Note on the slow motion of fluid

1936 ◽  
Vol 32 (4) ◽  
pp. 598-613 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Harmonic Function ◽  
Viscous Liquid ◽  
Stream Function ◽  
Slow Motion ◽  
Normal Derivative ◽  
Great Distance ◽  
Sharp Edge ◽  
Two Dimensional ◽  
Shearing Motion

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

Note on the slow motion of fluid

1940 ◽  
Vol 36 (3) ◽  
pp. 300-313 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Viscous Liquid ◽  
Stream Function ◽  
Special Form ◽  
Biharmonic Equation ◽  
Slow Motion ◽  
Simple Expression ◽  
Sharp Edge ◽  
Two Dimensional ◽  
Fixed Boundary ◽  
Uniform Shear

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into the stream, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary so that a method lately developed by N. Muschelišvili can be used in solving the biharmonic equation; a simple expression in finite terms is found for the stream function ψ1. Fig. 1 shows the section ABC of the fixed boundary of the liquid, the equation of the curve ABC is given in § 2, and ψ1 in § 3.

On the shearing motion of fluid past a projection

1944 ◽  
Vol 40 (1) ◽  
pp. 19-36 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Rational Function ◽  
Viscous Liquid ◽  
Stream Function ◽  
Special Form ◽  
Conformal Transformation ◽  
Biharmonic Equation ◽  
Two Dimensional ◽  
Uniform Shear ◽  
Shearing Motion

1. In this paper, a continuation of an earlier paper(1), we consider the two-dimensional motion of incompressible viscous liquid past a projection, the motion being one of uniform shear apart from the disturbance caused by the projection. A special form is assumed for the boundary, so that the area in the z-plane (Fig. 1) can be represented conformally on a circle in the ζ-plane by a rational function of ζ; this function contains a parameter a (0 < a ≤ 1), and by varying a the shape of the projection can be varied. Since a rational function is concerned in the conformal transformation a method lately developed by N. Muschelišvili(2) can be used in solving the biharmonic equation for the stream function, though the method actually used differs in some points of detail from that originally proposed by Muschelišvili and appears to be somewhat simpler.

Slow motion of viscous liquid in a semi-infinite channel

1951 ◽  
Vol 47 (1) ◽  
pp. 127-141 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Pressure Gradient ◽  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Slow Motion ◽  
Biharmonic Function ◽  
Two Dimensional ◽  
Calculated Velocity ◽  
Straight Lines ◽  
Stream Lines

1. A slow two-dimensional steady motion of liquid caused by a pressure gradient in a semi-infinite channel is considered. The medium is bounded by two parallel semi-infinite planes represented in Fig. 1 by the straight lines AB, DE. The stream-function ψ is a biharmonic function of x, y which exactly satisfies the condition that AB, DE must be stream-lines, but the condition that there must be no velocity of slip on these boundaries is satisfied only approximately, and the calculated velocity of slip gives a measure of the accuracy of the solution.

Note on the motion of viscous liquid past a parabolic cylinder

1954 ◽  
Vol 50 (1) ◽  
pp. 125-130 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Viscous Liquid ◽  
Stream Function ◽  
Incompressible Liquid ◽  
Parabolic Cylinder ◽  
Viscous Incompressible Liquid ◽  
Two Dimensional ◽  
Approximate Expressions

1. In §§ 2–4 of this paper approximate expressions are found for the stream function and pressure in the steady two-dimensional motion of viscous incompressible liquid past a fixed parabolic cylinder; exact expressions for the stream-function and pressure in a perfect liquid are derived as limits in § 5.

Note on the shearing motion of fluid past a projection

1944 ◽  
Vol 40 (3) ◽  
pp. 214-222 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Rational Function ◽  
Plane Strain ◽  
Viscous Liquid ◽  
Plane Stress ◽  
Two Dimensional ◽  
Close Approximation ◽  
Shearing Motion

1. A method has lately been developed by N. Muschelišvili (1) for the solution of problems of the slow two-dimensional motion of viscous liquid and of the corresponding problems of plane stress and plane strain, in cases in which the area in the x, y-plane that is concerned can be represented conformally on the interior of the circle |ζ| = 1 in the ζ-plane by a relation of the form z = x + iy = r(ζ), where r(ζ) is a rational function of ζ. In most problems in which the method has been used the function r(ζ) has been a simple one, but it is of importance to consider a rational function of as general a form as possible since, given any relation z = f(ζ), it will usually be possible to find a rational function that approximates to f(ζ) throughout the circle |ζ| = 1 and for a close approximation a complicated function r(ζ) will in general be required.

On the steady motion of viscous liquid in a corner

1949 ◽  
Vol 45 (3) ◽  
pp. 389-394 ◽  
Author(s):  
W. R. Dean ◽  
P. E. Montagnon
Keyword(s):  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Sharp Corner

1. In a steady two-dimensional motion of viscous liquid in the sharp corner formed by the rigid straight boundaries θ = 0, α, where r, θ are plane polar coordinates, it is found that, near enough to the corner, the most important term in the stream-function is of the form rmf(θ). The index m is evaluated in §§ 2–4 for values of α between 360 and 90°, and is found to be complex if α is less than about 146°; the limiting form of the stream-function when α is small is considered in § 5.

Note on the slow motion of fluid

1939 ◽  
Vol 35 (1) ◽  
pp. 27-43 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Viscous Fluid ◽  
Stream Function ◽  
Slow Motion ◽  
Simple Expression ◽  
Two Dimensional ◽  
Fixed Plane ◽  
Uniform Shear

In the first part of the paper a slow two-dimensional motion of viscous fluid is considered which approximates to a motion of uniform shear past an infinite fixed plane, and differs from this motion because there is a gap in the plane (Fig. 1). A simple expression in finite terms is found for the stream function.

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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