Axisymmetric Potential Research Articles

Axisymmetric potential flow calculations. Part 2: Design mode

The pure inverse problem of designing a body with a prescribed surface pressure in incompressible axisymmetric potential flow is solved using the streamfunction as a co-ordinate in the computational plane. The problem is formulated mathematically as a mixed boundary value problem and the solution is obtained using the finite-difference technique. The body shape is computed simultaneously with the entire flow field and does not have to be iteratively adjusted to meet the target surface pressure. Results are presented for elliptical, NACA 0012 and minimum-drag cross-sections.

Testing the gravitational field in elliptical galaxies - NGC 5077

NGC 5077 is an elliptical galaxy with a disk of ionized gas along its minor axis. Observations of the structure and kinematics of this gas disk allow us to investigate both the intrinsic shape and the mass distribution of the underlying galaxy. The rotation axis of the gas disk is displaced by 23-degrees +/- 5-degrees from the apparent major axis of the galaxy, so the gas cannot lie in the equatorial plane of an axisymmetric potential. If the galaxy is triaxial and the gas disk has settled into a principal plane then there are two possible configurations, both with the gas on elliptical orbits. One is an oblate triaxial system with the gas in a plane perpendicular to the longest axis, the other is a prolate triaxial galaxy with the gas in the plane perpendicular to the shortest axis. Alternatively, the gas may lie in the warped and precessing polar ring around an oblate galaxy - in this case the gas must have been acquired recently, probably less than 10(9) yr ago. The observed rotation curve of the gas rises slowly, reaching a peak at approximately 10 from the nucleus. For circular orbits in a spherical potential, this would imply a decrease of M/L toward the center, since the core radius of the light is 1 or less. For gas on elliptic orbits, however, the precise shape of the observed rotation curve will depend on the viewing angle. If the gas lies in a warped polar ring, material further out may be seen in projection close to the center so that the observed rotation velocity is much less than the true circular speed. For NGC 5077, we construct various triaxial and oblate mass models which have constant M/L and are consistent both with the observed velocity fields of gas and stars and with the central mass-to-light ratio implied by the observed stellar velocity dispersion. Thus the slowly rising rotation curve of the gas in NGC 5077 does not require the presence of dark matter in the inner regions.

Unsteady Forces on a Slender Body of Revolution in a Tube. (Effect of Tube and/or Body Deformation).

The problem under consideration is the axisymmetric unsteady potential flow past a slender body of revolution capable of arbitrary radial deformations in a tube which also performs wavy deformation, where the mean radius of the tube wall is the same order as that of the body. In the analysis, the Stokes stream function is introduced and the method of matched asymptotic expansions is used. The flow is examined for two different time parameters. As a result, unsteady contributions to the flow generated by a deformable slender body and tube are obtained up to the second-order approximation, and effects of various parameters on the axial force coefficients are examined in two cases.

Higher order resonant orbits

I present higher-order resonances of the tube and boomerang orbits in the co-rotating meridional plane of an axisymmetric galactic potential. Additionally, mirror forms of these orbits are given.

Stochasticity in galactic models

We study the effect of a spiral-like perturbation in the orbits of an axisymmetric galactic potential using surfaces of section. We find that the perturbation induces stochasticity and diffusion of orbits in phase space.

Zonal Schrödinger operators on then-sphere: Inverse spectral problem and rigidity

We study the Direct and Inverse Spectral Problems for a class of Schrodinger operatorsH=−Δ+V onSn withzonal (axisymmetric) potentials. Spectrum ofH is known to consist of clusters of eigenvalues {λkm=k(k+n-1)+μkm:m≦k}. The main result of the work is to derive asymptotic expansion of spectral shifts {μkm} in powers ofk−1, and to link coefficients of the expansion to certain transforms ofV. As a corollary we solve the Inverse Problem, get explicit formulae for the Weinsteinband-invariants of cluster distribution measures, and establishlocal spectral rigidity for zonal potential. The latter provides a partial answer to a long standing Spectral Rigidity Hypothesis of V. Guillemin.

Axisymmetric potential flow calculations. Part 1: Analysis mode

Using a natural body-fitted co-ordinate system, based on the streamlines, incompressible potential flow over an axisymmetric body at zero incidence is formulated as a Dirichlet boundary-value problem for the unknown co-ordinate r(x, ψ ). Flow over various body shapes is computed using a finite-difference scheme and SLOR, on both uniform and clustered grids. The results show excellent agreement with previous researchers. The coding is very simple, the formulation can be extended to compressible flows and can also be used to solve the inverse problem.

