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A Chebyshev-Fourier discretization with shock fitting is used to solve the unsteady Euler equations. The method is applied to shock interactions with plane waves and with a simple model of homogeneous isotropic turbulence. The plane wave solutions are compared to linear theory. Previously announced in STAR as N83-27952
This paper considers Vlasov instabilities driven by a very hot ion beam streaming parallel to a magnetic field. The linear theory of electromagnetic instabilities driven by such a beam in a homogeneous, collisionless, nonrelativistic plasma is used. Numerical solutions of the full linear dispersion equation for bi-Maxwellian distribution functions are presented. If the thermal speed of the beam is greater than its drift speed, the two dominant modes are the right-hand and left-hand resonant ion beam instabilities. The parametric dependencies of the threshold drift speeds and growth rates for both modes are presented. In particular, the thresholds are shown to be sensitive functions of the beam temperature anisotropies. Arguments are presented that these two modes are driven by the suprathermal ion component at interplanetary shocks, and that the growth of these modes is sufficient to account for the amplitudes of MHD-like waves observed at such shocks.
This paper considers the linear theory of electromagnetic instabilities driven by an electron beam in a homogeneous, nonrelativistic, Vlasov plasma. The beam is relatively hot, isotropic in its own frame, and streams parallel or antiparallel to a magnetic field B. Numerical solutions of the full dispersion equation for propagation parallel or antiparallel to B are presented, and the linear properties of the whistler heat flux and electron beam firehose instabilities are exhibited and compared. Under a broad range of parmameters the former mode has the lower beam speed threshold, and the larger maximum growth rate. In addition, it is demonstrated that, for a sufficiently large relative beam density, relative beam temperature, and plasma beta the whistler heat flux instability has a much lower beam speed threshold than the electrostatic electron beam instability. The application of these instabilities to first-order Fermi acceleration of electrons at space plasma shocks is discussed.
An analytical study is presented showing the effects of the notch tip geometry on the location and direction of crack growth from an existing notch in a unidirectional fibrous composite modeled as a homogeneous, anisotropic, elastic material. Anisotropic elasticity and the normal stress ratio theory are used to study crack growth from elliptical notches in unidirectional composites. Sharp cracks, circular holes, and ellipses are studied under far-field tension and shear loading. The capabilities of a previously developed design code was upgraded to handle more generalized plate geometries and laminates under a more generalized loading and boundary conditions. Discussion of the developments of the design code is presented.
An analytical study is presented showing the effects of the notch tip geometry on the location and direction of crack growth from an existing notch in a unidirectional fibrous composite modelled as a homogeneous, anisotropic, elastic material. Anisotropic elasticity and the normal stress ratio theory are used to study crack growth from elliptical notches in unidirectional composites. Sharp cracks, circular holes, and ellipses are studied under far-field tension and shear loading. Limited comparisons are made showing good correlation with experimental results.
In order to expand the predictive capability of single-point turbulence closure models to account for the early-stage transition regime, a methodology for the formulation and calibration of model equations for the ensemble-averaged disturbance kinetic energy and energy dissipation rate is presented. First the decay of laminar disturbances and turbulence in mean shear-free flows is studied. In laminar flows, such disturbances are linear superpositions of modes governed by the Orr-Sommerfeld equation. In turbulent flows, disturbances are described through transport equations for representative mean quantities. The link between a description based on a deterministic evolution equation and a probability based mean transport equation is established. Because an uncertainty in initial conditions exists in the laminar as well as the turbulent regime, a probability distribution must be defined even in the laminar case. Using this probability distribution, it is shown that the exponential decay of the linear modes in the laminar regime can be related to a power law decay of both the (ensemble) mean disturbance kinetic energy and the dissipation rate. The evolution of these mean disturbance quantities is then described by transport equations similar to those for the corresponding turbulent decaying flow. Second, homogeneous shear flow, where disturbances can be described by rapid distortion theory (RDT), is studied. The relationship between RDT and linear stability theory is exploited in order to obtain a closed set of modeled equations. The linear disturbance equations are solved directly so that the numerical simulation yields a database from which the closure coefficients in the ensemble-averaged disturbance equations can be determined.
