kondo effect is resolved.

I. Spectroscopic observation of discrete sites in simple liquids. II. Infrared intensity perturbations of hydrogen halide fundamentals in liquid xenon. III. Rotation of HCl in solid rare gases.

Part I

The spectrum of dissolved mercury atoms in simple liquids has been shown to be capable of revealing information concerning local structures in these liquids.

Part II

Infrared intensity perturbations in simple solutions have been shown to involve more detailed interaction than just dielectric polarization. No correlation has been found between frequency shifts and intensity enhancements.

Part III

Evidence for perturbed rotation of HCl in rare gas matrices has been found. The magnitude of the barrier to rotation is concluded to be of order of 30 cm^(-1).

A quantitative investigation of the solar modulation of cosmic-ray protons and helium nuclei

The differential energy spectra of cosmic-ray protons and He nuclei have been measured at energies up to 315 MeV/nucleon using balloon- and satellite-borne instruments. These spectra are presented for solar quiet times for the years 1966 through 1970. The data analysis is verified by extensive accelerator calibrations of the detector systems and by calculations and measurements of the production of secondary protons in the atmosphere.

The spectra of protons and He nuclei in this energy range are dominated by the solar modulation of the local interstellar spectra. The transport equation governing this process includes as parameters the solar-wind velocity, V, and a diffusion coefficient, K(r,R), which is assumed to be a scalar function of heliocentric radius, r, and magnetic rigidity, R. The interstellar spectra, j_D, enter as boundary conditions on the solutions to the transport equation. Solutions to the transport equation have been calculated for a broad range of assumed values for K(r,R) and j_D and have been compared with the measured spectra.

It is found that the solutions may be characterized in terms of a dimensionless parameter, ψ(r,R) = ^∞∫_r V dr'/(K(r',R). The amount of modulation is roughly proportional to ψ. At high energies or far from the Sun, where the modulation is weak, the solution is determined primarily by the value of ψ (and the interstellar spectrum) and is not sensitive to the radial dependence of the diffusion coefficient. At low energies and for small r, where the effects of adiabatic deceleration are found to be large, the spectra are largely determined by the radial dependence of the diffusion coefficient and are not very sensitive to the magnitude of ψ or to the interstellar spectra. This lack of sensitivity to j_D implies that the shape of the spectra at Earth cannot be used to determine the interstellar intensities at low energies.

Values of ψ determined from electron data were used to calculate the spectra of protons and He nuclei near Earth. Interstellar spectra of the form j_D α (W - 0.25m)^-2.65 for both protons and He nuclei were found to yield the best fits to the measured spectra for these values of ψ, where W is the total energy and m is the rest energy. A simple model for the diffusion coefficient was used in which the radial and rigidity dependence are separable and K is independent of radius inside a modulation region which has a boundary at a distance D. Good agreement was found between the measured and calculated spectra for the years 1965 through 1968, using typical boundary distances of 2.7 and 6.1 A.U. The proton spectra observed in 1969 and 1970 were flatter than in previous years. This flattening could be explained in part by an increase in D, but also seemed to require that a noticeable fraction of the observed protons at energies as high at 50 to 100 MeV be attributed to quiet-time solar emission. The turnup in the spectra at low energies observed in all years was also attributed to solar emission. The diffusion coefficient used to fit the 1965 spectra is in reasonable agreement with that determined from the power spectra of the interplanetary magnetic field (Jokipii and Coleman, 1968). We find a factor of roughly 3 increase in ψ from 1965 to 1970, corresponding to the roughly order of magnitude decrease in the proton intensity at 250 MeV. The change in ψ might be attributed to a decrease in the diffusion coefficient, or, if the diffusion coefficient is essentially unchanged over that period (Mathews et al., 1971), might be attributed to an increase in the boundary distance, D.

