Use of stokes’ stream function for discontinuities of potential at a spherical boundary

1960 ◽  
Vol 8 (1) ◽  
pp. 463-466 ◽  
Author(s):  
G. Power ◽  
H. L. W. Jackson
Keyword(s):  
Stream Function ◽  
Spherical Boundary

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.

A note on Stokes's stream function for motion with a spherical boundary

1953 ◽  
Vol 49 (1) ◽  
pp. 169-174 ◽  
Author(s):  
S. F. J. Butler
Keyword(s):  
Circular Cylinder ◽  
Stream Function ◽  
Complex Potential ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Irrotational Flow ◽  
Circle Theorem ◽  
Spherical Boundary

The circle theorem of Milne-Thomson(1) connecting the complex potential in a two-dimensional irrotational flow about a circular cylinder with that of the flow when the cylinder is absent has a three-dimensional counterpart in the result due to Weiss (3) for the perturbed velocity potential in an unlimited irrotational flow when the rigid spherical boundary r = a is inserted.

Calculations of rotational flows using stream function

1989 ◽  
Author(s):  
M. HAFEZ ◽  
C. YAM ◽  
K. TANG ◽  
H. DWYER
Keyword(s):  
Stream Function ◽  
Rotational Flows

New Exact Solutions of Steady Plane Flows of an In Compressible Fluid of Variable Viscosity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Sobia Younus
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Compressible Fluid ◽  
Variable Viscosity ◽  
Plane Motion ◽  
Transformation Technique ◽  
Vorticity Distribution ◽  
New Exact Solutions ◽  
Steady Plane ◽  
Uniform Stream

<span>Some new exact solutions to the equations governing the steady plane motion of an in compressible<span> fluid of variable viscosity for the chosen form of the vorticity distribution are determined by using<span> transformation technique. In this case the vorticity distribution is proportional to the stream function<span> perturbed by the product of a uniform stream and an exponential stream<br /><br class="Apple-interchange-newline" /></span></span></span></span>

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

Contracting ducts of finite length

1951 ◽  
Vol 2 (4) ◽  
pp. 254-271 ◽  
Author(s):  
L. G. Whitehead ◽  
L. Y. Wu ◽  
M. H. L. Waters
Keyword(s):  
Stream Function ◽  
Finite Length ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Two Dimensional ◽  
Dimensional Flow ◽  
Hodograph Plane ◽  
Two Dimensional Flow ◽  
Working Section

SummmaryA method of design is given for wind tunnel contractions for two-dimensional flow and for flow with axial symmetry. The two-dimensional designs are based on a boundary chosen in the hodograph plane for which the flow is found by the method of images. The three-dimensional method uses the velocity potential and the stream function of the two-dimensional flow as independent variables and the equation for the three-dimensional stream function is solved approximately. The accuracy of the approximate method is checked by comparison with a solution obtained by Southwell's relaxation method.In both the two and the three-dimensional designs the curved wall is of finite length with parallel sections upstream and downstream. The effects of the parallel parts of the channel on the rise of pressure near the wall at the start of the contraction and on the velocity distribution across the working section can therefore be estimated.

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