A biaxial graded-index fiber with a homogeneous cladding is studied. Two methods, wave equation and matrix differential equation, of formulating the problem and their respective solutions are discussed. For the wave equation formulation of the problem it is shown that for the case of a diagonal permittivity tensor the longitudinal electric and magnetic fields satisfy a pair of coupled second-order differential equations. Also, a generalized dispersion relation is derived in terms of the solutions for the longitudinal electric and magnetic fields. For the case of a step-index fiber, either isotropic or uniaxial, these differential equations can be solved exactly in terms of Bessel functions. For the cases of an istropic graded-index and a uniaxial graded-index fiber, a solution using the Wentzel, Krammers and Brillouin (WKB) approximation technique is shown. Results for some particular permittivity profiles are presented. Also the WKB solutions is compared with the vector solution found by Kurtz and Streifer. For the matrix formulation it is shown that the tangential components of the electric and magnetic fields satisfy a system of four first-order differential equations which can be conveniently written in matrix form. For the special case of meridional modes, the system of equations splits into two systems of two equations. A general iterative technique, asymptotic partitioning of systems of equations, for solving systems of differential equations is presented. As a simple example, Bessel's differential equation is written in matrix form and is solved using this asymptotic technique. Low order solutions for particular examples of a biaxial and uniaxial graded-index fiber are presented. Finally numerical results obtained using the asymptotic technique are presented for particular examples of isotropic and uniaxial step-index fibers and isotropic, uniaxial and biaxial graded-index fibers.
Exact image method, recently introduced for the exact solution of electromagnetic field problems involving homogeneous half spaces and microstrip-like geometries, is developed for the problem of homogeneous slab of dielectric and/or magnetic material in free space. Expressions for image sources, creating the exact reflected and transmitted fields, are given and their numerical evaluation is demonstrated. Nonradiating modes, guided by the slab and responsible for the loss of convergence of the image functions, are considered and extracted. The theory allows, for example, an analysis of finite ground planes in microstrip antenna structures.
Over nearly all of the surface of Venus the reflectivity and emissivity at centimeter wavelengths are about 0.15 and 0.85 respectively. These values are consistent with moderately dense soils and rock populations, but the mean reflectivity is about a factor of 2 greater than that for the Moon and other terrestrial planets. Pettingill and Ford, using Pioneer Venus reflectivities and emissivities, found a number of anomalous features on Venus that showed much higher reflectivities and much lower emissivities with both values approaching 0.5. These include Maxwell Montes, a number of high regions in Aphrodite Terra and Beta Regio, and several isolated mountain peaks. Most of the features are at altitudes above the mean radius by 2 to 3 km or more. However, such features have been found in the Magellan data at low altitudes and the anomalies do not exist on all high structures, Maat Mons being the most outstanding example. A number of papers have been written that attempt to explain the phenomena in terms of the geochemistry balance of weathering effects on likely surface minerals. The geochemists have shown that the fundamentally basaltic surface would be stable at the temperatures and pressures of the mean radius in the form of magnetite, but would evolve to pyrite and/or pyrrhotite in the presence of sulfur-bearing compounds such as SO2. Pyrite will be stable at altitudes above 4 or 5 km on Venus. Although the geochemical arguments are rather compelling, it is vitally important to rationally look at other explanations for radar and radio emission measurements such as that presented by Tryka and Muhleman. The radar reflectivity values are retrieved from the raw Magellan backscatter measurements by fitting the Hagfors' radar scattering model in which a surface roughness parameters and a normal incidence electrical reflectivity are estimated. The assumptions of the theory behind the model must be considered carefully before the results can be believed. These include that the surface roughness exists only at horizontal scales large compared to the wavelength, the vertical deviations are gaussianly distributed, there is no shadowing, and that the reflection occurs at the interface of two homogeneous dielectric half-spaces. Probably all these conditions are violated at the anomalous features under discussion. The most important of these is the homogeneity of the near surface of Venus, particularly in highlands. Under the assumptions of the theory, all of the radio energy is reflected by the impedance jump at the very boundary. However, in heterogeneous soil some fraction of the illuminating energy is propagated into the soil and then scattered back out by impedance discontinuities such as rock, voids, and cracks. In light soils, the latter effect can overwhelm the scattering effects of the true surface and greatly enhance the backscatter power, suggesting a much higher value of an effective dielectric constant that would be estimated from Hagfors' model.
Quantitative comparisons between theory and experiments on non-linear three-wave interactions in a nearly homogeneous plasma are presented in this paper. First, the theoretical spatial behaviour of signal and idler waves excited by a large amplitude pump wave is predicted, with damping of all waves included. These results are then specialized to allow detailed quantitative comparisons to be made with available experimental data on parametrically excited ion-acoustic waves. Where good agreement is obtained, the comparison indicates further measurements required to verify more completely the basic theory of three-wave interactions in plasmas. Where wide disagreement is obtained, the comparison indicates possible directions in which the theory might be improved.
Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.
