Paraboloid Of Revolution Research Articles

The influence of forced convection on solidification interfaces

The tendency of growing columnar dendrites to incline into the flow direction under the influence of some shear flow at the solidification front is a well-known phenomenon in directionally solidifying alloys. According to literature this effect is due to a deformation of the solute layer around a dendrite tip. It is assumed that solute-rich liquid is swept away from the upstream side of the dendrite tip to the downstream side, leading to a local enrichment of the solute and lowering of the solidus temperature on the downstream side. This gives preferential growth conditions on the upstream side, thus causing the crystal tips to incline upstream. On the basis of stirring experiments carried out at the Giesserei-Institut the deformation of the solute layer by forced convection at a 2-dimensional sine-wave-modulated interface and a 3-dimensional paraboloid of revolution was studied using a FEM-code (FIDAP) to solve the momentum and species transport equations under steady state conditions. The sine-wave-modulated interface could not account for the solute pile-up at the downstream side, especially after incorporating the influence of a solute partition coefficient at the interface. After adapting the 3-dimensional paraboloid to the analytic Ivantsov-solution describing the growth of an isoconcentrate dendrite tip a higher solute concentration could be observed at the downstream side of the dendrite tip. The results indicate that the sweep away effect actually occurs at the dendrite tip as long as the diffusion layers around the dendrite tips do not influence each other.

Dendritic solidification of xenon

Experiments on xenon dendrites growing in a three-dimensional volume of pure, supercooled melt have been performed. The shapes of the dendrites have been determined with high lateral resolution. The dendrites grow in a stable mode and do not show oscillations in the growth velocity nor in the curvature of the dendrite tip. The dendrites do not have the shape of a paraboloid of revolution. Surface anisotropies determine the shape of the dendrites. Four fins grow along a dendrite. The contour of these fins can be described by a power law z ∼ |x|β with β ≈ 1.67. These experimental results are in agreement with the analytical work of E. Brener [Phys. Rev. Lett. 71 (1993) 3653]. Both experiments and analytical work are compatible with the assumption that thermal fluctuations initiate side branching. Tip oscillations can be excluded.

Bending and stretching actions in shallow domes. Part 1. Analytical derivations

In this two-part set of articles the response of shallow axisymmetric shells with circular plan is investigated with the aid of the two-surface shell theory. Such an approach has the unique advantage of elucidating, both qualitatively, and quantitatively, the complex—and key—interaction of bending and stretching actions in such shells in a simple and direct manner. Although discussion centres upon a shallow paraboloid of revolution, the general features emerging from the study are, obviously, relevant for the generic family of shallow domes (e.g. spherical) covering the various limiting cases of support around their circumference. This first paper begins with a two-surface exposition of the set of equations describing the behaviour of the shell. Next, analytical bending solutions to the specific practical problem wherein the shell is subjected to uniformly distributed vertical loading (corresponding, say, to its self-weight) are presented; and this is followed by the working out of closed-form expressions for the distributions of bending and stretching effects in theshell under consideration.

A mathematician among the molasses barrels: Maclaurin's unpublished memoir on volumes Introduction: Maclaurin's memoir and its place in eighteenth-century Scotland

Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken?The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698–1746). This manuscript [14], the only original mathematical work by Maclaurin not previously printed, is published here for the first time, with the permission of the Trustees of the National Library of Scotland. (An almost identical copy [15] exists in the Edinburgh University Library.) In this work, Maclaurin proved that the difference between the cylinder constructed as above and the frustum of the given solid depends only on the height of the frustum, not the position of the frustum in the solid. When the solid is a cone, Maclaurin showed that its frustum exceeds the corresponding cylinder by one fourth the volume of a similar cone with the same height. For a sphere, the cylinder exceeds the frustum by one half the volume of the sphere whose diameter is equal to the height of the frustum; this holds, he observed, for all spheres. He derived analogous results for the ellipsoid and hyperboloid of revolution. Finally, for the paraboloid of revolution, he proved that the cylinder is precisely equal to the frustum.

