The stability of a viscous fluid between rotating cylinders with an axial flow

1964 ◽  
Vol 19 (4) ◽  
pp. 528-538 ◽  
Author(s):  
E. R. Krueger ◽  
R. C. Di Prima
Keyword(s):  
Viscous Fluid ◽  
Viscous Flow ◽  
Axial Flow ◽  
Rotating Cylinders ◽  
Counter Rotation ◽  
Theoretical Investigations ◽  
The Stability

The stability of viscous flow between rotating cylinders with an axial flow has been investigated theoretically by Goldstein (1937), Chandrasekhar (1960, 1962), and Di Prima (1960); and experimentally by Cornish (1933), Fage (1938), Kaye & Elgar (1957), Donnelly & Fultz (1960) and Snyder (1962a). As was pointed out by Di Prima (1960) there were a number of discrepancies in the early work of the 1930's which were clarified in part by the papers of the 1960's. In turn, there appear to be certain small detailed differences in the more recent papers. In part it is these differences with which the present paper is concerned. In addition, the results of the previous theoretical investigations which are limited to the case in which the cylinders rotate in the same direction, are extended to the case of counter rotation.

The stability of viscous flow between rotating cylinders in the presence of a magnetic field. II

1961 ◽  
Vol 262 (1311) ◽  
pp. 443-454 ◽  
Keyword(s):  
Magnetic Field ◽  
Viscous Flow ◽  
Rotating Cylinders ◽  
Counter Rotation ◽  
Angular Velocities ◽  
The Stability

The theory developed in an earlier paper (Chandrasekhar 1953) is extended to allow for counter-rotation of the two cylinders. Explicit results are given for the case when the two cylinders rotate in opposite directions with equal angular velocities.

Experiments on the stability of spiral flow at low axial Reynolds numbers

1962 ◽  
Vol 265 (1321) ◽  
pp. 198-214 ◽  
Keyword(s):  
Perturbation Theory ◽  
Viscous Flow ◽  
Hydrodynamic Stability ◽  
Outer Cylinder ◽  
Axial Flow ◽  
Reynolds Numbers ◽  
Rotating Cylinders ◽  
Spiral Flow ◽  
The Stability

The hydrodynamic stability of viscous flow between rotating cylinders with superposed axial flow has been studied experimentally. The experiments were confined to the case where the outer cylinder is at rest and the gap between cylinders is small. Particular attention has been given to small rates of axial flow. The results compare satisfactorily with Chandrasekhar’s perturbation theory valid under these conditions.

The stability of a viscous fluid between rotating cylinders with an axial flow

1960 ◽  
Vol 9 (4) ◽  
pp. 621-631 ◽  
Author(s):  
R. C. Diprima
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Axial Flow ◽  
Taylor Number ◽  
Small Reynolds Number ◽  
Rotating Cylinders ◽  
The Stability

The stability of a viscous fluid between two concentric rotating cylinders with an axial flow is investigated. It is assumed that the cylinders are rotating in the same direction and that the spacing between the cylinders is small. The critical Taylor number is computed for small Reynolds number associated with the axial flow. It is found that the critical Taylor number increases with increasing Reynolds number.

The stability of elastico-viscous flow between rotating cylinders Part 3. Overstability in viscous and Maxwell fluids

1966 ◽  
Vol 24 (2) ◽  
pp. 321-334 ◽  
Author(s):  
D. W. Beard ◽  
M. H. Davies ◽  
K. Walters
Keyword(s):  
Viscous Flow ◽  
Couette Flow ◽  
Taylor Number ◽  
Linear Perturbation ◽  
Rotating Cylinders ◽  
Initial Value ◽  
Viscous Liquids ◽  
Maxwell Fluids ◽  
Newtonian Liquids ◽  
The Stability

Consideration is given to the possibility of overstability in the Couette flow of viscous and elastico-viscous liquids. The relevant linear perturbation equations are solved numerically using an initial-value technique. It is shown that over-stability is not possible in the case of Newtonian liquids for the cases considered. In contrast, overstability is to be expected in the case of moderately-elastic Maxwell liquids. The Taylor number associated with the overstable mode decreases steadily as the amount of elasticity in the liquid increases, and it is concluded that highly elastic Maxwell liquids can be very unstable indeed.

