On the stability of viscous flow between parallel planes in the presence of a co-planar magnetic field

1954 ◽  
Vol 221 (1145) ◽  
pp. 189-206 ◽  
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Wave Number ◽  
Neutral Stability ◽  
Energy Relation ◽  
Geometrical Properties ◽  
Electrically Conducting ◽  
Neutral Curves ◽  
The Stability ◽  
The Magnetic Field

Linearized equations are derived which govern the stability of a viscous, electrically conducting fluid in motion between two parallel planes in the presence of a co-planar magnetic field. With one suitable approximation, which restricts the valid range of Reynolds number of the theory, the problem of stability is reduced to the solution of a fourth-order ordinary differential equation. The disturbances considered are neither amplified nor damped, but are neutral. Curves of wave number against Reynolds number for neutral stability are calculated for a range of values of a certain parameter, q , which represents the magnetic effects. For given physical and geometrical properties, the critical Reynolds number above which the flow is unstable rises with the strength of the magnetic field. These results are completely within the range of the approximation mentioned. In addition, an energy relation is derived which illustrates the balance between energy transferred from the basic flow to the disturbances, and that dissipated by viscosity and by the magnetic field perturbations.

The stability of the flow of an electrically conducting fluid between parallel planes under a transverse magnetic field

1955 ◽  
Vol 233 (1192) ◽  
pp. 105-125 ◽  
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Transverse Magnetic Field ◽  
Transverse Magnetic ◽  
Conducting Fluid ◽  
Wave Number ◽  
Neutral Stability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
The Stability

The stability under small disturbances is investigated of the two-dimensional laminar motion of an electrically conducting fluid under a transverse magnetic field. It is found that the dominating factor is the change in shape of the undisturbed velocity profile caused by the magnetic field, which depends only on the Hartmann number M . Curves of wave number against Reynolds number for neutral stability are calculated for a range of values of M ; for large values of M the calculations are similar to those which determine the stability of ordinary boundary-layer flow. The critical Reynolds number is found to rise very rapidly with increasing M , so that a transverse magnetic field has a powerful stabilizing influence on this type of flow.

Hydromagnetic instability of a free shear layer at small magnetic Reynolds numbers

1971 ◽  
Vol 49 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Kanefusa Gotoh
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Shear Layer ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Free Shear Layer ◽  
Electrically Conducting ◽  
The Stability ◽  
The Magnetic Field

The effect of a uniform and parallel magnetic field upon the stability of a free shear layer of an electrically conducting fluid is investigated. The equations of the velocity and the magnetic disturbances are solved numerically and it is shown that the flow is stabilized with increasing magnetic field. When the magnetic field is expressed in terms of the parameter N (= M2/R2), where M is the Hartmann number and R is the Reynolds number, the lowest critical Reynolds number is caused by the two-dimensional disturbances. So long as 0 [les ] N [les ] 0·0092 the flow is unstable at all R. For 0·0092 < N [les ] 0·0233 the flow is unstable at 0 < R < Ruc where Ruc decreases as N increases. For 0·0233 < N < 0·0295 the flow is unstable at Rlc < R < Ruc where Rlc increases with N. Lastly for N > 0·0295 the flow is stable at all R. When the magnetic field is measured by M, the lowest critical Reynolds number is still due to the two-dimensional disturbances provided 0 [les ] M [les ] 0·52, and Rc is given by the corresponding Rlc. For M > 0·52, Rc is expressed as Rc = 5·8M, and the responsible disturbance is the three-dimensional one which propagates at angle cos−1(0·52/M) to the direction of the basic flow.

Rayleigh–Bénard convection and pattern formation in magnetohydrodynamics

1998 ◽  
Vol 60 (3) ◽  
pp. 529-539 ◽  
Author(s):  
RENU BAJAJ ◽  
S. K. MALIK
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

A nonlinear thermal instability in a layer of electrically conducting fluid in the presence of a magnetic field is discussed. Steady-state bifurcation results in the formation of patterns: rolls, squares and hexagons. The stability of various patterns is also investigated. It is found that in the absence of a magnetic field only rolls are stable, but when the magnetic field strength exceeds a certain finite value, squares and hexagons also become stable.

