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

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.

The effect of a radial magnetic field on the stability of Couette flow

Canadian Journal of Physics ◽

10.1139/p94-038 ◽

1994 ◽

Vol 72 (5-6) ◽

pp. 258-265 ◽

Cited By ~ 1

Author(s):

M. A. Ali

Keyword(s):

Magnetic Field ◽

Eigenvalue Problem ◽

Incompressible Fluid ◽

Couette Flow ◽

Wave Number ◽

Rotating Cylinders ◽

Radial Magnetic Field ◽

Electrically Conducting ◽

Angular Velocities ◽

The effect of a radial magnetic field on the stability of an electrically conducting incompressible fluid between two concentric rotating cylinders is considered. The eigenvalue problem for determining the critical Taylor number TC and the corresponding wave number aC is solved numerically for different values of ±μ(= Ω2/Ω1), (where Ω1, and Ω2 are me angular velocities of the inner and outer cylinders, respectively) and for different gap sizes. It is observed that the radial magnetic field stabilizes the flow. This effect is more pronounced for cylinders that are corotating as compared with counter-rotating cylinders or the situation where only the inner one is rotating.

ONSET OF INSTABILITY IN HYDROMAGNETIC COUETTE FLOW

Analysis and Applications ◽

10.1142/s0219530504000059 ◽

2004 ◽

Vol 02 (02) ◽

pp. 145-159 ◽

Cited By ~ 6

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 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.

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

10.1017/s002211206400088x ◽

1964 ◽

Vol 19 (4) ◽

pp. 528-538 ◽

Cited By ~ 35

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 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.

Stability of nonaxisymmetric ferrofluid flow in rotating cylinders with magnetic field

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.3727 ◽

2005 ◽

Vol 2005 (23) ◽

pp. 3727-3737 ◽

Cited By ~ 6

Author(s):

Jitender Singh ◽

Renu Bajaj

Keyword(s):

Magnetic Field ◽

Applied Magnetic Field ◽

Critical Value ◽

Annular Space ◽

Rotating Cylinders ◽

Effect of an axially applied magnetic field on the stability of a ferrofluid flow in an annular space between two coaxially rotating cylinders with nonaxisymmetric disturbances has been investigated numerically. The critical value of the ratioΩ∗of angular speeds of the two cylinders, at the onset of the first nonaxisymmetric mode of disturbance, has been observed to be affected by the applied magnetic field.

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

The stability of viscous flow in a curved channel in the presence of a magnetic field

10.1098/rspa.1961.0190 ◽

1961 ◽

Vol 264 (1317) ◽

pp. 155-164 ◽

Cited By ~ 1

Keyword(s):

Magnetic Field ◽

Viscous Flow ◽

Uniform Magnetic Field ◽

Axial Direction ◽

Electrical Conductor ◽

Curved Channel ◽

Coaxial Cylinders ◽

Transverse Pressure ◽

The Stability ◽

The Magnetic Field

The stability of viscous flow between two coaxial cylinders maintained by a constant transverse pressure gradient is considered when the fluid is an electrical conductor and a uniform magnetic field is impressed in the axial direction. The problem is solved and the dependence of the critical number for the onset of instability on the strength of the magnetic field and the coefficient of electrical conductivity of the fluid is determined.

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

10.1017/s0022112066000673 ◽

1966 ◽

Vol 24 (2) ◽

pp. 321-334 ◽

Cited By ~ 60

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 ◽

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.

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

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

10.1098/rspa.1953.0023 ◽

1953 ◽

Vol 216 (1126) ◽

pp. 293-309 ◽

Cited By ~ 54

Keyword(s):

Magnetic Field ◽

Electrical Conductivity ◽

Viscous Flow ◽

Thermal Instability ◽

Horizontal Layer ◽

Rotational Stability ◽

Marginal Stability ◽

Inhibiting Effect ◽

The Stability ◽

The Magnetic Field

In this paper the theory of the stability of viscous flow between two rotating coaxial cylinders which has been developed by Taylor, Jeffreys and Meksyn is extended to the case when the fluid considered is an electrical conductor and a magnetic field along the axis of the cylinders is present. A differential equation of order eight is derived which governs the situation in marginal stability; and a significant set of boundary conditions for the problem is formulated. The case when the two cylinders are rotating in the same direction and the difference ( d ) in their radii is small compared to their mean (R 0 ) is investigated in detail. A variational procedure for solving the underlying characteristic value problem and determining the critical Taylor numbers for the onset of instability is described. As in the case of thermal instability of a horizontal layer of fluid heated below, the effect of the magnetic field is to inhibit the onset of instability, the inhibiting effect being the greater, the greater the strength of the field and the value of the electrical conductivity. In both cases, the inhibiting effect of the magnetic field depends on the strength of the field ( H ), the density ( ρ ) and the coefficients of electrical conductivity ( σ ), kinematic viscosity ( v ) and magnetic permeability ( μ ) through the same non-dimensional combination Q =μ 2 H 2 d 2 σ/ pv ; however, the effect on rotational stability is more pronounced than on thermal instability. A table of the critical Taylor numbers for various values of Q is provided.

THE STAEILITY OF COUETTE FLOW IN AN AXIAL MAGNETIC FIELD

Canadian Journal of Physics ◽

10.1139/p58-151 ◽

1958 ◽

Vol 36 (11) ◽

pp. 1509-1525 ◽

Cited By ~ 16

Author(s):

E. R. Niblett

Keyword(s):

Magnetic Field ◽

Boundary Conditions ◽

Theoretical Prediction ◽

Viscous Flow ◽

Rotation Speed ◽

Axial Magnetic Field ◽

Radioactive Isotopes ◽

Electrically Conducting ◽

Electrically Conducting Fluid ◽

Chandrasekhar's theory of the stability of viscous flow of an electrically conducting fluid between coaxial rotating cylinders with perfectly conducting walls is extended to include the case of non-conducting walls, and it is found that their effect is to reduce the critical Taylor numbers and increase the wavelength of the instability patterns by considerable amounts. An experiment designed to measure the values of magnetic field and rotation speed at the onset of instability in mercury between perspex cylinders is described. The radioactive isotopes Hg197 and Hg203 were used to trace the flow. The results support the theoretical prediction that the boundary conditions can have a large effect on the motion.

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

10.1017/s0022112064001033 ◽

1964 ◽

Vol 20 (1) ◽

pp. 95-101 ◽

Cited By ~ 59

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 ◽

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.

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

10.1017/s002211206400091x ◽

1964 ◽

Vol 19 (4) ◽

pp. 557-560 ◽

Cited By ~ 33

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 ◽

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.