scholarly journals 1. Stability of Fluid Motion.—Rectilineal Motion of Viscous Fluid between two Parallel Planes

1888 ◽  
Vol 14 ◽  
pp. 359-368 ◽  
Author(s):  
W. Thomson
Keyword(s):  
Viscous Fluid ◽  
Fluid Motion ◽  
Steady Motion ◽  
Philosophical Magazine ◽  
University Of Cambridge ◽  
The Stability ◽  
The University

Since the communication of the first of this series of articles to the Eoyal Society of Edinburgh in April, and its publication in the Philosophical Magazine in May and June, the stability or instability of the steady motion of a viscous fluid has been proposed as subject for the Adams Prize of the University of Cambridge for 1888.

On the disturbance of the steady flow of an inviscid liquid between parallel planes

1934 ◽  
Vol 234 (732) ◽  
pp. 43-78 ◽  
Keyword(s):  
Mathematical Proof ◽  
Fluid Motion ◽  
Steady Motion ◽  
Circular Cylinders ◽  
General Validity ◽  
Inviscid Liquid ◽  
Characteristic Value ◽  
Characteristic Values ◽  
The Stability ◽  
Liquid Flows

1—In a number of papers dealing with the stability of fluid motion, RAYLEIGH employed a certain method, which we may refer to as the “characteristic-value” method. For some problems this method gives results in agreement with observation. For example, it establishes that a heterogeneous inviscid liquid at rest under gravity is stable if the density decreases steadily as we pass upward; it establishes that an inviscid liquid rotating between concentric circular cylinders is stable if, and only if, the square of the circulation increases steadily as we pass outward. This result was stated by RAYLEIGH, and its validity appears to be confirmed by the experiments of TAYLOR, but a simple < d 2 u 0 /dy 2 retains the same sign throughout the liquid, u 0 being the velocity in the steady motion and y the distance from one of the planes. This result is deduced from the fact that mathematical proof by the characteristic value method was not given. I have recently supplied such a proof, extending the problem to include a heterogeneous liquid. But when the method is applied to some other problems, the situation is not so satisfactory. Among the results to which Rayleigh was led is the following. If an inviscid liquid flows between parallel planes, the motion is stable if the characteristic values of a parameter in a certain differential equation cannot be complex, the implication being that they are therefore real. Rayleigh further claimed that the method established the stability of a uniform shearing motion, for which d 2 u 0 /dy 2 =0. KELVIN and LOVE criticized the method, and a review of the situation in 1907 was given by ORR. In spite of the fact that its general validity remains obscure, the characteristic-value method has been widely employed. It is not the purpose of the present paper to attempt to justify or to discredit the characteristic-value method in general. The paper deals only with the simplest of all stability problems, that of an inviscid liquid flowing between fixed parallel planes. In §2 the method is discussed in some detail and in §3 an argument is developed to show that Rayleigh’s criterion for stability, mentioned above, cannot be legitimately deduced by his method. He proved that complex characteristic values are impossible, and I now prove that real characteristic values are also impossible. The conclusion to be drawn is that the characteristic-value method is not applicable to this case.

scholarly journals X. The resistance of a cylinder moving in a viscous fluid

1923 ◽  
Vol 223 (605-615) ◽  
pp. 383-432 ◽  
Keyword(s):  
Circular Cylinder ◽  
Viscous Fluid ◽  
Equations Of Motion ◽  
Fluid Motion ◽  
Slow Motion ◽  
Two Dimensions ◽  
Circular Cylinders ◽  
Flat Plates ◽  
Philosophical Magazine ◽  
Walled Channel