Flow in the neighborhood of the cusp of an axisymmetric cavity

The steady-state axisymmetric potential flow of an ideal incompressible fluid is considered. The structure of the singularity at the point where the free stream surface leaves the axis of symmetry is investigated.

On generalised toroidal-poloidal solutions of vector field equations

AbstractThe orthogonal coordinate systems ξi(i = 1, 2, 3) are determined, in which the gneralised toroidal and poloidal fields, defined respectively by T{T} = ∇ × {T∇ξ1} and S{S} = ∇ × T{S}, have the following three properties:GP1 Decoupling of the vector Helmholtz equation: There exist linear differential operators L1 and L2 such that Hu = 0, where H is the vector Helmholtz operator [see equation (1)] and u = T{T} + S {S}, if and only if L1T = 0 and L2S = 0.GP2 OrthogonalityGP3 Closure: ∇ × S{S} is a T field.Two choices of T and S fields are considered: type I fields with potentials T and S, which may depend on ξ1, ξ2 and ξ3, and type II fields with ξ1-independent potentials. It is shown that properties GP1–GP3 only hold for type I fields in spherical and cylindrical coordinate systems, and for type II fields in azimuthal and cylindrical coordinate systems with axisymmetric and two-dimensional potentials, respectively. Analogues of GP1 for the vector wave and diffusion equations, and the Navier equation of linear elasticity, are also only true in the same four cases. Generalisations of type I and II T and S fields to arbitrary coordinate systems are indicated.

A representation of the solutions of the generalized Cauchy-Riemann equations and its applications

A method of integral representations for the generalized Cauchy-Riemann system in terms of an arbitrary analytic function, similar to the well-known Whittaker-Polozhii representation /1/, is developed. The representation includes various well-known results as special cases, and the limiting case leads to the classical representation of the theory of a generalized axisymmetric potential. The representations established are used to reduce mixed problems for the system to paired equations and then to a Fredholm equation of the second kind. At the same time, a device is described for regularizing paired equations, and a case in which a closed solution exists is presented. The results are extended to a sytem of more-general form and also to second-order equations, whose type and dimensionality are not essential. It is shown that the integral operators constructed here convert the solution of a parabolic or hyperbolic equation with variable coefficients into a solution of the classical equations of heat conduction and wave propagation, thus furnishing an explicit representation for solutions of the corresponding Cauchy problems. The effectiveness of the approach is demonstrated with reference to the problem of inflow in a fissure in an inhomogeneous layer of finite width. Simple formulae for the pressure and discharge of fissures are presented.

Particular Integrable Cases in Rigid Body Dynamics

AbstractThe problem of motion of a rigid body about a fixed point under the action of axisymmetric potential and gyroscopic forces is considered. Particular solvable cases are found, which generalize the results due to Hess, Kharlamov and others.

A new kind of stellar orbit in a galactic potential

Numerical integrations of a star's motion in an axisymmetric galactic potential have unveiled a new kind of orbit in the meridional plane. Especially, neutron stars can be expected to move on such orbits.

Abundances of Classical Cepheids and Evidence for Secondary Star-Formation

In addition to the well-known radial abundance gradient in the disc of our Galaxy, there are indications of small-scale variations (Talent and Dufour 1979, Giridhar 1983). We have compiled spectroscopic abundances for long period Cepheids and to eliminate the effect of differences in ages, derived their places of formation by calculating their galactic orbits backwards in time. The stellar orbits were calculated considering the axisymmetric gravitational potential as well as the spiral potential due to the density wave. The parameters related to spiral waves are taken from Yuan (1969). The equations of motion in the (ζ, η, ξ) system were numerically integrated using the Runge-Kutta method and birthsites were transformed to (ζ, η) in the reference frame of the spiral pattern. The details of the computations are described by Giridhar (1985). In addition to the accepted value of 13.5 km s−1 kpc−1 for pattern velocity, birthsites were also calculated for ωp = 11.5 and 15.5 km s−1 kpc−1. Figure 1 shows the birthsites in the galactic plane for the three values of the pattern velocity. The Cepheids in the figure can be identified through the serial number in Table I which contains our compilation of Fe/H and derived birthsites. It is obvious from the figure that the systematic shifts in birthsites for different pattern velocities are not large enough to affect the assignment of birthsites to different arms.

A difference solution of the one-particle Schrödinger equation with axisymmetric potential

A difference method is given for solving the stationary Schrödinger equation with two independent variables. For the equation in cylindrical coordinates (ρ, z), the effect of the order of approximation of the derivatives with respect to ρ and of the difference forms of Richardsons's extrapolation on the accuracy of computing the energy eigenvalues is studied. The ionization energies of the hydrogen atom in a widely variable magnetic field are obtained to high accuracy.

