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.

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.

scholarly journals On the Exact Solutions of Couple Stress Fluids

2014 ◽  
Vol 1 ◽  
pp. 27-32 ◽  
Author(s):  
Waqar Khan ◽  
Faisal Yousafzai
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Couple Stress ◽  
Two Dimensional ◽  
Compatibility Equation ◽  
Flow Equations ◽  
Dimensional Flow ◽  
Two Dimensional Flow ◽  
Stream Functions ◽  
Couple Stress Fluids

Exact solutions of the momentum equations of couple stress fluid are investigated. Making use of stream function, the two-dimensional flow equations are transformed into non-linear compatibility equation, and then it is linearized by vorticity function. Stream functions and velocity distributions are discussed for various flow situations.

Steady two-dimensional viscous flow in a jet

1972 ◽  
Vol 55 (1) ◽  
pp. 49-63 ◽  
Author(s):  
K. Capell
Keyword(s):  
Point Source ◽  
Viscous Flow ◽  
Closed Form ◽  
Stream Function ◽  
Asymptotic Expansions ◽  
Far Field ◽  
Two Dimensional ◽  
Complete Structure ◽  
Dimensional Flow ◽  
Two Dimensional Flow

An idealized two-dimensional flow due to a point source ofxmomentum is discussed. In the far field the flow is modelled by a jet region of large vorticity outside which the flow is potential. After use of the transformation\[ \zeta^3 = (\xi + i\eta)^3 = x + iy, \]the equations suggest naively obvious asymptotic expansions for the stream function in these two regions, namely\[ \sum_{n=0}^{\infty}\xi^{1-n}f_n(\eta)\quad {\rm and}\quad\sum_{n=0}^{\infty}\xi^{1-n}F_n(\eta/\xi) \]respectively. Consistency in matching these expansions is achieved by including logarithmic terms associated with the occurrence of eigensolutions.Fnis easy to find andJncan be found in closed form so the inner and outer eigensolutions may be fully determined along with the complete structure of the expansions.

scholarly journals The iterated equation of generalized axially symmetric potential theory, VI General solutions

1974 ◽  
Vol 18 (3) ◽  
pp. 318-327
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Polar Coordinates ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Real Value ◽  
Image Position

The iterated equation of generalized axially symmetric potential theory [1] is the equation where, in its simplest form, the operator Lk is defined by the function f f(x, y) being assumed to belong to the class of C2n functions and the parameter l to take any real value. In appropriate circumstances, which will be indicated later, the operator can be generalized but as this can be done without altering the methods used, the operator will be taken in the form where r, θ are polar coordinates such that x = r cos θ, y = r sin μ = cosθ.

Two-dimensional flow under gravity in a jet of viscous liquid

1968 ◽  
Vol 31 (3) ◽  
pp. 481-500 ◽  
Author(s):  
N. S. Clarke
Keyword(s):  
Flow Region ◽  
Reynolds Numbers ◽  
Asymptotic Solutions ◽  
Perturbation Scheme ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Secondary Importance ◽  
Inertia Effects ◽  
Dimensional Flow ◽  
Two Dimensional Flow

This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.

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. III. Conjugate general solutions

1967 ◽  
Vol 7 (3) ◽  
pp. 290-300
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Image Position

The iterated equation of generalized axially symmetric potential theory (GASPT), in the notation of the first paper of this series [1] which will be designated I, is the equation where the operator Lk is defined by

scholarly journals The iterated equation of generalized axially symmetric potential theory. II. General solutions of Weinstein's type

1967 ◽  
Vol 7 (3) ◽  
pp. 277-289 ◽  
Author(s):  
J. C. Burns
Keyword(s):  
Potential Theory ◽  
Axially Symmetric ◽  
Symmetric Potential ◽  
General Solutions ◽  
Particular Solutions ◽  
Image Position

In the first paper of this series [1] which will be designated I, particular solutions of various kinds have been found for the iterated equation of generalized axially symmetric potential theory (GASPT) which, in the notation defined in I, is (1) where the operator is defined by

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