Experimental study of dendritic growth with an external flow

In this paper, we report new experimental results on dendritic growth in the presence of an imposed flow; we also summarise the theoretical approaches to the problem, and discuss previous results obtained on the subject. In a first part, we describe the available theoretical descriptions of the flow around a paraboloid of revolution, performed under various approximations: potential, uniform or Oseen flow. Each of these leads to an expression for the temperature or concentration field, and relates the supercooling or supersaturation to the Péclet number of the dendrite. Several predictions on the selected growth constant are also presented. The experiments themselves are performed on the growth of ammonium bromide with an external flow; we measure the concentration field around the crystal during the growth process. We also characterise the velocity field. We find supersaturations in good agreement with predictions based on potential or uniform flows. The growth constant is also determined in this system. We finally suggest that the available experimental results on dendritic growth in the presence of a flow can be interpreted within a unique theoretical framework.

A numerical model for the calculation of the growth velocity of nonisothermal parabolic dendrites

For a circular paraboloid of revolution growing in a shape preserving manner neither the surface curvature, nor the local interface velocity are constant on the freezing front. Yet within the widely quoted model [J. Lipton, W. Kurz and R. Trivedi, Acta Metall. 35, 957 (1987)] for the calculation of dendritic growth velocities the kinetic and GibbsThomson undercoolings evaluated at the dendrite tip are assumed to apply equally over the whole dendrite surface, approximating the non-isothermal dendrite as an isothermal dendrite with a reduced surface melting temperature. Reasons are discussed why this approach may seriously overestimate the growth velocity at high undercooling, where kinetic effects are important. Using a finite difference model the full, non-isothermal growth problem is solved for the solidification of pure Ni. The model shows that an undercooling of 175 K neglecting non-isothermal effects leads to a 35% error in the calculated growth velocity. Comparison with the available experimental data suggest an adequate fit to the data for Ni can be made without the need for adjustable parameters.

Dendritic Growth in Undercooled Melt with forced Convection: Theory.

Analytical solutions for melt flow with Oseen viscous flow approximation are obtained for the Navier-Stokes equations in the region near the tip of a dendrite of which shape is approximated to be a paraboloid of revolution. Temperature is also analyzed in the presence of flow in a melt. A theory of dendritic growth is proposed with flow in the undercooled pure melt. Local equilibrium condition at the interface is applied in the theory. The predicted growth rate of the tip of dendrite as functions of the melt undrcooling with and without forced flow velocity is successfully compared with the experimental results for the pure succinonitrile.

Photogrammetric tracking of tracer particles in modelled ocean flows

Three-dimensional particle tracking velocimetry (3D-PTV) has been used to quantify simulated ocean circulation and internal waves, in a 5 m diameter rotating tank. According to hydrostatic theory the water surface in the rotating tank will attain the surface of an elliptic paraboloid of revolution. Three “off the shelf” video cameras mounted above the water surface, rotating with the tank, monitored the 3D displacements of sparsely distributed surface and neutrally buoyant submerged particles (diameter of 0.01 m) inside a water volume of ca. 0.5 × 0.7 × 0.5 m 3. A numerical procedure for the calibrations of the photogrammetric system is described. The particles were automatically tracked through a time series of consistent image triplets using an epipolar approach. The experiments gave a standard error σ o = 0.33 pixels and a repetition accuracy, RMSE = 0.15 mm in X, Y and 0.5 mm in Z (the height).

The first and second fundamental boundary-value problems of elasticity theory for a transversally isotropic paraboloid of revolution

We give an exact solution of the interior and exterior problems of elasticity theory for a transversally isotropic paraboloid of revolution in the case when the stresses prescribed on its surface or the displacements along one variable can be represented by a Hankel integral and those along the other variable can be expanded in a trigonometric series. It is assumed that the roots of the characteristic equation are of multiplicity greater than one.

