Errata: "Variable-Mesh Difference Equation for the Stream Function in Axially Symmetric Flowy"

AIAA Journal ◽  
1964 ◽  
Vol 2 (7) ◽  
pp. 1359b-1359b
Author(s):  
JOHN C. LYSEN
Keyword(s):  
Difference Equation ◽  
Stream Function ◽  
Axially Symmetric ◽  
Variable Mesh

An Analysis of Power Law Viscous Materials Under a Plane-Strain Condition Using Complex Stream and Stress Functions

1981 ◽  
Vol 48 (3) ◽  
pp. 486-492 ◽  
Author(s):  
Y. S. Lee ◽  
L. C. Smith
Keyword(s):  
Stress Concentration ◽  
Stream Function ◽  
Power Law ◽  
Stress Concentration Factor ◽  
Stress Function ◽  
Concentration Factor ◽  
Viscous Medium ◽  
Biaxial Stress ◽  
Deformation Analysis ◽  
Axially Symmetric

The equilibrium and compatibility equations for nonlinear viscous materials described by the power law are solved by introducing the complex stream and stress function. The stresses, strain rates, and velocities derived from the summation form of the stream function and the product form of the stress function are identical to the results obtained from the axially symmetric field equation. The stream function solution is used in the deformation analysis of a viscous hollow cylindrical inclusion buried in an infinitely large viscous medium assuming an equal biaxial boundary stress. The stream function approach is used in determining the stress-concentration factor for a cavity in a viscous material subject to the identical boundary biaxial stress. The results agree with the results of Nadai. The effect of the strain-rate-hardening exponent, the geometry of the inclusion, and the material constants on the hoop stress-concentration factor in the interface between the inclusion and the matrix are discussed.

Potential and stream function of a vortex disk in the presence of a rigid sphere

1957 ◽  
Vol 53 (3) ◽  
pp. 717-727
Author(s):  
J. Martinek ◽  
G. C. K. Yeh ◽  
H. Zorn
Keyword(s):  
Stream Function ◽  
Integral Representations ◽  
Practical Significance ◽  
Analytic Description ◽  
Axially Symmetric ◽  
Sphere Theorem ◽  
Numerical Work ◽  
Symmetry Properties ◽  
Spherical Boundary ◽  
Transcendental Functions

In Sadowsky and Sternberg(1), elliptic integral representations of axially symmetric flows suggested essentially by Weinstein's work (2) on axially symmetric flows of an ideal incompressible fluid have been considered. The physical and practical significance of their investigation was twofold. First, a more transparent analytic description was obtained than that afforded by the representation through discontinuous integrals of Bessel functions originally used by Weinstein. Secondly, by superposition of such axially symmetric flows and appropriately chosen uniform streams, a variety of technically significant flows around solids, annular bodies and half-bodies of revolution can be constructed. On the other hand, in an attempt to utilize the symmetry properties of the potential field in reference to a spherical boundary, Weiss(3) has derived an (exterior) ‘sphere theorem’ by applying Kelvin's transformation and using the potential function. Butler (4) later obtained, by means of the Stokes stream function, a sphere theorem applicable to axially symmetric flows only. Ludford, Martinek and Yeh(5) found then the ‘interior sphere theorem’ as well as a theorem satisfying the general radiation condition. A general sphere theorem was consequently conceived, valid for all linear boundary conditions, and was recently published by Yeh, Martinek and Ludford (6). The significance of these theorems again lies in their application to physical problems. They often give closed form expressions of the disturbance potential in terms of higher transcendental functions whenever the undisturbed potential is given by means of transcendental functions. Furthermore, when the singularities (discrete or distributed) he near the perturbing spherical boundary the usual treatment by expansion in spherical harmonics leads to solutions in the form of infinite series which are, because of slow convergency, unsuited for numerical computation. For this situation the sphere theorems provide a remedy in the form of neat formulae readily adaptable to numerical work.

scholarly journals The Iterated Equation of Generalized Axially Symmetric Potential Theory

1969 ◽  
Vol 9 (1-2) ◽  
pp. 153-160
Author(s):  
J. C. Burns
Keyword(s):  
Perfect Fluid ◽  
Stream Function ◽  
Stokes Flow ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Sphere Theorem ◽  
Symmetric Potential ◽  
Irrotational Flow ◽  
Viscous Boundary ◽  
Viscous Boundary Conditions

Milne-Thomson's well-known circle theorem [1] gives the stream function for steady two-dimensional irrotational flow of a perfect fluid past a circular cylinder when the flow in the absense of the cylinder is known. Butler's sphere theorem [2] gives the corresponding result for axially symmetric irrotational flow of a perfect fluid past a sphere. Collins [3] has obtained a sphere theorem for axially symmetric Stokes flow of a viscous liquid which gives a stream function satisfying the appropriate viscous boundary conditions on the surface of a sphere when the stream function for irrotational flow in the absence of the sphere is known.