High pressure states in condensed matter: I. High pressure behavior of the iron-sulphur system with applications to the earth's core. II. Empirical equation of state for organic compounds at high pressures

Part I:

The earth's core is generally accepted to be composed primarily of iron, with an admixture of other elements. Because the outer core is observed not to transmit shear waves at seismic frequencies, it is known to be liquid or primarily liquid. A new equation of state is presented for liquid iron, in the form of parameters for the 4th order Birch-Murnaghan and Mie-Grüneisen equations of state. The parameters were constrained by a set of values for numerous properties compiled from the literature. A detailed theoretical model is used to constrain the P-T behavior of the heat capacity, based on recent advances in the understanding of the interatomic potentials for transition metals. At the reference pressure of 10⁵ Pa and temperature of 1811 K (the normal melting point of Fe), the parameters are: ρ = 7037 kg/m³, K_S0 = 110 GPa, K_S' = 4.53, K_S" = -.0337 GPa-1, and γ = 2.8, with γ α ρ^-1.17. Comparison of the properties predicted by this model with the earth model PREM indicates that the outer core is 8 to 10 % less dense than pure liquid Fe at the same conditions. The inner core is also found to be 3 to 5% less dense than pure liquid Fe, supporting the idea of a partially molten inner core. The density deficit of the outer core implies that the elements dissolved in the liquid Fe are predominantly of lower atomic weight than Fe. Of the candidate light elements favored by researchers, only sulfur readily dissolves into Fe at low pressure, which means that this element was almost certainly concentrated in the core at early times. New melting data are presented for FeS and FeS₂ which indicate that the FeS₂ is the S-hearing liquidus solid phase at inner core pressures. Consideration of the requirement that the inner core boundary be observable by seismological means and the freezing behavior of solutions leads to the possibility that the outer core may contain a significant fraction of solid material. It is found that convection in the outer core is not hindered if the solid particles are entrained in the fluid flow. This model for a core of Fe and S admits temperatures in the range 3450K to 4200K at the top of the core. An all liquid Fe-S outer core would require a temperature of about 4900 K at the top of the core.

Part II.

The abundance of uses for organic compounds in the modern world results in many applications in which these materials are subjected to high pressures. This leads to the desire to be able to describe the behavior of these materials under such conditions. Unfortunately, the number of compounds is much greater than the number of experimental data available for many of the important properties. In the past, one approach that has worked well is the calculation of appropriate properties by summing the contributions from the organic functional groups making up molecules of the compounds in question. A new set of group contributions for the molar volume, volume thermal expansivity, heat capacity, and the Rao function is presented for functional groups containing C, H, and O. This set is, in most cases, limited in application to low molecular liquids. A new technique for the calculation of the pressure derivative of the bulk modulus is also presented. Comparison with data indicates that the presented technique works very well for most low molecular hydrocarbon liquids and somewhat less well for oxygen-bearing compounds. A similar comparison of previous results for polymers indicates that the existing tabulations of group contributions for this class of materials is in need of revision. There is also evidence that the Rao function contributions for polymers and low molecular compounds are somewhat different.

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.

A molecular dynamics study of the microscopic properties of simple dense fluids

The microscopic properties of a two-dimensional model dense fluid of Lennard-Jones disks have been studied using the so-called "molecular dynamics" method. Analyses of the computer-generated simulation data in terms of "conventional" thermodynamic and distribution functions verify the physical validity of the model and the simulation technique.

The radial distribution functions g(r) computed from the simulation data exhibit several subsidiary features rather similar to those appearing in some of the g(r) functions obtained by X-ray and thermal neutron diffraction measurements on real simple liquids. In the case of the model fluid, these "anomalous" features are thought to reflect the existence of two or more alternative configurations for local ordering.

Graphical display techniques have been used extensively to provide some intuitive insight into the various microscopic phenomena occurring in the model. For example, "snapshots" of the instantaneous system configurations for different times show that the "excess" area allotted to the fluid is collected into relatively large, irregular, and surprisingly persistent "holes". Plots of the particle trajectories over intervals of 2.0 to 6.0 x 10^-12 sec indicate that the mechanism for diffusion in the dense model fluid is "cooperative" in nature, and that extensive diffusive migration is generally restricted to groups of particles in the vicinity of a hole.

A quantitative analysis of diffusion in the model fluid shows that the cooperative mechanism is not inconsistent with the statistical predictions of existing theories of singlet, or self-diffusion in liquids. The relative diffusion of proximate particles is, however, found to be retarded by short-range dynamic correlations associated with the cooperative mechanism--a result of some importance from the standpoint of bimolecular reaction kinetics in solution.

A new, semi-empirical treatment for relative diffusion in liquids is developed, and is shown to reproduce the relative diffusion phenomena observed in the model fluid quite accurately. When incorporated into the standard Smoluchowski theory of diffusion-controlled reaction kinetics, the more exact treatment of relative diffusion is found to lower the predicted rate of reaction appreciably.