Les familles exponentielles statistiques invariantes par les groupes du cône et du paraboloide de révolution

Statistical exponential families invariant with respect to the groups of the cone or the paraboloid of revolution are discussed. B(x, y) denotes the symmetric bilinear form on x 0 y 0 – x 1 y 1 – ·· ·– xdyd on ℝ d+1, C denotes the cone of revolution in ℝ d+1 {x; B(x,x) > 0 and x 0 > 0}, and, for p > ½(d−1), μ p is the positive measure on ℝ d+1 defined by its Laplace transform ⨍ exp (B(x,y))μ p (dy) = (B(y,y))−p for y on C. More precisely, if p > ½ (d−1) one has and μ½(d−1) concentrated on the boundary ∂C. This paper studies the natural exponential families

Les familles exponentielles statistiques invariantes par les groupes du cône et du paraboloide de révolution

Statistical exponential families invariant with respect to the groups of the cone or the paraboloid of revolution are discussed. B(x, y) denotes the symmetric bilinear form on x0y0 – x1y1 – ·· ·– xdyd on ℝd+1, C denotes the cone of revolution in ℝd+1 {x; B(x,x) > 0 and x0 > 0}, and, for p > ½(d−1), μ p is the positive measure on ℝd+1 defined by its Laplace transform ⨍ exp (B(x,y))μp(dy) = (B(y,y))−p for y on C. More precisely, if p > ½ (d−1) one has and μ½(d−1) concentrated on the boundary ∂C. This paper studies the natural exponential families

Free Vibration Analysis of Liquid Motion in a Rotating Circular Cylindrical Tank (Elliptic Mode).

Free vibration analysis is presented for liquid surface motion in a rigid circular cylindrical tank which is rotating around its symmetrical axis at constant angular velocity. It is considered that the equilibrium surface forms a paraboloid of revolution. Basic equations of liquid motion are derived by using the acceleration potential under the assumptions that the gravity. effect is dominant in comparison with the centrifugal force and that the liquid is inviscid and incompressible. It is confirmed that both elliptic and parabolic vibration modes of free surface oscillation appear due to tank rotation. The characteristic equations for the elliptic modes are derived by applying Galerkin's method. It is noted that the natural frequencies of the elliptic mode are separated by the Coriolis force. An experiment was conducted using a model tank. A good agreement was found between the theoretical and experimental results.

Surface-ray tracing on hybrid surfaces of revolution for UTD mutual coupling analysis

An angle invariance property based on Hertz's principle of particle dynamics is employed to facilitate the surface-ray tracing on nondevelopable hybrid quadric surfaces of revolution (h-QUASOR's). This property, when used in conjunction with a geodesic constant method, yields analytical expressions for all the ray-parameters required in the UTD formulation. Differential geometrical considerations require that some of the ray-parameters (defined heuristically in the UTD for the canonical convex surfaces) be modified before the UTD can be applied to such hybrid surfaces. Mutual coupling results for finite-dimensional slots have been presented as an example on a satellite launch vehicle modeled by general paraboloid of revolution and right circular cylinder. >

Three-Dimensional Analytical Temperature Field Around the Welding Cavity Produced by a Moving Distributed High-Intensity Beam

An analytical solution for the three-dimensional temperature field in the liquid and heat-affected zones around a welding cavity produced by a moving distributed low- or high-power-density-beam is provided. The incident energy rate distribution is assumed to be Gaussian and the cavity is idealized by a paraboloid of revolution in workpieces of infinite, semi-infinite, or finite thicknesses. The present study finds that temperature fields can be described by the Laguerre and confluent hypergeometric functions. By satisfying a momentum balance at the cavity base and utilizing a consequence of the second law of thermodynamics, the depth of penetration is uniquely determined. The results show that the predicted depths and temperatures of the cavity agree with available experimental data. Some crucial factors affecting the transition from low- to high-power-density-beam welding are presented.

