scholarly journals The influence of circulation on the stability of vortices to mode-one disturbances

1995 ◽  
Vol 451 (1943) ◽  
pp. 747-755 ◽  
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Initial Value Problem ◽  
Large Time ◽  
Asymptotic Methods ◽  
Two Dimensional ◽  
Initial Value ◽  
The Stability ◽  
Ultimate Fate ◽  
Solution Behaviour

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.

Excitation of surface waves in magnetohydrodynamics

1990 ◽  
Vol 43 (2) ◽  
pp. 183-188 ◽  
Author(s):  
Bhimsen K. Shivamoggi
Keyword(s):  
Magnetic Field ◽  
Gravitational Field ◽  
Asymptotic Behaviour ◽  
Pressure Distribution ◽  
Surface Waves ◽  
Initial Value Problem ◽  
Large Time ◽  
Far Field ◽  
Two Dimensional ◽  
Initial Value

A study is made of the transient development of two-dimensional linearized surface waves generated by a localized steady pressure distribution on the interface between a uniformly streaming, semi-infinite, infinitely conducting plasma subjected to a gravitational field and the confining vacuum magnetic field. The linearized equations associated with an initial-value problem are used to obtain the large-time asymptotic behaviour of the disturbance in the far field.

Stability of time-dependent rotational Couette flow Part 2. Stability analysis

1971 ◽  
Vol 48 (2) ◽  
pp. 365-384 ◽  
Author(s):  
C. F. Chen ◽  
R. P. Kirchner
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Kinetic Energy ◽  
Initial Value Problem ◽  
Basic Flow ◽  
The Other ◽  
Time Dependent ◽  
Stability Criteria ◽  
Initial Value ◽  
The Stability

The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results.

On the stability of the similarity solutions for swirling flow above an infinite rotating disk

1984 ◽  
Vol 144 ◽  
pp. 311-328 ◽  
Author(s):  
R. J. Bodonyi ◽  
B. S. Ng
Keyword(s):  
Large Time ◽  
Swirling Flow ◽  
Rotating Disk ◽  
Similarity Solutions ◽  
Time Behaviour ◽  
Rotating Fluid ◽  
Initial Value ◽  
Algebraic Decay ◽  
The Stability ◽  
Good Agreement

A stability theory for the steady swirling flow above an infinite rotating disk immersed in an otherwise unbounded rigidly rotating fluid is developed in order to corroborate the various numerical computations considered for this problem. An analysis of the initial-value problem for linearized time-dependent perturbations on the steady-state similarity solutions shows that the disturbance equations have a stable continuum spectrum which, under certain conditions, exhibits only algebraic decay in time. In addition, a numerical analysis on the discrete spectrum shows that there are unstable eigenvalues for certain rotational rates of the disk relative to the fluid at infinity. The results obtained are in good agreement with the large-time behaviour of the corresponding solutions of the unsteady similarity equations.

scholarly journals On the stability of the Schrödinger equation with time delay

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 759-766 ◽  
Author(s):  
Deniz Agirseven
Keyword(s):  
Hilbert Space ◽  
Time Delay ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Stability Estimates ◽  
Initial Value ◽  
Dinger Equation ◽  
The Stability

In the present paper, the initial value problem for the Schr?dinger equation with time delay in a Hilbert space is investigated. Theorems on stability estimates for the solution of the problem are established. The applications of theorems for three types of Schr?dinger problems are provided.

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