An earlier Paper by the same authors dealt with one aspect of the same problem and in its conclusions indicated the possibility of the present development. In order to deal with the equations of motion for a viscous fluid in two dimensions, the approximation for slow motion due to Stokes was used, with a consequent need for the introduction of a boundary limiting the expanse of the fluid. Whilst keeping the analysis as general as possible, the example given related to a circular cylinder centrally placed in a parallel-walled channel. The extension now to be described follows generally similar lines; the form of equation has been changed from that of Stokes to one proposed by Oseen, the change representing a closer approach to the full equations of motion by the introduction of terms dependent on the inertia of the fluid. A consideration of the differential equations by earlier writers has indicated a close agreement between the motions near a small sphere in the two cases, but a marked difference in the more remote parts of the fluid. Oseen has shown, for the sphere, that the resistance formulæ for the two cases are identical to the first order of small quantities. In the case of the two-dimensional motion of a cylinder the differences are rather more striking. In the Stores’ form of approximation it is not possible to satisfy all the essential conditions when the expanse of fluid is infinite, whilst with Oseen’ s type of equation this particular difficulty disappears. Having seen Oseen’ s solution for the motion of a sphere in a viscous fluid, Lamb applied a similar method to the circular cylinder, and an account of his analysis is given in the ‘Philosophical Magazine’ 'and his treatise on Hydrodynamics' (p. 605); a resistance formula is deduced which is applicable at low velocities. There are two approximations in this solution, one physical and implied in the original differential equation, and the second mathematical and introduced in the solution. There is a certain degree of inter-relation between the two approximations, but it has been found that the second of them may be removed. An estimate of the degree to which Oseen’ s approximation represents the complete equation of viscous fluid motion can be obtained by comparing the results of the new calculations with those of experiment. In the result it appears that the amount still to be accounted for by the remaining inertia terms is less than that already dealt with, in the case of both the resistance of circular cylinders and the skin friction of flat plates.

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

1923 ◽  
Vol 102 (718) ◽  
pp. 541-542 ◽  
Keyword(s):  
Viscous Fluid ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Outer Cylinder ◽  
Steady Motion ◽  
Annular Space ◽  
Rotating Cylinders ◽  
High Speeds ◽  
Approximate Form ◽  
The Stability

Part I .—The stability for symmetrical disturbances of a viscous fluid in steady motion between concentric rotating cylinders is investigated mathematically. It is shown that at slow speeds the motion is always stable, but that at high speeds the motion is only stable when the ratio of the speed of the outer cylinder to that of the inner one exceeds a certain value. When the ratio is less than this or when it is negative the motion becomes unstable at high speeds. The “criterion” for stability is found, and in cases suitable for experimental verification an approximate form for the “criterion” is developed which is useful for numerical computation. The type of instability which may be expected to appear when the speed of the cylinders is slowly increased is shown to consist of symmetrical ring-shaped vortices spaced at regular intervals along the length of the cylinders. These vortices rotate alternately in opposite directions. Their dimensions are calculated and it is shown that they are contained in partitions of rectangular cross-section. In the case when the instability arises while both cylinders are rotating in the same direction, these rectangles are squares, so that the vortices are spaced at distances apart equal to the thickness of the annular space between the two cylinders. In the case when the cylinders rotate in opposite directions the spacing, or distance between the centres of neighbouring vortices, is smaller than this; and at the same time two systems of vortices develop—an inner system which is similar to the system which appears when the two cylinders rotate in the same direction, and an outer system, which is much less vigorous and rotates in the opposite direction to the adjacent members of the inner system.

INTERVIEW WITH PETINA GAPPAH

Imbizo ◽  
2017 ◽  
Vol 7 (2) ◽  
pp. 92-98
Author(s):  
Faith Mkwesha
Keyword(s):  
Short Stories ◽  
Short Story ◽  
First Language ◽  
Short Story Collection ◽  
The Third ◽  
University Of Cambridge ◽  
Book Award ◽  
The University ◽  
The Guardian ◽  
University Of Zimbabwe

This interview was conducted on 16 May 2009 at Le Quartier Francais in Franschhoek, Cape Town, South Africa. Petina Gappah is the third generation of Zimbabwean writers writing from the diaspora. She was born in 1971 in Zambia, and grew up in Zimbabwe during the transitional moment from colonial Rhodesia to independence. She has law degrees from the University of Zimbabwe, the University of Cambridge, and the University of Graz. She writes in English and also draws on Shona, her first language. She has published a short story collection An Elegy for Easterly (2009), first novel The Book of Memory (2015), and another collection of short stories, Rotten Row (2016).  Gappah’s collection of short stories An Elegy for Easterly (2009) was awarded The Guardian First Book Award in 2009, and was shortlisted for the Frank O’Connor International Short Story Award, the richest prize for the short story form. Gappah was working on her novel The Book of Memory at the time of this interview.

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