A novel ray tracing on general paraboloids of revolution for UTD applications

Analytical expressions have been obtained for the ray-geometry parameters on the general paraboloid of revolution (GPOR) using a novel technique known as the geodesic constant method. These ray parameters are in a one-parameter form and can be readily employed in high-frequency EM antenna scattering problems. As an example, the uniform theory of diffraction (UTD) mutual admittance results are presented for two antennas located on the GPOR arbitrarily, in three dimensions. >

Growing tips of type I collagen fibrils formed in vitro are near-paraboloidal in shape, implying a reciprocal relationship between accretion and diameter.

Collagen fibrils generated in vitro at 37 degrees C by enzymic removal of C-terminal propeptides from type I pC-collagen (an intermediate in the normal processing of type I procollagen to collagen containing the C-terminal propeptides but not the N-terminal propeptides) display shape polarity, with one tip fine tapered and the other coarse tapered. Mass measurements by scanning transmission electron microscopy show that the mass per unit length along both kinds of tip increases roughly linearly over distances of approximately 100 D periods from the fibril end [D (axial periodicity) = 67 nm]. The fine tips of fibrils of widely differing lengths exhibit near-identical mass distributions, the mass in all cases increasing at the rate of approximately 17 molecules per D period, irrespective of fibril length. Coarse tips display less regular behavior. These results show that (i) the shape of a fine tip is not conical but resembles more closely a paraboloid of revolution, and (ii) for this shape to be maintained throughout growth, accretion (rate of mass uptake per unit area) cannot everywhere be the same on the surface of the tip but must decrease as the diameter increases. To a first approximation, accretion alpha (diameter)-1.

Energy-Beam redistribution and absorption in a drilling or welding cavity

Energy redistributed and absorbed in the cavity produced by a high-intensity beam during drilling and welding is systematically and quantitatively investigated. The incident energy flux is assumed to be a realistic Gaussian distribution, and the cavity is idealized by a paraboloid of revolution having both specular and diffuse reflectivities. By using a Monte Carlo method, the results show that the energy absorbed by the cavity wall deviates markedly from the Gaussian distribution. Effects of the cavity depth-to-opening radius ratio, absorptivity, and specular reflectivity on the absorbed energy are presented. A comparison of incident energy fluxes of uniform and Gaussian distributions is also made.

Finite element analysis of laminated composite paraboloid of revolution shells

A generalized formulation for the doubly curved laminated composite shell is attempted using eight-noded curved quadratic isoparametric finite elements with all three radii of curvature. The formulation is also applied to the isotropic material as a special case. In the present investigation, only the paraboloid of revolution is taken up for computing the deflections and stress resultants. Various parametric studies are carried out and the current results for both isotropic and laminated composite shells are compared with those available in the published literature. The shape functions are obtained from interpolation polynomial and the element stiffness matrices are formed on the basis of macromechanical analysis of laminates using the principle of minimum potential energy.

Finite element analysis of laminated composite conoidal shell structures

A general finite element formulation with the eight-noded isoparametric curved quadratic element applicable for the analysis of doubly curved laminated composite shell structures has been reported in another paper. The bending behaviour of composite shells having the form of paraboloid of revolution has been investigated. In the present paper the finite element formulation is extended to analyse the conoidal shells. Numerical results are obtained for two isotropic conoidal shell problems and one laminated composite shell problem with eight different ply lay-ups and layers. The authors' results of the two isotropic problems compare well with those available in published literature. The numerical results of deflection and stress resultants of eight different types of the third problem are discussed in order to estimate their relative performance for the uniformly distributed loading. The authors further propose a comparative performance matrix to study the relative performance of two laminated composite materials which will be of great help in design.

Novel geodesic splitting on paraboloid of revolution with applications to ray-theoretic analysis

Computer simulations have shown a novel geodesic splitting on the paraboloid of revolution leading to a multiplicity of surface ray paths. Such a phenomenon would have wide ramifications for wave propagation problems in general, besides applications in target-detection problems and the computational requirements of ray-theoretic formulations such as the UTD, in computing the antenna characteristics in the high-frequency domain.