On the stability of a magnetically driven rotating fluid flow

1974 ◽  
Vol 63 (3) ◽  
pp. 593-605 ◽  
Author(s):  
A. T. Richardson
Keyword(s):  
Cell Structure ◽  
Low Frequency ◽  
Azimuthal Velocity ◽  
Magnetic Reynolds Number ◽  
Taylor Number ◽  
Marginal Stability ◽  
Cylinder Wall ◽  
Neutral Stability ◽  
The Stability ◽  
Direct Numerical Integration

After making the laboratory approximation of small magnetic Reynolds number, the steady, axisymmetric and purely azimuthal velocity profile that in principle can be generated in an incompressible viscous electrically conducting fluid contained in a fixed infinitely long circular cylinder by a magnetic field transverse to the cylinder axis and uniformly rotating with low frequency is subjected to infinitesimal axisymmetric perturbations. The principle of the exchange of stabilities is assumed to hold and the marginal-stability problem becomes a sixth-order eigenvalue problem involving the magnetic Taylor number and the axial wavenumber. An asymptotic analysis, based on the assumption that the magnetic Taylor number is large, and using solutions of the comparison equation d6y/dz6 = zy, is presented in order to obtain first approximations to the neutral-stability curves of the first and second eigenmodes, and compared with the results of direct numerical integration. It is found that at the onset of instability the secondary motions have a multi-cell structure, the motions in the region, near the cylinder wall, of adversely distributed angular momentum driving through weak viscous action the cells in the interior.

The Effects of Main-Flow Radial Velocity on the Stability of Developing Laminar Pipe Flow

1976 ◽  
Vol 43 (2) ◽  
pp. 209-212 ◽  
Author(s):  
F. C. T. Shen ◽  
T. S. Chen ◽  
L. M. Huang
Keyword(s):  
Radial Velocity ◽  
Main Flow ◽  
Integration Scheme ◽  
Neutral Stability ◽  
Radial Velocity Component ◽  
Numerical Integration Scheme ◽  
Critical Reynolds Numbers ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Sommerfeld Equation

In studying the stability due to axisymmetric disturbances of the developing flow of an incompressible fluid in the entrance region of a circular tube, a generalized version of the Orr-Sommerfeld equation was derived which takes account of the radial velocity component in the main flow. The new terms in the generalized Orr-Sommerfeld equation are inversely proportional to the Reynolds number. The resulting eigenvalue problem consisting of the disturbance equation and the boundary conditions was solved by a direct numerical integration scheme along with an iteration procedure. Neutral stability curves and critical Reynolds numbers at various axial locations are presented. A comparison of the present results with those from the conventional Orr-Sommerfeld equation in which the effect of the main-flow radial velocity is neglected, shows that inclusion of the radial velocity contributes to a destabilization of the main flow.

On the stability of slowly varying flow between concentric cylinders

1977 ◽  
Vol 355 (1681) ◽  
pp. 209-224 ◽  
Keyword(s):  
Growth Rate ◽  
Outer Cylinder ◽  
Low Frequency ◽  
Base Flow ◽  
Taylor Number ◽  
Sinusoidal Modulation ◽  
Concentric Cylinders ◽  
The Stability

We have applied the W.K.B. type of approximation to the stability of flow between concentric cylinders when the speed of rotation is varying slowly. For the case of a fixed outer cylinder we have shown that if the speed of the inner cylinder is increasing slowly then the growth rate for an axisymmetric disturbance is reduced considerably compared with that calculated by using a steady base flow. This leads to an increase of the Taylor number at which growth would first occur of the order of 20 % in cases where the theory seems applicable. We have also obtained results for a small sinusoidal modulation of low frequency to the inner cylinder velocity. These confirm Hall’s (1975) results of slight destabilization, almost independent of the frequency.

Low-frequency instabilities of a warm plasma in a magnetic field: Part 1. Instabilities driven by field-aligned currents

1977 ◽  
Vol 17 (1) ◽  
pp. 105-122 ◽  
Author(s):  
Dean F. Smith ◽  
Joseph V. Hollweg
Keyword(s):  
Magnetic Field ◽  
Stability Analysis ◽  
Low Frequency ◽  
Marginal Stability ◽  
Numerical Techniques ◽  
Acoustic Instability ◽  
Ion Acoustic ◽  
Warm Plasma ◽  
The Stability ◽  
Thermal Speed

The marginal stability of a plasma carrying current along the static magnetic field with isotropic Maxwellian ions and isotropic Maxwellian electrons drifting relative to the ions is investigated. The complete electromagnetic dispersion relation is studied using numerical techniques; the electron sums are restricted to three terms which limits the analysis to frequencies much less than the electron gyro-frequency, but includes frequencies somewhat above the ion gyro-frequency. A ‘kink-like’ instability and an instability of the Alfvén mode are found to have the lowest threshold drift velocities in most cases. In fact the threshold drift for the kink-like instability can be significantly less than the ion thermal speed. Electrostatic and electromagnetic ion-cyclotron instabilities are also found as well as the electro-static ion-acoustic instability. No instability of the fast magnetosonic mode was found. The stability analysis provides only threshold drift velocities and gives no information about growth rates.