A Method for the Study of Gas Dynamics in Galactic Potentials

AbstractA particle method for the simulation of the global evolution of cold gas in galactic potentials is described. It incorporates an efficient artificial viscosity, based on the smoothed particle hydrodynamics approach, which conserves both linear and angular momenta and in the continuum limit is equivalent to a Navier-Stokes viscosity in a compressible gas. Parameters of this viscosity have been calibrated in such a way as to make a gas disk, embedded in an axisymmetric galactic potential and originally inclined to its equatorial plane, settle to the preferred plane on the time scale characterizing the differential precession. After a careful analysis of the previously published settling times we find that our results are consistent with Steiman-Cameron’s (1984) lower limits, and we argue that Simonson’s (1982) time scales have been seriously underestimated. We also demonstrate that two dimensional methods based on rigid rings do not model the evolution of a differentially precessing settling gas disk realistically and cannot be used to study the morphology of gas distribution.

Evolution of the velocity distribution in galactic disks

This paper presents a numerical investigation of the evolution of the three-dimensional velocity distribution and of the vertical density distribution in galactic disks. The physical mechanism is the scattering of stars off giant molecular clouds (GMCs) which heat up the stellar populations. A population of GMCs is embedded in the fixed potential of a constant scale height exponential disk and a nearly isothermal halo. The stars orbit in the fixed axisymmetric potential and are scattered by the GMCs which are modeled as soft particles. The equations of motion are integrated directly. Initially, the three velocity dispersions each grow as sigma(t)proportional(1+(t/tau))/sup 0.38/. Asymptotically the velocity dispersions in the plane grow as the quarter power of time while the vertical dispersion grows as sigma/sub z/(t)proportionalt/sup 0.31/. The velocity ellipsoid grows slowly rounder with time. Typically it has an axial ratio of about 0.6. The functional form and growth rate for the velocity dispersions are in fairly good agreement with published theoretical predictions based on the Fokker-Planck equation. The numerical growth rates are sufficient to explain the velocity dispersions of young stellar populations in the solar neighborhood. It is, however, necessary to postulate higher masses or larger numbers of GMCs at earlier timesmore » in order to explain the velocity dispersion of the hottest disk populations. The velocity distribution is vertically isothermal and Gaussian. The vertical density distribution is also Gaussian.« less

An algorithm for the numerical extension of a generalized axially-symmetric potential from the symmetry axis

An ill-posed problem of extending a generalized axisymmetric potential from the axis of symmetry is considered. A conditionally correct formulation of the problem and a stability estimate on a compact set of correctness are given. An algorithm for numerical approximation, which in the uniform metric conditionally approximates the solutions belonging to the compact set is constructed.

Iterative and Direct Determinations of the Gravitational Potential of Axisymmetric Mass Distributions in Spherical Coordinates

AbstractHighly flattened objects may appear in early stages as well as in late stages of stellar evolution. The calulation of their axisymmetric gravitational potential in spherical coordinates requires a lot of computer time. In this paper we compare iterative solutions of the Poisson differential equation with direct evaluations of the Poisson integral with respect to accuracy and time consumption. further, we derived a fast direct method which is superior in computer time to the discussed methods whereas its accuracy remains nearly unchanged.

ON A DARCY-FLOW FORMULA AND THE POTENTIAL OF SLENDER AXIAL SOURCES

Abstract A slender-body theory is developed for the axisymmetric Darcy flow from a large reservior to an isolated well bore. The resulting analysis serves to eliminate from the standard petroleum-engineering formula: the generally undefined parameter re , representing fictitious distance from an injection or production well to the undisturbed reservoir. In particular, we show that, upon equating re , to the well length l, one can interpret the above formula as an exact asymptotic result under fairly general circumstances including variations of the local well-bore pressure pw and radial injection velocity v 0 with axial position. We also present higher-order corrections to the asymptotic result, in terms of the well slenderness ratio re /l, by making use of a previous perturbation analysis of Handelsman and Keller (1967) for axisymmetric potential flow. Our results have application to other problems involving the potential field around slender sources.

Isokinetic probe sampling from slurry pipelines

AbstractA theoretical and experimental study of the errors arising in sampling through probes projecting into the flow has been conducted. Water–sand slurries with particles of surface‐mean diameter between 0.165 and 0.72 mm were considered. Theoretical predictions were performed assuming the mixture could be described by the axisymmetric potential flow relationship. Satisfactory agreement in the important intermediate K value region was achieved for tapered or thin‐walled probes. However, significant errors arise when thick‐walled blunt probes are used.