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 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.

Very slow grow-up of solutions of a semi-linear parabolic equation

2011 ◽  
Vol 54 (2) ◽  
pp. 381-400 ◽  
Author(s):  
Marek Fila ◽  
John R. King ◽  
Michael Winkler ◽  
Eiji Yanagida
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Large Time ◽  
Global Solutions ◽  
Linear Parabolic Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
The Cauchy Problem ◽  
Supercritical Nonlinearity ◽  
Singular Steady State

AbstractWe consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.

Polarographic studies of indium(III) in aqueous medium of sodium azide

1995 ◽  
Vol 73 (2) ◽  
pp. 232-240 ◽  
Author(s):  
Roberto Tokoro ◽  
Mauro Bertotti ◽  
Lúcio Angnes
Keyword(s):  
Large Time ◽  
Time Window ◽  
Hydrazoic Acid ◽  
Catalytic Wave ◽  
Controlled Potential ◽  
The Mean ◽  
The Stability ◽  
Controlled Potential Coulometry ◽  
Strength Medium ◽  
Good Agreement

In this paper, some aspects of the polarography of indium(III) in azide/hydrazoic acid buffer are presented. The electrode process is mostly governed by diffusion in the complexing medium. Nevertheless, for controlled-potential coulometry (involving a large time window) a catalytic process was observed, ascribed to the reaction of the hydrazoic acid and indium amalgam. The stability of azide complexes of In(III) was studied polarographically, and the formation of four mononuclear species was confirmed by computational analysis of data obtained from polarograms of In(III) in different azide concentrations. The values of the global constants obtained are: β1 = 3.7 × 103 M−1, β2 = 8.6 × 105 M−2, β3 = 5.0 × 107 M−3, β4 = 2.1 × 109 M−4, in a constant ionic strength medium held at 2.0 M with NaClO4. These data are in good agreement with those obtained by potentiometry. The value found for E0′ of the In(III)/In couple was −512.2 mV vs. calomel electrode (3 M NaCl), the mean diffusion coefficient of the complex being (6.1 ± 0.2) × 10−6 cm2 s−1 in perchlorate medium (T = 25 °C). Keywords: polarography, indium(III), azide, stability constants, catalytic wave.

Large-time behaviour of solutions of Burgers' equation

2002 ◽  
Vol 132 (4) ◽  
pp. 843-878 ◽  
Author(s):  
Daniel B. Dix
Keyword(s):  
Initial Data ◽  
Asymptotic Behaviour ◽  
Initial Value Problem ◽  
Large Time ◽  
Burgers Equation ◽  
Time Behaviour ◽  
Initial Value ◽  
Large Time Behaviour ◽  
Image Position ◽  
Sharp Error

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x| , where n R. There are two constants of the motion that affect the large-time behaviour: Hopf considered the case n = 0 (i.e. u0L1(R)), and the case sufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only on and n, such that uniformly in . When n 1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

scholarly journals About the Use of Entropy Production for the Landau–Fermi–Dirac Equation

2021 ◽  
Vol 183 (1) ◽  
Author(s):  
R. Alonso ◽  
V. Bagland ◽  
L. Desvillettes ◽  
B. Lods
Keyword(s):  
Dirac Equation ◽  
Classical Limit ◽  
Large Time ◽  
A Priori Estimates ◽  
Landau Equation ◽  
A Priori ◽  
Time Behaviour ◽  
Entropy Dissipation ◽  
Priori Estimates ◽  
Potential Case

AbstractIn this paper, we present new estimates for the entropy dissipation of the Landau–Fermi–Dirac equation (with hard or moderately soft potentials) in terms of a weighted relative Fisher information adapted to this equation. Such estimates are used for studying the large time behaviour of the equation, as well as for providing new a priori estimates (in the soft potential case). An important feature of such estimates is that they are uniform with respect to the quantum parameter. Consequently, the same estimations are recovered for the classical limit, that is the Landau equation.

OPTIMAL VELOCITY MODEL WITH RELATIVE VELOCITY

2006 ◽  
Vol 17 (01) ◽  
pp. 65-73 ◽  
Author(s):  
SHIRO SAWADA
Keyword(s):  
Nonlinear Analysis ◽  
Relative Velocity ◽  
Velocity Model ◽  
Fundamental Diagram ◽  
Optimal Velocity ◽  
Traffic Jam ◽  
Optimal Velocity Model ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Good Agreement

The optimal velocity model which depends not only on the headway but also on the relative velocity is analyzed in detail. We investigate the effect of considering the relative velocity based on the linear and nonlinear analysis of the model. The linear stability analysis shows that the improvement in the stability of the traffic flow is obtained by taking into account the relative velocity. From the nonlinear analysis, the relative velocity dependence of the propagating kink solution for traffic jam is obtained. The relation between the headway and the velocity and the fundamental diagram are examined by numerical simulation. We find that the results by the linear and nonlinear analysis of the model are in good agreement with the numerical results.

Global Stability Analysis of the Wake behind a Circular Cylinder Based on the POD-Chiba Method

2011 ◽  
Vol 137 ◽  
pp. 72-76
Author(s):  
Wei Zhang ◽  
Xian Wen ◽  
Yan Qun Jiang
Keyword(s):  
Stability Analysis ◽  
Circular Cylinder ◽  
Global Stability ◽  
Dimensional Model ◽  
Global Stability Analysis ◽  
The Stability ◽  
Low Dimensional ◽  
Proper Orthogonal ◽  
Calculated Amount ◽  
Good Agreement

A proper orthogonal decomposition (POD) method is applied to study the global stability analysis for flow past a stationary circular cylinder. The flow database at Re=100 is obtained by CFD software, i.e. FLUENT, with which POD bases are constructed by a snapshot method. Based on the POD bases, a low-dimensional model is established for solving the two-dimensional incompressible NS equations. The stability of the flow solution is evaluated by a POD-Chiba method in the way of the eigensystem analysis for the velocity disturbance. The linear stability analysis shows that the first Hopf bifurcation takes place at Re=46.9, which is in good agreement with available results by other high-order accurate stability analysis methods. However, the calculated amount of POD is little, which shows the availability and advantage of the POD method.

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