The rotation class of a flow

Generalizing a construction of A. Weil, we introduce a topological invariant for flows on compact, connected, finite dimensional, Abelian, topological groups. We calculate this invariant for some examples.     

Interaction between internal and external friction in rotation of vane with viscous filling

Rotation of the four-blade axisymmetric vane around the fixed point is considered in the constant flow of medium. The vane contains the cavity that is filled by viscous filling. Aerodynamic force acts upon the blades of the vane. The quasi-steady approach is used for modeling an aerodynamic load. A case for blades with low fineness is considered. Nonlinear approximation of the aerodynamic fineness function is used. Internal friction is modeled as the linear function between the angular speed of the vane and the vortex vector of the filling. This is a mechanical system with variable dissipation. Corresponding dynamic system is simulated for different values of the density of external medium and the coefficient of the internal friction. Two types of the vane configuration (oblate and oblong) are considered. Interaction between the external aerodynamic load and the internal friction is observed. Areas are found, where the steady axial rotation is not globally attracting.     

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 a smooth flow in a Laval nozzle

A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle. The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.     

Stability of a flame front in a divergent flow

We study the evolution of perturbations on the surface of a stationary plane flame front in a divergent flow of a combustible mixture incident on a plane wall perpendicular to the flow. The flow and its perturbations are assumed to be two-dimensional; i.e., the velocity has two Cartesian components. It is also assumed that the front velocity relative to the gas is small; therefore, the fluid can be considered incompressible on both sides of the front; in addition, it is assumed that in the presence of perturbations the front velocity relative to the gas ahead of it is a linear function of the front curvature. It is shown that due to the dependence (in the unperturbed flow) of the tangential component of the gas velocity on the combustion front on the coordinate along the front, the amplitude of the flame front perturbation does not increase infinitely with time, but the initial growth of perturbations stops and then begins to decline. We evaluate the coefficient of the maximum growth of perturbations, which may be large, depending on the problem parameters. It is taken into account that the characteristic spatial scale of the initial perturbations may be much greater than the wavelengths of the most rapidly growing perturbations, whose length is comparable with the flame front thickness. The maximum growth of perturbations is estimated as a function of the characteristic spatial scale of the initial perturbations.     

Stability of a cylindrical shell in a stratified flow

Analysis of the Kelvin stability of a horizontal plane flow of two fluid layers of different densities and one moving relative to the other is extended to the case of longitudinal coaxial flow of a two-layered fluid within a circular cylindrical shell. It is shown that the loss of stability of the whole system sets in for low velocities of the layer motion, one relative to the other. A comparison is made with classical flutter for which the flow is not stratified.     

Stability properties of a flow process in graphs

We study a flow process in infinite graphs where vertices with large resources tend to attract resources from neighbors. The initial resources are random. An interesting question is whether in each finite region all motion stops after a finite time. Under certain assumptions, we prove that this is true. For some other cases, we prove a weaker stability result. We pay attention mostly to the case of Z², but several results can be easily generalized to Z^d.     

Stability of the flow behind a pressure pulse in a tube

If a long tube with a slight overpressure is suddenly opened at one end, a pressure signal moves into the tube and a flow towards the open end is generated. The stability of this flow is investigated by solving the Orr-Sommerfeld equation for quasi-steady, onedimensional velocity profiles. Amplified disturbances are found for the boundary-layer type of flow close to the open end for a Reynolds number Re=u _mR/₀>17400 whereu _m=p ₀/ ₀ a ₀ is the maximum speed attained by the outflowing gas. A comparison of the time scales and of the amplification suggests that laminar flow can exist up to Re=100000.Experiments indicate that small disturbances due to the experiment can trigger transition at much lower Reynolds numbers.

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Zusammenfassung Ein langes Rohr mit einem schwachen Überdruck wird plötzlich an einem Ende geöffnet. Dadurch entsteht eine Druckwelle, die ins Rohr hinein läuft. Zwischen Druckwelle und Rohrende strömt das Fluid gegen das Rohrende. Die Stabilität dieser Strömung wird untersucht, indem die Orr-Sommerfeld Gleichung gelöst wird für quasistationäre und eindimensionale Geschwindigkeitsprofile. Die grenzschichtähnlichen Profile nahe dem Rohrende werden zuerst instabil für eine Reynoldszahl Re=u _mR/₀>17400 Dabei istu _m=p ₀/ ₀ a ₀ die maximale Geschwindigkeit, die das ausströmende Gas ereicht. Ein Vergleich von Zeitmaßstäben und Anfachung läßt erwarten, daß laminare Strömung bis gegen Re=100000 existieren kann.Versuche zeigen, daß kleinere Fehler im Versuchsaufbau den Umschlag bei viel kleineren Reynoldszahlen auslösen können.

     

Stability of transonic shocks in supersonic flow past a wedge

In this paper we study the stability of transonic shocks in steady supersonic flow past a wedge. We take the potential flow equation as the mathematical model to describe the compressible flow. It is known that in generic case such a problem admits two possible location of shock, connecting the flow ahead it and behind it. They can be distinguished as supersonic-supersonic shock and supersonic-subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine-Hugoniot conditions and entropy condition. In this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled.     

