Average Radius Of Curvature Research Articles

Curvature Determination of Spinodal Interface in a Condensed Matter System

Laser scanning confocal microscopy has been used to study the time evolution of a three dimensional spatially bicontinuous structure of a phase-separated polymer blend undergoing the late stage of spinodal decomposition, as a model example of condensed matter systems. Both the mean and Gaussian curvatures of the interface between two coexisting phases have been directly measured for the first time from the reconstructed 3D image. The Gaussian curvature is negative, clearly demonstrating that the interface of the bicontinuous structure is anticlastic, while the mean curvature is close to zero. Self-similar growth of the structure is inferred from the time evolution of the average radius of curvature of the interface.

Non-intrusive measurements of bubble size and velocity

A non-intrusive measuring technique based on video-imaging has been developed for the measurement of bubble size, velocity and frequency. Measurements carried out with this method have been compared to those obtained by an optimized phase-Doppler system in standard configuration, for a wide range of bubble sizes produced from single injectors in a quiescent environment. The two measuring techniques have yielded velocities and frequencies that are in very good agreement while the size of spherical bubbles was consistently measured by both methods. The phase-Doppler system was also used to size oblate-spheroidal bubbles moving with their equatorial plane parallel to the scattering plane, yielding measurements reasonably close to the average radius of curvature of the bubbles in the neighborhood of the equatorial plane, as calculated from the video-imaging data. Both methods were used for detailed velocity measurements of the bubble-stream in the neighborhood of the injector tip. The observed bubble-velocity variation with the distance from the injector tip does not always display the usual increasing trend leading into the terminal velocity. When injection conditions are near the transition from discrete to jet injection mode and the bubbles are small, the latter decelerate into a terminal velocity due to direct interaction of successive bubbles at the injector tip. The measured terminal velocities of bubble-chains for a variety of bubble sizes and injection frequencies, are successfully predicted by using a far-field wake approximation to account for the drafting effect which is responsible for bubble-chain velocities higher than those of single bubbles.

Membrane curvature studied using two-dimensional NMR in fluid lipid bilayers.

The method of two-dimensional exchange spectroscopy in deuterium ${(}^{2}$H) nuclear magnetic resonance (NMR), originally developed for the study of slow, reorientational molecular motions in solids [S. Wefing and H. W. Spiess, J. Chem. Phys. 89, 1219 (1988)], has been applied to the study of multilamellar vesicle samples of phospholipid bilayer systems in their fluid (${\mathit{L}}_{\mathrm{\ensuremath{\alpha}}}$) phase. As is well known, the $^{2}\mathrm{H}$ NMR quadrupolar splitting in such fluid membranes is proportional to (3 ${\mathrm{cos}}^{2}$\ensuremath{\theta}-1), where \ensuremath{\theta} is the angle between the local bilayer surface normal and the external magnetic field. Measurements of two-dimensional exchange $^{2}\mathrm{H}$ NMR spectra using a mixing time ${\mathit{t}}_{\mathit{m}}$ were analyzed to give the reorientational angle distribution function for the local surface normals of individual lipid molecules comprising the membranes. A quantitative analysis of this correlation function taking into account the lateral diffusion of the lipid molecules demonstrated the sensitivity of the technique to the local curvature and shapes of the multilayers, which were shown to exhibit surface roughness on the mesoscopic length scale between about 10 nm and 1 \ensuremath{\mu}m. The results were consistent with an average radius of curvature between 1 and 2 \ensuremath{\mu}m, but approximately 15% of the molecules were located on relatively flat membranes having larger radii of curvature.

The normal glenohumeral relationships. An anatomical study of one hundred and forty shoulders.