We use a recent computer implementation of the first principles theory of electromagnetic scattering to compute far-field extinction by a spherical particle embedded in an absorbing unbounded host. Our results show that the suppressing effect of increasing absorption inside the host medium on the ripple structure of the extinction efficiency factor as a function of the size parameter is similar to the well-known effect of increasing absorption inside a particle embedded in a nonabsorbing host. However, the accompanying effects on the interference structure of the extinction efficiency curves are diametrically opposite. As a result, sufficiently large absorption inside the host medium can cause negative particulate extinction. We offer a simple physical explanation of the phenomenon of negative extinction consistent with the interpretation of the interference structure as being the result of interference of the field transmitted by the particle and the diffracted field due to an incomplete wave front resulting from the blockage of the incident plane wave by the particle's geometrical projection.
A combined experimental/theoretical study is presented of the onset conditions for gas phase reaction and particle nucleation in hot substrate/cold gas CVD of transition metal oxides. Homogeneous reaction onset conditions are predicted using a simple high activation energy reacting gas film theory. Experimental tests of the basic theory are underway using an axisymmetric impinging jet CVD reactor. No vapor phase ignition has yet been observed in the TiCl4/O2 system under accessible operating conditions (below substrate temperature Tw = 1700 K). The goal of this research is to provide CVD reactor design and operation guidelines for achieving acceptable deposit microstructures at the maximum deposition rate while simultaneously avoiding homogeneous reaction/nucleation and diffusional limitations.
Stress derivatives for perfectly elastic homogeneous and isotropic shell in equilibrium with forces acting along edges, applying theory of elliptic partial differential equations
The question of the significance of the galaxies for the structure and the evolution of the universe is considered, giving attention to the possibility that galaxies are merely local perturbations, superficialities in an otherwise homogeneous universe. Means of manufacturing galaxies in the context of Einstein's theory are examined. It appears that a wide range of possible anisotropic universes should generate a residual vorticity. Density fluctuations occur as a basis for the origin of galaxies.
Scattering theory is applied to derive the equilibrium rate coefficient for a general homogeneous chemical reaction involving ideal gases. The reaction rate is expressed in terms of the product of a number of normalized momentum distribution functions, the product of the number of molecules with a given internal energy state, and the spin-averaged T-matrix elements. An expression for momentum distribution at equilibrium for an arbitrary molecule is presented, and the number of molecules with a given internal-energy state is represented by an expression which includes the partition function.
Gradient theory converts the molecular theory of inhomogeneous fluid into nonlinear boundary value problems for density and stress distributions in fluid interfaces, contact line regions, nuclei and microdroplets, and other fluid microstructures. The relationship between the basic patterns of fluid phase behavior and the occurrence and stability of fluid microstructures was clearly established by the theory. All the inputs of the theory have molecular expressions which are computable from simple models. On another level, the theory becomes a phenomenological framework in which the equation of state of homogeneous fluid and sets of influence parameters of inhomogeneous fluids are the inputs and the structures, stress tensions and contact angles of menisci are the outputs. These outputs, which find applications in the science and technology of drops and bubbles, are discussed.
A sandwich construction consists of a low-density core material with high strength face sheets bounded to the top and bottom surfaces. The construction has been widely used in the aerospace and marine industries due to its outstanding characteristics such as noise absorption, weight minimization, heat insulation, and better bending stiffness. In sandwich structures used in high-performance aircraft, the face sheets are often made of fiber-reinforced composite materials and the core is made of honeycomb. The structures may also have variable thickness so as to satisfy aerodynamic requirements. In the stress analysis, the constant-thickness face sheets are usually considered as membrane and the core is assumed to be inextensible but deformable in the thickness direction. The static behavior of variable-thickness, isotropic and homogeneous sandwich beams was successfully studied by employing a constant-thickness theory but allowing stiffnesses to vary in accordance with local thickness variations. It has been recently found in a refined theory that the analyses based on the constant thickness theory locally can lead to significant errors in structural responses if the sandwich beam is thickness-tapered and the cores are deformable in transverse shear. The errors arise mainly from two factors: (1) the transverse shear components of the membrane forces in the face sheets alter the transverse shears carried by the core; and (2) the face-sheet membrane strains arise from transverse shear deformation of the core. In practice the variable thickness may not only exist in core but also in face sheets. The thickness-variations may even be a type of step function. In this case the transverse shear stress in the face sheets and bending stress in the core should be taken into account in the refined theory mentioned. In the present study, energy principles are employed in deriving governing equations for general bending of anisotropic sandwich beams with variable thickness in both face sheets and cores. Solutions to these equations are based on a finite difference scheme. As an example in application, a simply supported thickness-tapered sandwich beam subject to a concentrated load at its center is considered. Let W' be the maximum deflection of the beam in which face sheets are considered as membrane, while W'' is that based on using the modified refined theory. It is found that W' is always larger than W'', however, the magnitude of (W'- W'') appears to be insensitive to the change of the taper of the beam.