Stability of a steady flow guided by flexible walls

Flexible channels conveying fluid may loose stability by divergence or flutter, thus leading to undesirable dynamic behaviour. A lot of investigations have already been done on this subject, each involving structural models expressing a specific type of stiffness. The aim of this contribution is to investigate and compare systematically the influence of various types of structural stiffness on the stability behaviour of the coupled problem. Therefore, a simple, yet general model is built. The fluid is bounded by a rigid and an elastical supported, flexible wall, which exhibits bending and extensional stiffness and considers effects of pre-load. The influence of the various characteristics on the stability of the steady state is discussed and a perturbation approach is used to investigate the influence of small nonlinearities. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of a two-dimensional stationary rotating flow in an exterior cylinder

We investigate the stability of an exact stationary flow in an exterior cylinder. The horizontal velocity is the two-dimensional rotating flow in an exterior disk with a critical spatial decay, for which the L² stability is known under smallness conditions. We prove its stability property for three-dimensional perturbations although the Hardy type inequalities are absent as in the two-dimensional case. The proof uses a large time estimate for the linearized equations exhibiting different behaviors in the Fourier modes, namely, the standard L²-$L^q$ decay of the two-dimensional mode and an exponential decay of the three-dimensional modes.     

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 of transonic shock for supersonic flow past a wedge

When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C¹. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability of non-isothermal phase transitions in a steady van der Waals flow

This paper is concerned with the multidimensional stability of non-isothermal subsonic phase transitions in a steady supersonic flow of van der Waals type. For the sake of seeking physical admissible planar waves, the viscosity–capillarity criterion (Slemrod in Arch Ration Mech Anal 81(4):301–315, 1983) is chosen to be the admissible criterion. By showing the Lopatinski determinant being non-zero, we prove that subsonic phase transitions are uniformly stable in the sense of Majda (Mem Am Math Soc 41(275):1–95, 1983) under both one-dimensional and multidimensional perturbations.     

Mean curvature flow through singularities for surfaces of rotation

In this paper, we study generalized “viscosity” solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.     

Steady flow generated by the differential rotation of a porous disk of finite thickness

When a body of fluid bounded by a porous disk of finite thickness is disturbed from a state of rigid rotation by an enhanced (or reduced) angular velocity of the disk, a few authors followed Darcys model and observed that the centrifugal pumping occurs through the entire porous layer regarded as a convection zone. The shear stress can develop only at the edge of the porous layer. We use a porous disk of high permeability that allows the fluid in the porous disk to deform in response to the changing angular velocity. Based on the Birkmans model, we solve for the steady non-linear flow and observe that there arises (i) a convection zone of nearly uniform angular velocity at the boundary (within the porous layer) and (ii) a transition zone adjacent to the convection zone which provides a smooth transition to the interior. This makes the model relevant to some astrophysical situations as described by some authors [1, 3]. The two point boundary value problem is solved subject to the boundary conditions, the far field conditions, and the matching conditions at the fluid-porous medium interface. The solution is obtained using a numerical procedure known as the method of Adjoints.Received: June 13, 2002; revised: July 7, 2003     

Steady flow generated by the differential rotation of a porous disk of finite thickness

When a body of fluid bounded by a porous disk of finite thickness is disturbed from a state of rigid rotation by an enhanced (or reduced) angular velocity of the disk, a few authors followed Darcys model and observed that the centrifugal pumping occurs through the entire porous layer regarded as a convection zone. The shear stress can develop only at the edge of the porous layer. We use a porous disk of high permeability that allows the fluid in the porous disk to deform in response to the changing angular velocity. Based on the Birkmans model, we solve for the steady non-linear flow and observe that there arises (i) a convection zone of nearly uniform angular velocity at the boundary (within the porous layer) and (ii) a transition zone adjacent to the convection zone which provides a smooth transition to the interior. This makes the model relevant to some astrophysical situations as described by some authors [1, 3]. The two point boundary value problem is solved subject to the boundary conditions, the far field conditions, and the matching conditions at the fluid-porous medium interface. The solution is obtained using a numerical procedure known as the method of Adjoints.     

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 $ (0,ar + br^{{ - 1}} ,0,0,0,\alpha + \beta \log r). $ (0,ar + br^{{ - 1}} ,0,0,0,\alpha + \beta \log r). It is shown that this equilibrium is Ljapounov stable under small perturbations in $ \mathcal{L}^{2} (\Gamma ), $ \mathcal{L}^{2} (\Gamma ), where $ \Gamma = \{ (r,\varphi ,z)/r_{1} < r < r_{2} ,\varphi \in [0,2\pi ],z \in \mathbb{R}\} , $ \Gamma = \{ (r,\varphi ,z)/r_{1} < r < r_{2} ,\varphi \in [0,2\pi ],z \in \mathbb{R}\} , 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.     

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.