Axisymmetric Potential Research Articles

Potentials and flows associated with a line segment

Integral transformations are developed to construct three and five axisymmetric potentials for a needle or straight line segment. These potentials are applied to flow past a needle and one of the main results is for streaming flow past a line segment in which the fluid velocity vanishes on the boundary. This solution may also be regarded as a Stokes flow or an inviscid potential flow.

Axisymmetric potential flow in a circular tube

Axisymmetric Green's functions and the associated stream functions, which satisfy the boundary condition that the wall of the tube is a stream surface, are presented for various singularities. These include a source on the axis an axially-oriented doublet, a source ring, a source disk, and a vortex ring. Results are expressed as integrals of the modified Bessel functions. These can be applied to formulate Fredholm integral equations of either the first or second kind for determining the axisymmetric, irrotational flow about bodies of revolution in a tube. In the present work, only those of the first kind are treated, including integral equations for axial source and doublet distributions, and a vortex sheet on the surface of the body. Three different methods for solving each of these three integral equations are examined: the method of piecewise-constant singularity elements of von Karman, Kaplan's method of expanding the unknown distribtuion as a series of Legendre polynomials, and solutions by a technique of eliminating peaks in the kernel (representing integrals by means of a quadrature formula) and solving the resulting set of linear equations by means of a suitable iteration formula. Numerical results for three of the methods, applied to a spheroid, are presented. The resulting added masses and a comparison with predictions from slenderbody theory are given in Appendixes.

Gravitational perturbations of equatorial orbits

Asymptotic solutions are developed for the motion of a geocentric satellite in the equatorial plane due to gravitational perturbations such as nonsphericity (especially oblateness) of the primary body. Axisymmetric potentials are considered. A class of transformations is developed and the equations of motion are solved by the method of generalized multiple scales. Further it is shown that the equations of motion can be transformed into the required form to within any specified degree of accuracy. The transformations form an Abelian group of infinite order which leaves the differential equations of motion invariant. Solutions are developed in terms of elementary functions instead of elliptic or other higher transcendental functions and are shown to agree with known results.

On the applicability of the third integral of motion

Discussion of the resonance case of the motion of a star in an axisymmetric galactic potential where the frequencies alpha sub 1 and alpha sub 2 are rationally dependent. A rigorous procedure is set forth to demonstrate the existence of a family of invariant curves on which a mapping of this motion can be based, with the aid of a third integral of motion.

Value distribution of potentials in three real variables

The object of this paper is to study the value distribution of potentials in three real variables by means of the Bergman integral operator with methods drawn from the analytic theory of polynomials. In this paper we obtain results on the value distribution of nonaxisymmetric potentials which generalize recent papers of the author [3] and of Morris Marden [2] which treat the value distribution of axisymmetric potentials, generalized axisymmetric potentials, and in Marden's paper, some special classes of harmonic polynomials in three real variables. Let (x, y, z) be a point in E3. A function 1i=T(x, y, z) with continuous second partial derivatives in a domain G shall be referred to as a potential if it satisfies (1) Tcx +Tyy + Tzz = O in G. Bergman has shown [1, p. 43] that potentials are generated by the integral operator (2) T(x, y, z) = t f f(, ()4` dt where the functionf(r, 4) which is the associate of T is an analytic function of T for all T over some region Q in the complex plane and continuous in ;, for 4 over the contour Y with (3) z = x + 1(iy + Z) + (iy-Z) The Bergman operator (2) shall be designated by (4) 'F(x, y, z) = B(f, Y, X), X = (x, y, z). Presented to the Society, June 17, 1972; received by the editors February 7, 1972 and, in revised form, May 8, 1972. AMS (MOS) subject classifications (1970). Primary 31B05, 44A15; Secondary 30A08, 30A70. 1 This paper is based on a part of the author's Ph.D. dissertation which was directed by Professor Morris Marden at the University of Wisconsin-Milwaukee. ? American Mathematical Society 1973

Inverse formulation and finite difference solution for flow from a circular orifice

The problem of flow from a large reservoir through a circular orifice is formulated by considering the velocity potential and Stokes's stream function as the independent variables and the radial and axial dimensions as the dependent variables, and a finite difference solution is obtained to the resulting boundary-value problem. This inverse formulation has the advantage over a finite difference solution in the physical plane that the region of flow is rectangular and consequently well adapted for minimum logic in programming a digital computer. The inverse finite difference solution is more readily obtained than a comparable solution in the physical plane, even though the inverse partial differential equation and associated boundary conditions are non-linear. The results from the inverse finite difference solution are in close agreement with other most recent results from approximate solutions to this problem.The inverse method of solution is applicable to other free streamline as well as confined axisymmetric potential flow problems. The essential difference in other problems will be in the boundary conditions.Keywords: Orifice, Finite Differences, Non-linear Partial Differential Equation, Potential Flow.

Motion of a Satellite of a Very Oblate Planet

view Abstract Citations (20) References (1) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Motion of a Satellite of a Very Oblate Planet Danby, J. M. A. Abstract An axisymmetric potential with only the second harmonic included is investigated. The question of the integrability of the resulting dynamical system is open. But by considering large oblateness and using numerical methods, all the symptoms of nonintegrability appear. However the manner of their appearance indicates that for oblateness of astronomical interest, possible nonintegrability would be unimportant. Publication: The Astronomical Journal Pub Date: December 1968 DOI: 10.1086/110765 Bibcode: 1968AJ.....73.1031D full text sources ADS |

On a problem of the theory of axisymmetric potential, and on a system of circular and annular dies

On a problem of the theory of axisymmetric potential, and on a system of circular and annular dies

The applicability of the third integral of motion: Some numerical experiments

view Abstract Citations (1289) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The applicability of the third integral of motion: Some numerical experiments Henon, Michel ; Heiles, Carl Abstract The problem of the existence of a third isolating integral of motion in an axisymmetric potential is in- vestigated by numerical experiments. I t is found that the third integral exists for only a limited range of initial conditions. Publication: The Astronomical Journal Pub Date: February 1964 DOI: 10.1086/109234 Bibcode: 1964AJ.....69...73H full text sources ADS |

Line source distributions and slender-body theory

A systematic procedure is presented for the determination of uniformly valid successive approximations to the axisymmetric incompressible potential flow about elongated bodies of revolution meeting certain shape requirements. The presence of external disturbances moving with respect to the body under study is admitted. The accuracy of the procedure and its extension beyond the scope of the present study—e.g. to problems in plane flow - are discussed.