scholarly journals The iterated equation of generalized axially symmetric potential theory. I. Particular solutions

1967 ◽  
Vol 7 (3) ◽  
pp. 263-276 ◽  
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Stream Function ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Symmetric Flow ◽  
Symmetric Potential ◽  
Dimensional Flow ◽  
Physical Problems ◽  
Two Dimensional Flow ◽  
Image Position

The iterated equation of generalized axially symmetric potential theory (GASPT) [1] is defined by the relations (1) where (2) and Particular cases of this equation occur in many physical problems. In classical hydrodynamics, for example, the case n = 1 appears in the study of the irrotational motion of an incompressible fluid where, in two-dimensional flow, both the velocity potential φ and the stream function Ψ satisfy Laplace's equation, L0(f) = 0; and, in axially symmetric flow, φ and satisfy the equations L1 (φ) = 0, L-1 (ψ) = 0. The case n = 2 occurs in the study of the Stokes flow of a viscous fluid where the stream function satisfies the equation L2k(ψ) = 0 with k = 0 in two-dimensional flow and k = −1 in axially symmetric flow.

The Stokes flow problem for a class of axially symmetric bodies

1960 ◽  
Vol 7 (4) ◽  
pp. 529-549 ◽  
Author(s):  
L. E. Payne ◽  
W. H. Pell
Keyword(s):  
Stream Function ◽  
Stokes Flow ◽  
Flow Problem ◽  
Fluid Motion ◽  
Explicit Calculation ◽  
Axially Symmetric ◽  
Flow Problems ◽  
Fundamental Operator ◽  
Axis Of Symmetry

The Stokes flow problem is concerned with fluid motion about an obstacle when the motion is such that inertial effects can be neglected. This problem is considered here for the case in which the obstacle (or configuration of obstacles) has an axis of symmetry, and the flow at distant points is uniform and parallel to this axis. The differential equation for the stream function ψ then assumes the form L2−1ψ = 0, where L−1 is the operator which occurs in axiallysymmetric flows of the incompressible ideal fluid. This is a particular case of the fundamental operator of A. Weinstein's generalized axially symmetric potential theory. Using the results of this theory and theorems regarding representations of the solutions of repeated operator equations, the authors (1) give a general expression for the drag of an axially symmetric configuration in Stokes flow, and (2) indicate a procedure for the determination of the stream function. The stream function is found for the particular case of the lens-shaped body.Explicit calculation of the drag is difficult for the general lens, without recourse to numerical procedures, but is relatively easy in the case of the hemispherical cup. As examples illustrative of their procedures, the authors briefly consider three Stokes flow problems whose solutions have been given previously.

Solar Internal Rotation and Dynamo Waves: A Two-Dimensional Asymptotic Solution in the Convection Zone

2000 ◽  
Vol 179 ◽  
pp. 379-380
Author(s):  
Gaetano Belvedere ◽  
Kirill Kuzanyan ◽  
Dmitry Sokoloff
Keyword(s):  
Magnetic Field ◽  
Asymptotic Solution ◽  
Internal Rotation ◽  
Convection Zone ◽  
General Idea ◽  
Dynamo Model ◽  
Axially Symmetric ◽  
Dynamo Action ◽  
Regeneration Rate ◽  
Dynamo Waves

Extended abstractHere we outline how asymptotic models may contribute to the investigation of mean field dynamos applied to the solar convective zone. We calculate here a spatial 2-D structure of the mean magnetic field, adopting real profiles of the solar internal rotation (the Ω-effect) and an extended prescription of the turbulent α-effect. In our model assumptions we do not prescribe any meridional flow that might seriously affect the resulting generated magnetic fields. We do not assume apriori any region or layer as a preferred site for the dynamo action (such as the overshoot zone), but the location of the α- and Ω-effects results in the propagation of dynamo waves deep in the convection zone. We consider an axially symmetric magnetic field dynamo model in a differentially rotating spherical shell. The main assumption, when using asymptotic WKB methods, is that the absolute value of the dynamo number (regeneration rate) |D| is large, i.e., the spatial scale of the solution is small. Following the general idea of an asymptotic solution for dynamo waves (e.g., Kuzanyan & Sokoloff 1995), we search for a solution in the form of a power series with respect to the small parameter |D|–1/3(short wavelength scale). This solution is of the order of magnitude of exp(i|D|1/3S), where S is a scalar function of position.

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