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.

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.

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 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.

scholarly journals VIII. Stability of a viscous liquid contained between two rotating cylinders

1923 ◽  
Vol 223 (605-615) ◽  
pp. 289-343 ◽  
Keyword(s):  
Viscous Liquid ◽  
Relative Velocity ◽  
Steady Motion ◽  
Mathematical Representation ◽  
Rotating Cylinders ◽  
Fluid Instability ◽  
Small Disturbances

In recent years much information has been accumulated about the flow of fluids past solid boundaries. All experiments so far carried out seem to indicate that in all cases steady motion is possible if the motion be sufficiently slow, but that if the velocity of the fluid exceeds a certain limit, depending on the viscosity of the fluid and the configuration of the boundaries, the steady motion breaks down and eddying flow sets in. A great many attempts have been made to discover some mathematical representation of fluid instability, but so far they have been unsuccessful in every case. The case, for instance, in which the fluid is contained between two infinite parallel planes which move with a uniform relative velocity has been discussed by Kelvin, Rayleigh, Sommerfeld, Orr, Mises, Hope, and others. Each of them cam e to the conclusion that the fundamental small disturbances of this system are stable. Though it is necessarily impossible to carry out experiments with infinite planes, it is generally believed that the motion in this case would be turbulent, provided the relative velocity of the two planes were sufficiently great.

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.

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