Finally, an entirely new approach to an understanding of the liquid state is suggested. Our experience in dealing with the simulation data--and especially, graphical displays of the simulation data--has led us to conclude that many of the more frustrating scientific problems involving the liquid state would be simplified considerably, were it possible to describe the microscopic structures characteristic of liquids in a concise and precise manner. To this end, we propose that the development of a formal language of partially-ordered structures be investigated.

Phase transitions from the solid state of monatomic elements

The principle aims of this thesis include the development of models of sublimation and melting from first principles and the application of these models to the rare gases.

A simple physical model is constructed to represent the sublimation of monatomic elements. According to this model, the solid and gas phases are two states of a single physical system. The nature of the phase transition is clearly revealed, and the relations between the vapor pressure, the latent heat, and the transition temperature are derived. The resulting theory is applied to argon, krypton, and xenon, and good agreement with experiment is found.

For the melting transition, the solid is represented by an anharmonic model and the liquid is described by the Percus-Yevick approximation. The behavior of the liquid at high densities is studied on the isotherms kT/∈ = 1.3, 1.8, and 2.0, where k is Boltzmann's constant, T is the temperature, and e is the well depth of the Lennard-Jones 12-6 pair potential. No solutions of the PercusYevick equation were found for ρσ³ above 1.3, where ρ is the particle density and σ is the radial parameter of the Lennard-Jones potential. The liquid structure is found to be very different from the solid structure near the melting line. The liquid pressures are about 50 percent low for experimental melting densities of argon. This discrepancy gives rise to melting pressures up to twice the experimental values.

Geometrical effects on the resonance absorption of neutrons

An investigation was conducted to estimate the error when the flat-flux approximation is used to compute the resonance integral for a single absorber element embedded in a neutron source.

The investigation was initiated by assuming a parabolic flux distribution in computing the flux-averaged escape probability which occurs in the collision density equation. Furthermore, also assumed were both wide resonance and narrow resonance expressions for the resonance integral. The fact that this simple model demonstrated a decrease in the resonance integral motivated the more detailed investigation of the thesis.

An integral equation describing the collision density as a function of energy, position and angle is constructed and is subsequently specialized to the case of energy and spatial dependence. This equation is further simplified by expanding the spatial dependence in a series of Legendre polynomials (since a one-dimensional case is considered). In this form, the effects of slowing-down and flux depression may be accounted for to any degree of accuracy desired. The resulting integral equation for the energy dependence is thus solved numerically, considering the slowing down model and the infinite mass model as separate cases.

From the solution obtained by the above method, the error ascribable to the flat-flux approximation is obtained. In addition to this, the error introduced in the resonance integral in assuming no slowing down in the absorber is deduced. Results by Chernick for bismuth rods, and by Corngold for uranium slabs, are compared to the latter case, and these agree to within the approximations made.

On the transport properties of fluid-particle flow

The hydrodynamic forces acting on a solid particle in a viscous, incompressible fluid medium at low Reynolds number flow is investigated mathematically as a prerequisite to the understanding of transport processes in two-phase flow involving solid particles and fluid. Viscous interaction between a small number of spherical particles and continuous solid boundaries as well as fluid interface are analyzed under a “point-force” approximation. Non-spherical and elastic spherical particles in a simple shear flow area are then considered. Non-steady motion of a spherical particle is briefly touched upon to illustrate the transient effect of particle motion.

A macroscopic continuum description of particle-fluid flow is formulated in terms of spatial averages yielding a set of particle continuum and bulk fluid equations. Phenomenological formulas describing the transport processes in a fluid medium are extended to cases where the volume concentration of solid particles is sufficiently high to exert an important influence. Hydrodynamic forces acting on a spherical solid particle in such a system, e.g. drag, torque, rotational coupling force, and viscous collision force between streams of different sized particles moving relative to each other are obtained. Phenomenological constants, such as the shear viscosity coefficient, and the diffusion coefficient of the bulk fluid, are found as a function of the material properties of the constituents of the two-phase system and the volume concentration of solid. For transient heat conduction phenomena, it is found that the introduction of a complex conductivity for the bulk fluid permits a simple mathematical description of this otherwise complicated process. The rate of heat transfer between particle continuum and bulk fluid is also investigated by means of an Oseen-type approximation to the energy equation.

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.

I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (1-10 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.

The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.

Part II

A general method of treating inelastic collisions quantum mechanically is developed and applied to several one-dimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.

The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.

Several models for the exchange reaction involving three identical particles are also treated: (1) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.