We measured the dimensions of the humeral and glenoid articular surfaces in 140 shoulders that were representative of a given population of patients, and also evaluated several glenohumeral relationships. Ninety-six measurements were made in the shoulders of cadavera and forty-four, on magnetic resonance-imaging studies of living patients. Eighty-five per cent of the humeral measurements fell within eight fixed combinations of the radius of curvature and the thickness of the humeral head, in two-millimeter increments. The average radius of curvature of the humeral head in the coronal plane was 24 +/- 2.1 millimeters (range, nineteen to twenty-eight millimeters). The average thickness of the humeral head was 19 +/- 2.4 millimeters (range, fifteen to twenty-four millimeters). There was a wide variability in the size of the humeral head and a direct correlation between the differences in size and the heights in both men and women. The humeral articular surface was spherical in the center; however, the peripheral radius was two millimeters less in the axial plane than in the coronal plane. Thus, the peripheral contour of the articular surface was elliptical (ratio, 0.92). The radius of curvature of the glenoid, measured in the coronal plane, was an average of 2.3 +/- 0.2 millimeters greater than that of the humeral head. The average dimensions of the glenoid in the superior-inferior and anterior-posterior (lower half) directions were 39 +/- 3.5 millimeters (range, thirty to forty-eight millimeters) and 29 +/- 3.2 millimeters (range, twenty-one to thirty-five millimeters). The anterior-posterior dimension of the glenoid was pear-shaped, the lower half being larger than the top half. The ratio of the lower half to the top half was 1:0.80 +/- 0.01. There was a strong linear correlation between the lateral humeral offset and the size of the humeral head (radius of curvature and thickness). The average lateral humeral offset was 56 +/- 5.7 millimeters (range, forty-three to sixty-seven millimeters). The superior most point on the humeral articular surface was an average of 8 +/- 3.2 millimeters (range, three to twenty millimeters) cephalad to the top of the greater tuberosity. Our data show that reconstruction of the lateral humeral offset is important in optimization of the moment arm of the deltoid and rotator cuff and of the normal tension of the soft tissue after prosthetic reconstruction.

Defining the effective radius of curvature for a nonideal optical beam

A simple expression is given for evaluating the effective or average radius of curvature R(z) of an arbitrarily wrinkled, distorted, or nondiffraction-limited optical beam. This real-beam radius of curvature obeys the same free-space propagation equation as does the wavefront curvature of an ideal Gaussian beam. Extracting this correctable curvature from the beam at any plane simultaneously collimates the beam, makes that location a waist for the real beam, and minimizes the far-field beam divergence starting from that position.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Equation in complex variable of axisymmetrical deformation problems for a general shell of revolution

In this paper, the equation of axisymmetrical deformation problems for a general shell of revolution is derived in one complex variable under the usual Love-Kirchhoff assumption. In the case of circular ring shells, this equation may be simplified into the equation given by F.Tolke(1938)[3], R.A.Clark(1950)[4]and V.V.Novozhilov(1951)[5]. When the horizontal radius of the shell of revolution is much larger than the average radius of curvature of meridian curve, this equation in complex variable may be simplified into the equation for slander ring shells. If the ring shell is circular in shape, then this equation can be reduced into the equation in complex variable for slander circular ring shells given by this author (1979)[6]. If the form of elliptic cross-section is near a circle, then the equation of slander ring shell with near-circle ellipitic cross-section may be reduced to the complex variable equation similar in form for circular slander ring shells.

An improvement of distributed Bloch line model for low-q bubble materials

A way of improving the intrinsic accuracy of the distributed Bloch line model is proposed for use in practical simulations of vertical Bloch lines in small-diameter bubble materials with relatively small q value. The model is based on approximate equations of wall motion derived from the Gilbert equation. As the q value is decreased, the inexact nature of the equations of wall motion gives rise to an unbalanced residual component in the effective reversible force. The residual component is found to concentrate around a Bloch line and causes the calculated domain motion to be quite unreliable. It is also found that the density of the residual force is proportional to the inverse square of the average radius of curvature of the domain wall in the neighborhood of a Bloch line. A simple expression is derived for a correction force density which cancels the residual force. The derivation of the correction force density is presented together with numerical results showing the effectiveness of the correction. >

A REFINED CHORD-ARC METHOD OF CALCULATING GEODESICS

AbstractThe traditional methods of calculating geodesics by means of a chord-arc system use an average radius of curvature, based either on a general average for the region concerned, an average at the midlatitude, or an average using the values at the endpoints. Both the latter two methods are decidedly superior to the first, but are not the best that could be done, for when the geodesic is such that its midpoint occurs where it crosses the meridian due east-west, it is seen that both the above averages are inferior to one based on one at the true midpoint. We thus seek a better method.