Hydromagnetic Stability of Two Rivlin-Ericksen Elastico-Viscous Superposed Conducting Fluids

1997 ◽  
Vol 52 (6-7) ◽  
pp. 528-532
Author(s):  
R. C. Sharma ◽  
P. Kumar
Keyword(s):  
Magnetic Field ◽  
Wave Number ◽  
Horizontal Magnetic Field ◽  
Number Range ◽  
Wave Number Range ◽  
Unstable Configuration ◽  
The Stability ◽  
Superposed Fluids ◽  
The Magnetic Field ◽  
Conducting Fluids

Abstract The stability of the plane interface separating two Rivlin-Ericksen elastico-viscous superposed fluids of uniform densities when the whole system is immersed in a uniform horizontal magnetic field has been studied. The stability analysis has been carried out, for mathematical simplicity, for two highly viscous fluids of equal kinematic viscosities and equal kinematic viscoelasticities. It is found that the stability criterion is independent of the effects of viscosity and viscoelasticity and is dependent on the orientation and magnitude of the magnetic field. The magnetic field is found to stabilize a certain wave-number range of the unstable configuration. The behaviour of growth rates with respect to kinematic viscosity and kinematic viscoelasticity parameters are examined numerically.

Experimental investigations of the stability of channel flows. Part 2. Two-layered co-current flow in a rectangular channel

1972 ◽  
Vol 52 (3) ◽  
pp. 401-423 ◽  
Author(s):  
Timothy W. Kao ◽  
Cheol Park
Keyword(s):  
Reynolds Number ◽  
Current Flow ◽  
Rectangular Channel ◽  
Shear Waves ◽  
Transition To Turbulence ◽  
Wave Number ◽  
Neutral Stability ◽  
Mean Flow ◽  
Transition Range ◽  
The Stability

The stability of the laminar co-current flow of two fluids, oil and water, in a rectangular channel was investigated experimentally, with and without artificial excitation. For the ratio of viscosity explored, only the disturbances in water grew in the beginning stages of transition to turbulence. The critical water Reynolds number, based upon the hydraulic diameter of the channel and the superficial velocity defined by the ratio of flow rate of water to total cross-sectional area of the channel, was found to be 2300. The behaviour of damped and growing shear waves in water was examined in detail using artificial excitation and briefly compared with that observed in Part 1. Mean flow profiles, the amplitude distribution of disturbances in water, the amplification rate, wave speed and wavenumbers were obtained. A neutral stability boundary in the wave-number, water Reynolds number plane was also obtained experimentally.It was found that in natural transition the interfacial mode was not excited. The first appearance of interfacial waves was actually a manifestation of the shear waves in water. The role of the interface in the transition range from laminar to turbulent flow in water was to introduce and enhance spanwise oscillation in the water phase and to hasten the process of breakdown for growing disturbances.

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.

Instability of a surface of discontinuity of velocity in a parallel uniform magnetic field

1963 ◽  
Vol 16 (2) ◽  
pp. 187-196 ◽  
Author(s):  
D. D. Mallick
Keyword(s):  
Magnetic Field ◽  
Secular Equation ◽  
Magnetic Reynolds Number ◽  
Wave Number ◽  
Inviscid Fluid ◽  
Dimensionless Wave Number ◽  
Electrically Conducting ◽  
Two Parameters ◽  
Two Sides ◽  
The Magnetic Field

The problem described by the title is investigated when the magnetic field is uniform and parallel to the velocity on the two sides of a surface of discontinuity of velocity in an electrically conducting inviscid fluid. The secular equation depends on two parameters β and N, where β is the ratio of magnetic Reynolds number to dimensionless wave number and N is the ratio of the magnetic to the kinetic energy of the fluid. It is found that the flow is unstable for all values of β and N.

