Paraboloid Of Revolution Research Articles

Nonlinear isochronous oscillations of a fluid in a paraboloid: theory and experiment

Within the framework of the shallow-water model, the nonlinear axisymmetric oscillations of a fluid in a paraboloid of revolution and in an unbounded parabolic channel are investigated. It is established that in the paraboloid of revolution the oscillation period does not depend on the amplitude, that is, the oscillations are isochronous. Experimental investigations of free fluid oscillations in a paraboloid confirm this theoretical result.

On the sharpness of a needle

In a recent paper [Oct. 2003], Van Bladel discussed in detail the behavior of fields and surface charges near singularities of metallic structures, such as a hollow circular cylinder, a circular cone, a thin prolate spheroid, and a uniformly charged segment. At times, the reasoning in [Van Bladel, Oct. 2003] leads to conclusions not as definitive as one might wish because of the various approximations introduced, such as the electrostatic limit for the field or idealized singularities for the conductors (e.g., the tip of a cone). In this work, an example is provided that may assist in shedding some light on the questions raised in [Van Bladel, Oct. 2003], because it consists of an exact, closed-form, simple solution to a boundary-value problem that is valid at all frequencies. The problem consists of a plane wave propagating in free space and axially incident on the convex side of a paraboloid of revolution. This problem was solved by Schensted [1955] for a perfectly conducting (PEC) paraboloid, and by Roy and Uslenghi [Oct. 1997] for an isorefractive paraboloid, which comprises the PEC paraboloid as a particular case. A remarkable fact is that these exact solutions are also the geometrical optics solutions.

Indentation-induced deformation at ultramicroscopic and macroscopic contacts

Depth-sensing indentation at ultramicroscopic and macroscopic contacts (“nanoindentation” and “macroindentation,” respectively) was performed on four brittle materials (soda-lime glass, alumina titanium carbide, sapphire, and silicon) and the resulting load–displacement traces examined to provide insight to the elastic and plastic deformation scaling with contact size. The load–displacement traces are examined in terms of the unloading stiffness, the energies deposited during loading and recovered on unloading, and the effect of the indenter tip radius on the loading curve. The results of the analyses show that the elastic and plastic deformation during loading and unloading is invariant with the scale of the contact, and the unloading curve is best described by neither a conical tip nor a paraboloid of revolution, but of some compromise.

Approximate Solution of an Axisymmetric Contact Problem with Allowance for Tangential Displacements on the Contact Surface

A structurally nonlinear contact problem of a punch shaped like a paraboloid of revolution is studied. An equation for the contact‐pressure density is derived with allowance for the radial tangential displacements of the boundary points of an elastic half‐space. A method for constructing a closed‐form approximate solution is proposed. The effect of the tangential displacements on the main contact parameters is discussed.

Modelling and characterisation of Mo2C precipitation and cementite dissolution during tempering of Fe–C–Mo martensitic steel

The precipitation and Ostwald ripening behaviour of needle shaped Mo2C particles during the tempering of a ternary Fe-C-Mo martensitic steel have been characterised and modelled, taking account of local equilibrium, the capillarity effect, and the simultaneous enrichment and dissolution of cementite. Particles of Mo2C are represented as paraboloids of revolution, with the tip radius chosen to yield the maximum lengthening rate. Transmission electron microscopy has been used to validate the theory; measurements of the average length, volume fraction, and number density of particles showed good agreement with experimental observations.

Investigation of the mutual coupling between apertures on doubly curved convex surfaces: theory and measurements

A hybrid method of high-frequency approximation (UTD) and method of moments is used to calculate the mutual coupling between circular apertures on a doubly curved surface. This requires that the geodesics on doubly curved surfaces be found. To represent the aperture fields, a single waveguide mode approximation is used. In order to verify the theoretical results, a general paraboloid of revolution with circular waveguide-fed apertures was built at Ericsson Microwave Systems AB, Molndal, Sweden. This has given a possibility to verify the theoretical model for a general doubly curved surface against measurements. To the authors' knowledge, no such verification is available in the literature. It was found that measured and theoretical results in terms of mutual coupling agree very well for copolar coupling. As expected, the mutual coupling results are very dependent on the polarization. In fact, the definition of polarization directions and the need for polarization control are important issues for doubly curved conformal array antennas.

Embedding Misner and Brill–Lindquist Initial Data for Black-Hole Collisions

In this article we consider the isometrical immersions into Euclidean three-space of two-dimensional slices of the Misner and the Brill–Lindquist initial data for black-hole collisions. We show negativity of curvature and deduce other geometric properties of the slices. Under the assumption that ends behave strongly like paraboloid of revolution, we prove that Misner and the Brill–Lindquist slices cannot be isometrically immersed into R3. This condition on an end is natural in general relativity because it holds for each end of a slice of the Schwartzschild metric where it is embedded as a paraboloid of revolution.

