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.

The stability of Couette flow in the presence of an axial magnetic field

1963 ◽  
Vol 17 (1) ◽  
pp. 52-60 ◽  
Author(s):  
Ulrich H. Kurzweg
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Couette Flow ◽  
Axial Magnetic Field ◽  
Taylor Number ◽  
Asymptotic Formulas ◽  
Magnetic Prandtl Number ◽  
The Mean ◽  
The Stability ◽  
Gap Spacing

The stability of Couette flow between concentric, co-rotating cylinders in an axial magnetic field is examined for fluids of arbitrary magnetic Prandtl number Pm = ν/η, where ν is the kinematic and η the magnetic viscosity of the fluid. It is assumed that the gap spacing d between the cylinders is small compared to the mean radius and that no magnetic disturbances penetrate into the cylinder walls. The critical Taylor number at which non-oscillatory disturbances are marginally stable is determined as a function of the magnetic Prandtl number and the dimensionless parameter S = (Vad/v)2, where Va is the Alfvén velocity. Asymptotic formulas relating the critical Taylor number to the magnitude of the magnetic field are derived for the limiting conditions of very small and very large magnetic Prandtl number.

THE STAEILITY OF COUETTE FLOW IN AN AXIAL MAGNETIC FIELD

1958 ◽  
Vol 36 (11) ◽  
pp. 1509-1525 ◽  
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 ◽  
The Stability

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.

An exact solution in the stability of m. h. d. Couette flow

1972 ◽  
Vol 327 (1569) ◽  
pp. 269-278 ◽  
Keyword(s):  
Magnetic Field ◽  
Exact Solution ◽  
Couette Flow ◽  
Stability Problem ◽  
Axial Magnetic Field ◽  
Narrow Gap ◽  
Mhd Stability ◽  
Arbitrary Magnetic Field ◽  
The Stability ◽  
Conducting Cylinders

The MHD stability problem for dissipative Couette flow in a narrow gap between corotating, conducting cylinders with an axial magnetic field is solved exactly. Results are presented for an arbitrary magnetic field; in particular, previous results on the zero and infinite magnetic field limits are verified.

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.

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.

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

1994 ◽  
Vol 72 (5-6) ◽  
pp. 258-265 ◽  
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 Stability

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.

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.

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