The stability of a two-dimensional laminar jet

1958 ◽  
Vol 4 (3) ◽  
pp. 261-275 ◽  
Author(s):  
T. Tatsumi ◽  
T. Kakutani
Keyword(s):  
Reynolds Number ◽  
Basic Assumption ◽  
Wave Number ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Lower Branch ◽  
Laminar Jet ◽  
Numerical Result ◽  
The Stability ◽  
Different Parts

This paper deals with the stability of a two-dimensional laminar jet against the infinitesimal antisymmetric disturbance. The curve of the neutral stability in the (α, R)-plane (α, the wave-number; R, Reynolds number) is calculated using two different methods for the different parts of the curve; the solution is developed in powers of (αR)−1 for obtaining the upper branch of the curve and in powers of αR for the lower branch.The asymptotic behaviour of these branches is that for branch I,$\alpha \rightarrow 2, \;\; c \rightarrow \frac{2}{3}$ for $R \rightarrow \infty$; and for branch II, $R \sim 1\cdot12\alpha^{-1|2},\; c \sim 1\cdot 20 \alpha^2$ for α → 0. Some discussion is given on the validity of the basic assumption of the stability theory in relation to the numerical result obtained here.

Hydrothermal wave instability of thermocapillary-driven convection in a coplanar magnetic field

1997 ◽  
Vol 347 ◽  
pp. 141-169 ◽  
Author(s):  
JĀNIS PRIEDE ◽  
GUNTER GERBETH
Keyword(s):  
Magnetic Field ◽  
Marangoni Number ◽  
Travelling Waves ◽  
Return Flow ◽  
Electrically Conducting ◽  
Flux Lines ◽  
Magnetic Flux Lines ◽  
Critical Marangoni Number ◽  
The Stability ◽  
The Magnetic Field

We study the linear stability of surface-tension-driven unidirectional shear flow in an unbounded electrically conducting liquid layer heated from the side and subjected to a uniform magnetic field in the plane of the layer. The threshold of convective instability with respect to oblique travelling waves is calculated depending on the strength and orientation of the magnetic field. For longitudinal waves the critical Marangoni number and the corresponding wavelength are found to increase directly with the induction of a sufficiently strong magnetic field. In general, a coplanar magnetic field causes stabilization of all disturbances except those aligned with the field, which are not influenced at all. With increase of the magnetic field this effect results in the alignment of the most unstable disturbance along the magnetic flux lines. The maximal stabilization is ensured by the magnetic field being imposed spanwise to the basic flow. The corresponding critical Marangoni number is found to be almost insensitive to the thermal properties of the bottom. The strength of the magnetic field necessary to attain the maximal stabilization for a thermally well-conducting bottom is considerably lower than that for an insulating bottom. The basic return flow is found to be linearly stable with respect to purely hydrodynamic disturbances. This effect determines the stability of the basic state with respect to transverse hydrothermal waves at Prandtl number Pr<Prc=0.018. For such a small Pr no alignment of the critical perturbation with a spanwise magnetic field is possible, and the critical Marangoni number can be increased almost directly with the strength of the magnetic field without limit.

On the stability of the screw pinch in the CGL model

1976 ◽  
Vol 16 (3) ◽  
pp. 261-283 ◽  
Author(s):  
Krishna M. Srivastava ◽  
F. Waelbroeck
Keyword(s):  
Magnetic Field ◽  
Wave Number ◽  
Larmor Radius ◽  
Anisotropic Pressure ◽  
Finite Larmor Radius ◽  
Main Column ◽  
First Order Correction ◽  
The Stability ◽  
The Magnetic Field ◽  
Ideal Plasma

We have investigated the stability of the screw pinch with the help of the double adiabatic (CGL) equations including the finite Larmor radius effects through the anisotropic pressure tensor. The calculations are approximate, with FLR treated as a first-order correction to the ideal plasma equations. The dispersion relation has been solved for various values of R2 = p∥/p⊥ and α for the rale and imaginary part of the frequency (ω = ωR ± iωI) in three particular cases: (a) μ = 0, the θ-pinch, (b) μ = ∞, the Z-pinch, (c) μ = -α/m, field distubances parallel to the equilibrium field. Here μ is the pitch of the magnetic field in the pressureless plasma surrounding the main column, α is the wave number, m is the azimuthal number, p∥ and p⊥ are plasma pressures along and perpendicular to the magnetic field.

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