Modelling of the Electromagnetic Field in the Interplanetary Space and in the Earth’s Magnetosphere

A magnetohydrodynamic model of the solar wind flow is constructed using a kinematic approach. It is shown that a phenomenological conductivity of the solar wind plasma plays a key role in the forming of the interplanetary magnetic field (IMF) component normal to the ecliptic plane. This component is mostly important for the magnetospheric dynamics which is controlled by the solar wind electric field. A simple analytical solution for the problem of the solar wind flow past the magnetosphere is presented. In this approach the magnetopause and the Earth’s bow shock are approximated by the paraboloids of revolution. Superposition of the effects of the bulk solar wind plasma motion and the magnetic field diffusion results in an incomplete screening of the IMF by the magnetopause. It is shown that the normal to the magnetopause component of the solar wind magnetic field and the tangential component of the electric field penetrated into the magnetosphere are determined by the quarter square of the magnetic Reynolds number. In final, a dynamic model of the magnetospheric magnetic field is constructed. This model can describe the magnetosphere in the course of the severe magnetic storm. The conditions under which the magnetospheric magnetic flux structure is unstable and can drive the magnetospheric substorm are discussed. The model calculations are compared with the observational data for September 24–26, 1998 magnetic storm (Dst min = -205 nT) and substorm occurred at 02:30 UT on January 10, 1997.

Exact Analytic Solutions of the Nonlinear Long-Wave Equations in the Case of Axisymmetric Fluid Vibrations in a Parabolic Rotating Vessel

A class of exact analytic solutions of the system of nonlinear long-wave equations is found. This class corresponds to the axisymmetric vibrations of an ideal incompressible homogeneous fluid in a rotating vessel in the shape of a paraboloid of revolution. The radial velocity of these motions is a linear function, and the azimuthal velocity and free surface displacements are polynomials in the radial coordinate with time-dependent coefficients. The nonlinear vibration frequency is equal to the frequency of the lowest mode of linear axisymmetric standing waves in the parabolic vessel.

Theoretical Modeling of Surface Asperity Depression Into an Elastic Foundation Under Static Loading

A theoretical analysis is carried out in closed-form to quantitatively describe the pressing of an individual surface asperity into its elastic bulk when subjected to normal loads. To this end, a single asperity is simulated by a paraboloid of revolution of an arbitrary even power. The investigation is based on theory of elastic contact as originally developed by Shtaerman. It is shown that additional pressing of an individual asperity into the elastic bulk essentially depends upon four parameters: the elastic compression of its apex, the initial magnitude of the height of the asperity, a constant that characterizes the shape of the asperity peak, and the elastic properties of the materials involved in the contact. The analysis shows that the impression of the asperity into the elastic bulk increases for decreasing smoothness of the paraboloid. It will be demonstrated that the impression of the asperity into the elastic bulk, if both are made of the same material, typically reaches 50 percent of the value of elastic compression of the asperity peak.

Theory for growth of needle-shaped particles in multicomponent systems

A solution is presented for the growth of needle-shaped particles (paraboloids of revolution) in multicomponent systems that obey Henry’s law. Interface kinetics and capillarity effects are incorporated, and it is demonstrated that the maximum velocity hypothesis cannot be sustained if it is assumed that there is local equilibrium at the interface. The particle is unable to grow with equilibrium for small supersaturations when capillarity effects are prominent and for large supersaturations when the interface kinetics effect is large. A method to obtain the lengthening rate and tip radius is provided.

Increase of Microhardness of Single Crystals. Cases of Silicon and Aluminum.

Relation of microhardness to the indentation depth has been experimentally investigated for two single crystalline materials of silicon and aluminum. For the case of silicon, the microhardness is strongly affected by the coupled behavior between phase transition induced by the hydrostatic pressure from the diamond structure to the β-tin and dislocation emission. Increase of hardness at the depth less than about 2μm is observed at the case of aluminum, which has already been reported for the other ductile materials discussing with the evolution of the dislocation network. The onset of the increase is, however, dependent on the surface characterization. The elastic constants estimated from the unloading curves don't show the drastic change according to the indentation depth. The assumption on a paraboloid of revolution as an indenter in the process defining the hardness is also verified and its error is found to be small within the present measurable range.

On the thermodynamics of contact interaction in an atomic force microscope

Contact interaction in an atomic force microscope is considered in terms of the thermodynamic approach. It is shown that hysteresis observed when a sample is probed in the vertical direction is due to the surface energy-work thermodynamic cycle. The force of sample-tip interaction is calculated for the case when the tip is a paraboloid of revolution. Fluctuations of the basic thermodynamic parameters are found. The role of electrocapillary forces is considered. A new method of spectroscopy in the lateral force mode is suggested.

