Stability of the magnetic Couette-Taylor flow

In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form It is shown that this equilibrium is Ljapounov stable under small perturbations in where provided that the parameters a, b, , are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.Received: February 5, 2003; revised: September 29, 2003Dedicated to Prof. H. Amann on the occasion of his 65. birthday     

Taylor-Couette流稳定性的谱方法研究

该文根据ｓｔｏｋｅｓ算子特征函数,利用谱方法研究了由轴对称Ｔａｙｌｏｒ Ｃｏｕｅｔｔｅ流导出的多模态方程．给出了三模态方程平衡点存在的条件,证明了它的吸引子的存在性,并给出其Ｈａｕｓ ｄｏｒｆｆ维数的上界的估计．     

Numerical and experimental study of the flow in an eccentric Couette-Taylor system with small gap

Various papers were dedicated to the technical and industrial aspects of the Taylor–Couette flow, by analyzing the flow structure in dependence of characteristic parameters. Also some studies concerning the influence of an eccentrical position of the rotating inner cylinder relatively to the fixed outer cylinder exist in the case of a large gap width. The principal technical application of such studies is the flow in journal bearings where the gap width is very small, of the order 0.1\% relative to the inner cylinder radius. Analytical and numerical investigations of the flow in journal bearings are generally based on the two–dimensional Reynolds approximation. This investigation in aiming at a validation of the range where the Reynolds equation is applicable in relation to the characteristical parameters: gap width, eccentricity and Reynolds number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Couette-Taylor流的谱Galerkin逼近

利用谱方法对轴对称的旋转圆柱间的Couette-Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式,利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式,证明其正交性,并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数,给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性,给出了解谱Galerkin逼近的误差估计,并展示了数值计算结果．     

Couette-Taylor流的力学机理与能量转换

同轴圆筒间Couette-Taylor流问题是典型的旋转流动问题,它是层流到湍流过渡的范例,国内外众多学者对其进行了深入的研究.该文探讨Couette-Taylor流问题的力学机理与能量转换,通过将Couette-Taylor流三模混沌系统转换成Kolmogorov形系统,把系统的力矩分为四种类型:惯性力矩,内力矩,耗散力矩和外力矩.通过不同力矩的结合分析和研究了Couette-Taylor流产生混沌的关键因素和物理意义.研究了哈密顿能量,动能和势能之间的相互转换.讨论了能量与雷诺数之间的关系.研究表明四种力矩的耦合是产生混沌的必要条件,而且只有耗散力矩和驱动力矩(外力矩)相匹配时,系统才能产生混沌,其中任何三种力矩耦合均不可能产生混沌.圆筒旋转产生的外力矩供给系统能量,能量增长导致流动失稳,从而产生泰勒漩涡和混沌,进而得出了Couette-Taylor流的能量转换和物理意义.引进Casimir函数分析系统的动力学行为和能量转换,并估计混沌吸引子的界.Casimir函数反映了能量转换和轨道与平衡点间的距离,数值结果仿真出它们之间的关系.     

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.     

Stability of a pseudo-periodic flow

Stability of the free oscillatory flow in a U-shaped tube is considered by the use of a linear stability analysis. A previous initial-value problem approach concerning the stability of time-dependent rotational Couette flow is extended to this pseudo-periodic flow. The theoretical results obtained are in good agreement with experimental observations.

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Résumé La stabilié du mouvement oscillatoire libre d'un fluide pesant et visquex dans un tube en U est étudiée en utilisant une analyse linéaire. Une méthode développée précédemment pour l'étude de la stabilité de l'écoulement instationnaire de Couette entre deux cylindres coaxiaux, est étendue à cet écoulement pseudo-périodique. Les résultats théoriques obtenus confirment pleinement de précédentes observations expérimentales.

     

Stability of non-rotationally symmetric disturbances for inviscid flow between rotating cylinders in the presence of an axial magnetic field

The stability of an inviscid fluid between two concentric rotating cylinders in the presence of an axial magnetic field to nonrotationally symmetric disturbances is investigated under “small gap” approximation. The growth rate of unstable disturbances has been examined for co-rotating cylinders. It has been found that the growth rate of rotationally symmetric disturbances is greater than that of nonrotationally symmetric disturbances and the magnetic field strength which stabilizes the axisymmetric disturbances is also sufficient to stabilize non-symmetric disturbances.     