On the stability of viscous flow between rotating cylinders Part 2. Numerical analysis

1964 ◽  
Vol 20 (1) ◽  
pp. 95-101 ◽  
Author(s):  
D. L. Harris ◽  
W. H. Reid
Keyword(s):  
Numerical Analysis ◽  
Eigenvalue Problem ◽  
Viscous Flow ◽  
Numerical Method ◽  
Couette Flow ◽  
Stream Function ◽  
Oscillatory Behaviour ◽  
Rotating Cylinders ◽  
The Stability

A simple numerical method is presented for solving the eigenvalue problem which governs the stability of Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. When the cylinders rotate in opposite directions, these eigenfunctions exhibit an exponentially damped oscillatory behaviour for sufficiently large values of − μ, where μ = Ω2/Ω1. In terms of the stream function which describes the motion in planes through the axis of the cylinders, this means that weak, viscously driven cells appear in the outer layes of the fluid which, according to Rayleigh's criterion, are dynamically stable. For μ = − 3, for example, four cells are present, the amplitudes of which are in the ratios 1·0:0·0172:0·013:0·00125.

ONSET OF INSTABILITY IN HYDROMAGNETIC COUETTE FLOW

2004 ◽  
Vol 02 (02) ◽  
pp. 145-159 ◽  
Author(s):  
ISOM H. HERRON
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Viscous Flow ◽  
Couette Flow ◽  
Axial Magnetic Field ◽  
Narrow Gap ◽  
Rotating Cylinders ◽  
Magnetic Prandtl Number ◽  
The Stability

The stability of viscous flow between rotating cylinders in the presence of a constant axial magnetic field is considered. The boundary conditions for general conductivities are examined. It is proved that the Principle of Exchange of Stabilities holds at zero magnetic Prandtl number, for all Chandrasekhar numbers, when the cylinders rotate in the same direction, the circulation decreases outwards, and the cylinders have insulating walls. The result holds for both the finite gap and the narrow gap approximation.

Stability of spiral flow between concentric circular cylinders at low axial Reynolds number

1965 ◽  
Vol 21 (4) ◽  
pp. 635-640 ◽  
Author(s):  
Subhendu K. Datta
Keyword(s):  
Reynolds Number ◽  
Perturbation Theory ◽  
Viscous Liquid ◽  
Axial Flow ◽  
Circular Cylinders ◽  
Rotating Cylinders ◽  
Spiral Flow ◽  
The Stability

The stability of a viscous liquid between two concentric rotating cylinders with an axial flow has been investigated. Attention has been confined to the case when the cylinders are rotating in the same direction, the gap between the cylinders is small and the axial flow is small. A perturbation theory valid in the limit when the axial Reynolds number R → 0 has been developed and corrections have been obtained for Chandrasekhar's earlier results.

The stability of elastico-viscous flow between rotating cylinders. Part 2

1964 ◽  
Vol 19 (4) ◽  
pp. 557-560 ◽  
Author(s):  
R. H. Thomas ◽  
K. Walters
Keyword(s):  
Viscous Flow ◽  
Stability Problem ◽  
Narrow Channel ◽  
Taylor Number ◽  
Wave Number ◽  
Rotating Cylinders ◽  
Orthogonal Functions ◽  
Determinantal Equation ◽  
Coaxial Cylinders ◽  
The Stability

Further consideration is given to the stability of the flow of an idealized elasticoviscous liquid contained in the narrow channel between two rotating coaxial cylinders. The work of Part 1 (Thomas & Walters 1964) is extended to include highly elastic liquids. To facilitate this, use is made of the orthogonal functions used by Reid (1958) in his discussion of the associated Dean-type stability problem. It is shown that the critical Taylor number Tc decreases steadily as the amount of elasticity in the liquid increases, until a transition is reached after which the roots of the determinantal equation which determines the Taylor number T as a function of the wave-number ε become complex. It is concluded that the principle of exchange of stabilities may not hold for highly elastic liquids.

Experiments on the stability of viscous flow between rotating cylinders III. Enhancement of stability by modulation

1964 ◽  
Vol 281 (1384) ◽  
pp. 130-139 ◽  
Keyword(s):  
Viscous Flow ◽  
Couette Flow ◽  
Rotating Cylinders ◽  
The Stability

The onset of instability in Couette flow can be inhibited by modulating the rate of rotation of the inner cylinder. The origin of the inhibition has been shown experimentally to be due to the viscous wave propagated across the annulus by the modulated cylinder.

Sign in / Sign up

Export Citation Format

Share Document