Spontaneous grain refinement in alloy systems at low undercooling

In a previous paper [J. Appl. Phys. 82 (1997) 3783] we presented an analysis of the stability of a dendrite against a small perturbation to the tip velocity. This model demonstrated that dendritic growth in pure metals and alloys would become unstable above some upper critical undercooling ΔT 2 ∗ , while in alloys above a critical concentration, dendritic growth may also become unstable below a lower critical undercooling, ΔT 1 ∗ . However, for ΔT 1 ∗ quantitative agreement between the example system studied, Ni–Cu, and the available data was poor. In this paper, we present an improved computational scheme which allows us to relax the assumption that dendrites propagate as isothermal paraboloids of revolution. This results in much improved agreement with the experimental data.

Growth of needle and plate shaped particles: theory for small supersaturations, maximum velocity hypothesis

A solution to the diffusion controlled growth of needle and plate shaped particles is presented as their shape approaches respectively a paraboloid of revolution or a parabolic cylinder, under small supersaturation values, when capillarity and interface kinetic effects are present. The solutions show that as supersaturation decreases, the growth rate and needle tip radius approach a common value regardless of interfacial kinetics effects as capillarity is the main factor that retards particle growth. Simple asymptotic expressions are thus obtained to predict the growth rate and tip radius at low supersaturations, assuming a maximum velocity hypothesis. These represent the circumstances during solid state precipitation reactions which lead to secondary hardening in steels.

Five-Hundred-Meter Aperture Spherical Telescope Project

AbstractA Five hundred meter Aperture Spherical Telescope (FAST) is proposed to be built in the unique karst area of southwest China, and will act, in a sense, as a prototype for the Square Kilometer Array (SKA). It will be over twice as large as the Arecibo telescope coupled with much wider sky coverage. Some results from site surveys for such a SKA concept are briefly reported. Technically, FAST is not simply a copy of the existing Arecibo telescope but has rather a number of innovations. Firstly, the proposed main spherical reflector, by conforming to a paraboloid of revolution in real time through actuated active control, enables the realization of both wide bandwidth and full polarization capability while using standard feed design. Secondly, a feed support system which integrates optical, mechanical and electronic technologies will effectively reduce the cost of the support structure and control system. Pre-research on FAST has become a key project in the CAS.

Five-hundred-meter Aperture Spherical Telescope project

A Five hundred meter Aperture Spherical Telescope (FAST) is proposed to be built in the unique karst area of southwest China, and will act, in a sense, as a prototype for the Square Kilometer Array (SKA). It will be over twice as large as the Arecibo telescope coupled with much wider sky coverage. Some results from site surveys for such a SKA concept are briefly reported. Technically, FAST is not simply a copy of the existing Arecibo telescope but has rather a number of innovations. Firstly, the proposed main spherical reflector, by conforming to a paraboloid of revolution in real time through actuated active control, enables the realization of both wide bandwidth and full polarization capability while using standard feed design. Secondly, a feed support system which integrates optical, mechanical and electronic technologies will effectively reduce the cost of the support structure and control system. Pre-research on FAST has become a key project in the CAS.

Extending the Observable Zenith Angle of FAST Using an Offset Feed

AbstractThe five hundred meter aperture spherical radio telescope is will use an active spherical reflector. When the zenith scan angle is changed, the illuminated part of the reflecting surface is made to fit a paraboloid of revolution in real time by active control. The maximum zenith scan angle | ψmax | of FAST is 30° under conditions of the geometry selected in order that the feed does not illuminate the ground. The result of this paper shows that the maximum zenith scan angle | ψmax | can be extended to 69° by offsetting the feed.

Drift of spiral waves on nonuniformly curved surfaces

The evolution of spiral waves on nonuniformly curved surfaces is theoretically investigated in the framework of kinematic approach. We predict the existence of the drift proportional to the gradient of Gaussian curvature of the surface. In the excitable media with equal diffusion coefficients of activator and inhibitor the direction of the drift is perpendicular to the gradient of Gaussian curvature. If the diffusion coefficients are different the component of the velocity drift parallel to the gradient of Gaussian curvature appears. In the particular case of the paraboloid of revolution the spiral wave will “climb up” onto the top of the paraboloid. This theoretical prediction is confirmed by computer simulations. The drift of spiral waves towards the top of parabolic surface was observed in experiments with BZ reaction. The experimental results are also presented.

Steady-state shapes of growing crystals in the field of local nonequilibrium diffusion

It is analytically found that in the local nonequilibrium diffusion field the growing crystals have the following steady-state isosolutal shapes: the elliptical paraboloid, the paraboloid of revolution, the parabolic cylinder, and the parabolic platelet. The crystal interfaces have an arbitrary configuration if the growth velocity is equal to or greater than the speed of solute diffusion in the bulk system.