The stability of unsteady cylinder flows

1975 ◽  
Vol 67 (1) ◽  
pp. 29-63 ◽  
Author(s):  
P. Hall
Keyword(s):  
Angular Velocity ◽  
Broad Band ◽  
Low Frequency ◽  
Theoretical Work ◽  
Finite Amplitude ◽  
Taylor Number ◽  
Alternative Formulation ◽  
Concentric Cylinders ◽  
Stabilizing Effect ◽  
The Stability

First, the linear stability of the flow between two concentric cylinders when the outer one is a t rest and the inner has angular velocity Ω{1+ εcosωt} is considered. In the limit in which ε and ω tend to zero it is found that the critical Taylor number a t which instability first occurs is decreased by an amount of order ε2from its unmodulated value, the stabilizing effect a t order ε2ω2being slight. The limit in which ω tends to infinity with ε arbitrary is then studied. In this case it is found that the critical Taylor number is decreased by an amount of order ε2ω−3from its unmodulated value.Second, the effect of taking nonlinear terms into account is investigated. It is found that equilibrium perturbations of small but finite amplitude can exist under slightly supercritical conditions in both the high and low frequency limits. Some comparisons with experimental results are made, but these indicate that further theoretical work is needed for a broad band of values of ω. In appendix B it is shown how this can be done by an alternative formulation of the problem.

scholarly journals Azimuthal magnetorotational instability with super-rotation

2018 ◽  
Vol 84 (1) ◽  
Author(s):  
G. Rüdiger ◽  
M. Schultz ◽  
M. Gellert ◽  
F. Stefani
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Numerical Solutions ◽  
Hartmann Number ◽  
Outer Cylinder ◽  
Magnetic Reynolds Number ◽  
Neutral Stability ◽  
Magnetorotational Instability ◽  
Magnetic Prandtl Number ◽  
The Stability

It is demonstrated that the azimuthal magnetorotational instability (AMRI) also works with radially increasing rotation rates contrary to the standard magnetorotational instability for axial fields which requires negative shear. The stability against non-axisymmetric perturbations of a conducting Taylor–Couette flow with positive shear under the influence of a toroidal magnetic field is considered if the background field between the cylinders is current free. For small magnetic Prandtl number $Pm\rightarrow 0$ the curves of neutral stability converge in the (Hartmann number,Reynolds number) plane approximating the stability curve obtained in the inductionless limit $Pm=0$. The numerical solutions for $Pm=0$ indicate the existence of a lower limit of the shear rate. For large $Pm$ the curves scale with the magnetic Reynolds number of the outer cylinder but the flow is always stable for magnetic Prandtl number unity as is typical for double-diffusive instabilities. We are particularly interested to know the minimum Hartmann number for neutral stability. For models with resting or almost resting inner cylinder and with perfectly conducting cylinder material the minimum Hartmann number occurs for a radius ratio of $r_{\text{in}}=0.9$. The corresponding critical Reynolds numbers are smaller than $10^{4}$.

Stability of bounded rapid shear flows of a granular material

1996 ◽  
Vol 308 ◽  
pp. 31-62 ◽  
Author(s):  
Chi-Hwa Wang ◽  
R. Jackson ◽  
S. Sundaresan
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Granular Material ◽  
Bulk Density ◽  
Particulate Material ◽  
Dominant Mode ◽  
Marginal Stability ◽  
Sheared Layer ◽  
The Stability ◽  
Unusual Type

This paper presents a linear stability analysis of a rapidly sheared layer of granular material confined between two parallel solid plates. The form of the steady base-state solution depends on the nature of the interaction between the material and the bounding plates and three cases are considered, in which the boundaries act as sources or sinks of pseudo-thermal energy, or merely confine the material while leaving the velocity profile linear, as in unbounded shear. The stability analysis is conventional, though complicated, and the results are similar in all cases. For given physical properties of the particles and the bounding plates it is found that the condition of marginal stability depends only on the separation between the plates and the mean bulk density of the particulate material contained between them. The system is stable when the thickness of the layer is sufficiently small, but if the thickness is increased it becomes unstable, and initially the fastest growing mode is analogous to modes of the corresponding unbounded problem. However, with a further increase in thickness a new mode becomes dominant and this is of an unusual type, with no analogue in the case of unbounded shear. The growth rate of this mode passes through a maximum at a certain value of the thickness of the sheared layer, at which point it grows much faster than any mode that could be shared with the unbounded problem. The growth rate of the dominant mode also depends on the bulk density of the material, and is greatest when this is neither very large nor very small.

