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.

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 liquid near a position of separation

1950 ◽  
Vol 46 (2) ◽  
pp. 293-306 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Double Series ◽  
Constant Density

1. A steady motion of viscous liquid of constant density is considered in §§ 2–7; full allowance is made for the inertia of the motion, and it is assumed that the stream-function can be expanded in a double series in the coordinates x, y.

scholarly journals The rotation of two circular cylinders in a viscous fluid

1922 ◽  
Vol 101 (709) ◽  
pp. 169-174 ◽  
Keyword(s):  
Viscous Fluid ◽  
Boundary Conditions ◽  
Stream Function ◽  
Elastic Stress ◽  
Steady Motion ◽  
Elastic Equilibrium ◽  
Circular Cylinders ◽  
Two Dimensional ◽  
Concentric Circles ◽  
Hydrodynamical Problem

In a previous communication we employed the solution of the equation ∇ 4 ψ = 0 in bipolar co-ordinates defined by α + iβ = log x + i ( y + a )/ x + i ( y - a ) (1) to discuss the problem of the elastic equilibrium of a plate bounded by any two non-concentric circles. There is a well-known analogy between plain elastic stress and two-dimensional steady motion of a viscous fluid, for which the stream-function satisfies ∇ 4 ψ = 0. The boundary conditions are, however, different in the two cases, and the hydrodynamical problem has its own special difficulties.

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.

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.

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.

Boundary Singularities in Steady Potential Compressible Flow Through Plane Two-Dimensional Channels

1991 ◽  
Vol 205 (6) ◽  
pp. 389-398 ◽  
Author(s):  
C A C Streeter
Keyword(s):  
Stream Function ◽  
Flow Direction ◽  
Polar Coordinates ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Perfect Gas ◽  
Boundary Singularity ◽  
Convergent Channel ◽  
Choked Flow ◽  
Concave Corner

Boundary singularity solutions are reviewed, which relate to the steady, transonic potential flow of a perfect gas, through plane, two-dimensional convergent channels. Two singularities occur in the physical plane, one at upstream infinity and the other at the concave corner in channels that have a parallel entry section followed by a convergent section that ends at the throat plane. Solutions of these singularities are obtained from the general solution of Chaplygin's stream function equation in the hodograph system of coordinates. Uniform, uni-directional subsonic entry flow, in a finite-length channel, gives rise to a boundary singularity in the hodograph plane. The stream function is multi-valued at this singularity, where it varies between one and zero, while the flow direction is zero. This condition is removed by introducing circular polar coordinates (r, α) with origin at the singular point. For a sufficiently small value of r, a second-order partial differential equation of the stream function is obtained in the new coordinates. A particular solution of this equation is given, when r is small, rs say, in which the stream function gradient in the radial direction is taken as zero for all values of α at rs. The full series solution of the new equation is given in a paper which is to follow. Finally, consideration is given to the problem of matching the subsonic and supersonic flow regions at the sonic surface. A new system of equations is given, which determine the stream function gradient at sonic points from values of the stream function in the expansion region. Computed results are given that relate to the steady, choked flow of a perfect gas through a 15° convergent channel. The overall solution is shown to converge.

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.

scholarly journals The steady broadside motion of an anchor ring in an infinite viscous liquid

1931 ◽  
Vol 134 (823) ◽  
pp. 47-57 ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Viscous Liquid ◽  
Stream Function ◽  
Steady Motion ◽  
Second Order ◽  
Meridian Plane ◽  
Legendre Functions ◽  
Order Partial Differential Equation ◽  
Partial Differential

The ring is translated along its axis of revolution with constant velocity in an infinite viscous liquid. The motion of the liquid is due to the motion of the ring, each particle moving in a meridian plane to which the vector vorticity is perpendicular. The analysis is conducted in orthogonal curvilinear “ring coordinates” using vectors, and the condition of continuity leads to a stream function which is connected with the vorticity by a partial differential equation of the second order. The equation of steady motion, on ignoring the inertia terms, is a partial differential equation of the second order in which the dependent variable is the vorticity. The motion thus comes to depend on a fourth-order partial differential equation in which the dependent variable is the stream function. Two independent types of solution of this equation are obtained in trigonometrical series involving associated Legendre functions of degree half an odd integer, the solutions tending to zero at infinity. The arbitrary constants are determined from the boundary conditions of no slip at the surface of the ring. By means of the usual dyadics an expression, is obtained for the resistance to the motion. Numerical values are omitted in the absence of the necessary tables, a defect which it is hoped to remedy in the near future.

scholarly journals III. On the motion of a sphere in a viscous liquid

1888 ◽  
Vol 179 ◽  
pp. 43-63 ◽  
Keyword(s):  
Viscous Liquid ◽  
Steady Motion ◽  
Constant Force ◽  
Two Dimensional ◽  
Sufficient Time ◽  
Small Oscillations ◽  
Uniform Velocity ◽  
Straight Line ◽  
Principal Axes ◽  
The Subject

1. The first problem relating to the motion of a solid body in a viscous liquid which was successfully attacked was that of a sphere, the solution of which was given by Professor Stokes in 1850, in his memoir “On the Effect of the Internal Friction of Fluids on Pendulums,” ‘Cambridge Phil. Soc. Trans.,’ vol. 9, in the following cases: (i.) when the sphere is performing small oscillations along a straight line; (ii.) when the sphere is constrained to move with uniform velocity in a straight line; (iii.) when the sphere is surrounded by an infinite liquid and constrained to rotate with uniform angular velocity about a fixed diameter: it being supposed, in the last two cases, that sufficient time has elapsed for the motion to have become steady. In the same memoir he also discusses the motion of a cylinder and a disc. The same class of problems has also been considered by Meyer and Oberbeck, the latter of whom has obtained the solution in the case of the steady motion of an ellipsoid, which moves parallel to any one of its principal axes with uniform velocity. The torsional oscillations about a fixed diameter, of a sphere which is either filled with liquid or is surrounded by an infinite liquid when slipping takes place at the surface of the sphere, forms the subject of a joint memoir by Helmholtz and Piotrowski. Very little appears to have been effected with regard to the solution of problems in which a viscous liquid is set in motion in any given manner and then left to itself. The solution, when the liquid is bounded by a plane which moves parallel to itself, is given by Professor Stokes at the end of his memoir referred to above; and the solutions of certain problems of two-dimensional motion have been given by Stearn. In the present paper I propose to obtain the solution for a sphere moving in a viscous liquid in the following cases:—(i.) when the sphere is moving in a straight line under the action of a constant force, such as gravity; (ii.) when the sphere is surrounded by viscous liquid and is set in rotation about a fixed diameter and then left to itself.

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