Studies of the validity of the Kirchhoff approximation for rough‐surface scattering

The accuracy of the Kirchhoff approximation for randomly rough, pressure‐release surfaces has been examined through comparison with exact numerical results based on solving an integral equation. A Gaussian roughness spectrum was chosen for the surfaces, and a Monte Carlo procedure was used to obtain the average bistatic scattering cross section. Results will be presented that illustrate the breakdown of the Kirchhoff approximation as the incident grazing angle is reduced and as the average radius of curvature is reduced. The accuracy of shadowing corrections to the Kirchhoff approximation will also be discussed. [Work supported by ONR.]

Rise velocity of large bubbles in stagnant liquid in non-circular ducts

A method for estimating the rise velocity of large bubbles in non-circular vertical ducts is proposed. It is suggested here that the bubble rise velocity is controlled by the effective average radius of curvature at the bubble cap. This curvature is estimated on the basis of the suggested bubble shape, which is related to the duct geometry.

Interface Motion and Nonequilibrium Properties of the Random-Field Ising Model

The dynamics of a continuum interface model in a random medium are studied and the results applied to the random-field Ising model. We find that if the dimensionality $d&lt;5$, the interface will move only if a force beyond a finite depinning threshold ${F}_{c}\ensuremath{\sim}{h}^{\frac{4}{(5\ensuremath{-}d)}}$ is applied, where $h$ is the random field strength. Thus, when the random-field Ising model is quenched to low temperatures, there is a critical value ${R}_{c}\ensuremath{\sim}\frac{1}{{h}^{\frac{4}{(5\ensuremath{-}d)}}}$ for the average radius of curvature $R$ of the domain walls. If $R&gt;{R}_{c}$, the domain structure is frozen. If $R&lt;{R}_{c}$, the domain structure evolves until $R\ensuremath{\sim}{R}_{c}$.

Diffraction evidence for distorted graphite-like ribbons in an activated carbon of very large surface area

A carbon prepared from petroleum coke which possesses a very large active surface area (3500 m 2/gm) has been examined with X-ray diffraction techniques. Simultaneous analysis of the low and high angle scattering suggests that the material is composed of distorted ribbons of graphite-like layers. On the average, each ribbon contains only one or a very few graphite-like layers, and these layers are distorted with an average radius of curvature of ~ 25A. This very open structure results in a large surface area and gives rise to extensive small angle scattering.

The importance of non-planar bilayer regions in photosynthetic membranes and their stabilisation by galactolipids

Photosynthetic membranes contain considerable regions of high surface curvature, notably at their margins, where the average radius of curvature is about 10 nm. The proportion of total membrane lipid in the outer and inner thylakoid margin monolayers is estimated at 21% and 13%, respectively. The major thylakoid lipid, monogalactosyldiacylglycerol, is roughly cone-shaped and will not form complete lamellar bilayer phases, even in combination with other thylakoid lipids. It is proposed that this galactolipid plays a role in: (a) stabilising regions of concave curvature in thylakoids; and (b) packaging hydrophobic proteins in planar bilayer regions by means of inverted micelles. This model predicts substantial asymmetries in the distribution of lipids both across and along the thylakoid bilayer plane.

Scattering of an electromagnetic wave from a very rough semi-infinite dielectric plane (exact treatment of the boundary conditions)

The reflected and refractive fields are calculated for a very rough surface which is assumed to be a set of tangent planes with small corrections due to the local finite radii of curvature. A perturbation technique similar to those used in the moderately rough surface calculations is developed to treat the solution for the tangent planes as the unperturbed solution with the curvature correction as the perturbation. Our results are expressed in terms of the slope distribution function of the tangent planes and the average radius of curvature. Numerical results for scattering of both polarizations are obtained. The depolarization calculation is shown to be consistent with the experimental data. Further, the relationship between the surface model and the conventional Gaussian model is discussed, and a comparison with previous works is given.

Explanation of the experimental results of light backscattered from a very rough surface*

We have calculated reflected and refracted electromagnetic waves from a rough surface with the boundary conditions exactly satisfied. The surface model consists of three parameters: the dimension of the independent cells, h; the parameter in the slope distribution function of the tangent plane t0; and the average radius of curvature of the surface, R. Within the reasonable range of these parameters, we have calculated the polarized component P and the depolarized component D measured by Renau et al. and have found the following features of the scattering of the reflected wave from a rough surface that have not been previously explained theoretically. (i) P and D have very different angular dependences. (ii) Whereas P is a sensitive function of t0, D is almost independent of t0. (iii) D does not vanish at all angles. A qualitative comparison between the data and the calculation is given.