Stability of the flow of an elastico-viscous fluid

Summary The eigenvalues of the generalized Orr-Sommerfeld equation for an elasticoviscous fluid are found for a constant velocity profile. The result shows that the effect of elasticity is to stabilize the flow.It is also shown that the second-order equation of state is not a suitable model for a fluid when the stability of that fluid is to be analysed.

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Zusammenfassung Die Eigenwerte der allgemeinen Orr-Sommerfeld-Gleichung für eine elastisch-viskose Flüssigkeit werden für ein Profil mit konstanter Geschwindigkeit angegeben. Das Ergebnis beweist, daß der Einfluß der Elastizität eine Stabilisierung der Strömung bewirkt.Es wird auch gezeigt, daß die Zustandsgleichung zweiter Ordnung kein geeignetes Modell für eine Flüssigkeit ist, deren Stabilität analysiert werden soll.

     

Stability of the Kolmogorov flow and its modifications

Recurrence formulas are obtained for the kth term of the long wavelength asymptotics in the stability problem for general two-dimensional viscous incompressible shear flows. It is shown that the eigenvalues of the linear eigenvalue problem are odd functions of the wave number, while the critical values of viscosity are even functions. If the velocity averaged over the long period is nonzero, then the loss of stability is oscillatory. If the averaged velocity is zero, then the loss of stability can be monotone or oscillatory. If the deviation of the velocity from its period-average value is an odd function of spatial variable about some x ₀, then the expansion coefficients of the velocity perturbations are even functions about x ₀ for even powers of the wave number and odd functions about for x ₀ odd powers of the wave number, while the expansion coefficients of the pressure perturbations have an opposite property. In this case, the eigenvalues can be found precisely. As a result, the monotone loss of stability in the Kolmogorov flow can be substantiated by a method other than those available in the literature.     

Stability of oscillatory two-phase Couette flow

The authors investigate the stability of two-phase Couette flowof different liquids bounded between plane parallel plates.One of the plates has a time-dependent velocity in its own plane,which is composed of a constant steady part and a time-harmoniccomponent. In the absence of time-harmonic modulations, theflow can be unstable to an interfacial instability if the viscositiesare different, and the more viscous fluid occupies the thinnerof the two layers. Using Floquet theory, it is shown analyticallyin the limit of long waves that time-periodic modulations inthe basic flow can have a significant influence on flow stability.In particular, flows which are otherwise unstable for extensiveranges of viscosity ratios can be stabilized completely by theinclusion of background modulations, a finding that can haveuseful consequences in many practical applications.     

Stability of flow past cylinder cascade

The global stability analysis of the two–dimensional incompressible unsteady flow past a circular clinder cascade is studied using linear stability analysis. A new numerical technique is used to find the value of critical Reynolds number that causes the flow to undergo Hopf bifurcation. An attempt has also been made to study the blockage effect on critical Reynolds number and associated Strouhal number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of invariant measures and continuity of the KdV flow

We prove continuity of the KdV flow on spaces of probability measures with respect to some Wasserstein metrics on H^s, s > 0 and obtain, as a consequence, stability of the invariant measure built by Bourgain in [2]. The main reference is paper [7], joint work with A.S. de Suzzoni.     

Stability of thermocapillary-driven flow in rectangular cavities

The flow in an infinitely extended cavity with rectangular cross section is considered. The upper boundary is a free surface where thermocapillary forces are driving a circulation within the cavity (Marangoni effect). If the driving force (temperature difference) exceeds a critical threshold, the two dimensional flow becomes unstable towards an infinitesimal three-dimensional perturbation. The critical mode depends on the aspect ratio and may be steady or oscillatory. Results are presented for vanishing Prandtl number and varying aspect ratios of the cavity. These are compared to the classical lid-driven cavity problem and the mechanism leading to instability is discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis of asset flow differential equations

I study the stability analysis of the solutions for the dynamical system of nonlinear asset flow differential equations (AFDEs) in three versions. I show that the previous two versions are not structurally stable mathematically because there are infinitely many critical points. It is important to reformulate a problem in order to eliminate any hypersensitivity in the mathematical model. I find that there is no critical point in the new version unless the chronic discount over the past finite time interval is zero.     

Stability of viscous flow between rotating cylinders

Zusammenfassung Ein Beweis des Rayleighschen Kriteriums der Stabilität einer zähen Strömung zwischen zwei unendlichen, coaxialen, rotierenden Zylindern, der auf einer einfachen Abänderung des Energieintegrals für die Perturbationsgeschwindigkeit begründet ist, wird dargestellt.