Resonance in a mathematical model of baroreflex control: arterial blood pressure waves accompanying postural stress

2005 ◽  
Vol 288 (6) ◽  
pp. R1637-R1648 ◽  
Author(s):  
Peter E. Hammer ◽  
J. Philip Saul
Keyword(s):  
Blood Pressure ◽  
Mathematical Model ◽  
Blood Volume ◽  
Low Frequency ◽  
Pressure Waves ◽  
Central Blood Volume ◽  
Arterial Baroreflex ◽  
Arterial Blood ◽  
Gain Margin ◽  
The Stability

A mathematical model of the arterial baroreflex was developed and used to assess the stability of the reflex and its potential role in producing the low-frequency arterial blood pressure oscillations called Mayer waves that are commonly seen in humans and animals in response to decreased central blood volume. The model consists of an arrangement of discrete-time filters derived from published physiological studies, which is reduced to a numerical expression for the baroreflex open-loop frequency response. Model stability was assessed for two states: normal and decreased central blood volume. The state of decreased central blood volume was simulated by decreasing baroreflex parasympathetic heart rate gain and by increasing baroreflex sympathetic vaso/venomotor gains as occurs with the unloading of cardiopulmonary baroreceptors. For the normal state, the feedback system was stable by the Nyquist criterion (gain margin = 0.6), but in the hypovolemic state, the gain margin was small (0.07), and the closed-loop frequency response exhibited a sharp peak (gain of 11) at 0.07 Hz, the same frequency as that observed for arterial pressure fluctuations in a group of healthy standing subjects. These findings support the theory that stresses affecting central blood volume, including upright posture, can reduce the stability of the normally stable arterial baroreflex feedback, leading to resonance and low-frequency blood pressure waves.

scholarly journals Evolution of a narrow-band spectrum of random surface gravity waves

2003 ◽  
Vol 478 ◽  
pp. 1-10 ◽  
Author(s):  
KRISTIAN B. DYSTHE ◽  
KARSTEN TRULSEN ◽  
HARALD E. KROGSTAD ◽  
HERVÉ SOCQUET-JUGLARD
Keyword(s):  
Three Dimensional ◽  
Low Frequency ◽  
Three Dimensions ◽  
Band Spectrum ◽  
Nls Equation ◽  
Random Surface ◽  
Narrow Bandwidth ◽  
Quasi Stationary State ◽  
The Stability ◽  
Three Dimensional Simulations

Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

FABRICATION OF NANOPOROUS CHITOSAN MEMBRANES

NANO ◽  
2010 ◽  
Vol 05 (01) ◽  
pp. 53-60 ◽  
Author(s):  
XIAOLIANG WANG ◽  
XIANG LI ◽  
ELEANOR STRIDE ◽  
MOHAN EDIRISINGHE
Keyword(s):  
Low Frequency ◽  
Drug Carriers ◽  
Structural Features ◽  
Wound Dressings ◽  
Ftir Analysis ◽  
Tissue Engineering Scaffolds ◽  
Nanoscale Structures ◽  
Low Frequency Ultrasound ◽  
Chitosan Membranes ◽  
The Stability

Naturally derived biopolymers have been widely used for biomedical applications such as drug carriers, wound dressings, and tissue engineering scaffolds. Chitosan is a typical polysaccharide of great interest due to its biocompatibility and film-formability. Chitosan membranes with controllable porous structures also have significant potential in membrane chromatography. Thus, the processing of membranes with porous nanoscale structures is of great importance, but it is also challenging and this has limited the application of these membranes to date. In this study, with the aid of a carefully selected surfactant, polyethyleneglycol stearate-40, chitosan membranes with a well controlled nanoscale structure were successfully prepared. Additional control over the membrane structure was obtained by exposing the suspension to high intensity, low frequency ultrasound. It was found that the concentration of chitosan/surfactant ratio and the ultrasound exposure conditions affect the structural features of the membranes. The stability of nanopores in the membrane was improved by intensive ultrasonication. Furthermore, the stability of the blended suspensions and the intermolecular interactions between chitosan and the surfactant were investigated using scanning electron microscope and Fourier transform infrared spectroscopy (FTIR) analysis, respectively. Hydrogen bonds and possible reaction sites for molecular interactions in the two polymers were also confirmed by FTIR analysis.

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