Directional slope and curvature distributions of wind waves

The slope and curvature distributions of wind waves along two principal axes (upwind-downwind and cross-wind) have been measured in a laboratory tank under various wind velocities. In both directions, the slope distributions are very closely Gaussian, and the components of the mean-square water-surface slope vary loga,rithmically with the friction velocity of the wind. As the windvelocityincreases, the ratio of the upwind-downwind and cross-wind components increases and lies between 0.5 and 0.6 at high wind velocities in the gravity-governed regime of wind-wave interaction. The radius of .water-surface curvature, along either direction of measurement, is generally found to be greater at a steeper viewing angle from the normal to the mean water surface. The average radius of curvature of the disturbed surface varies inversely with the friction velocity of the wind. The ratio of the upwind-downwind and cross-wind components of the average radius of curvature is unity at all wind velocities, indicating that the wind-disturbed water surface is isotropic on the smallest scale. Other results show that both the slope and the curvature distributions are asymmetric along the upwind-downwind direction, either because of the presence of parasitic capillaries or because of the occurrence of wave breaking. The results also indicate that even the high frequency portion of the spectrum is saturated locally but the spectrum is not universal, and that the long waves suppress the growth of the nearly saturated ripples.

Optimization of Tension Level and Stinger Length for Offshore Pipeline Installation

Offshore pipelines have been installed for a period of some 10 to 20 years using a variety of methods. This paper presents a new analysis method for the optimization of certain operation parameters in laying continuous pipe under tension from a barge. This method of optimization is now being used by one of the major offshore contractors. The solution is given by matching geometrical parameters through graphical overlays of network charts obtained by a computerized finite-element iterative solution. The solution is based on the requirement that the tension level be optimized with respect to either the minimum stinger length or the maximum average radius of curvature of the stinger.

Losses and Impulse Response of a Parabolic Index Fiber With Random Bends

The coupling coefficients of the modes of a parabolic index fiber with randomly curved axis are derived and are used to compute its excess losses and impulse response. It is found that bends with a period comparable to the natural ray oscillation period in the parabolic index medium are catastrophic. The average radius of curvature R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> of a guide composed of circular sections with an average length of 1 cm must not decrease below approximately R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> = 1 m. Mode coupling by random bends has the tendency to reduce the width of the impulse response function. However, this improvement is accompanied by losses. Reducing the width of the impulse response for coupled mode operation to half its uncoupled width causes 0.7 dB additional loss, a ten-fold reduction of the pulse width costs 18 dB.

An application of topographical analysis to the wear of polymers

Profiles of abraded metal surfaces have been analysed in terms of a random surface model to determine the average radius of curvature of the asperities, R av . The wear rates of polymers against these surfaces vary inversely with R av , and the form of the variation is consistent with a fatigue theory of wear. Analyses have also been made of the topography of metal surfaces modified, during sliding against carbon fibre reinforced polymers, by abrasion, corrosion or transfer. In all cases the wear rates of the composites again vary inversely with R av . The results provide quantitative support for the conclusion that the topography of the metal counterface is the predominant factor in determining the magnitude of the wear rate of polymers and composites in widely differing conditions of sliding.

The phylogenetic allometry of the semicircular canals of small fishes

The working frequency range of the semicircular canals is a function of several variables including their tube radii and radii of curvature. The average radius of curvature \(\bar R\), and the outer tube radius squared, \(\bar r_{\text{0}}^{\text{2}} \), were determined from preserved specimens of 33 species (19 families) of small (0.4–400 g) fishes. These had the following allometries with respect to body mass: $$\bar R = {\text{0}}{\text{.671 }}M^{{\text{0}}{\text{.264}} \pm {\text{0}}{\text{.038}}} $$ $$\bar r_{\text{0}}^{\text{2}} = {\text{0}}{\text{.011 }}M^{{\text{0}}{\text{.0348}} \pm {\text{0}}{\text{.077}}} $$ 02=0.011 M0.348±0.077 where M is given in g, \(\bar R\) and \(\bar r_{\text{0}} \) in mm and the error limits include 95% of all observations.