Note on static and dynamic instability of a non-uniform beam

The instability of a non-uniform beam analyzed by Lee and Reissner from a static point of view is reanalyzed by a dynamic approach. The beam is loaded by a parallel or tangential compressive force respectively. As expected the load causing static instability, obtained by both approaches, is exactly the same. A peculiar behaviour of the frequency for both cases of loading is revealed by the dynamic method. The tangential load causing dynamic instability is obtained.

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Zusammenfassung Die Instabilität eines Stabes mit konstantem Querschnitt, welche durch Lee und Reissner vom statischen Standpunkt aus untersucht worden ist, wird einer dynamischen Analyse unterzogen. Der Stab steht unter einer axialen bzw. tangentialen Drucklast. Wie erwartet, ist die Belastung, welche statische Instabilität erzeugt, bei beiden Untersuchungsmethoden dieselbe. Ein besonderes Verhalten der Frequenz für beide Lastfälle wird durch die dynamische Methode aufgezeigt, und es wird die tangentiale Last erhalten, welche dynamische Instabilität erzeugt.

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Preparation of this note has been supported by Sherutei Teuffa, Ltd., Haifa, Israel.     

Numerical studies of laminar flow in ducts and pipes

In part I a method is introduced for the numerical solution of swirling axisymmetric laminar flow situations. Consideration is given to finite difference formulation, boundary condition implementation, iterative solution strategies and pressure calculation. Part II is devoted to the application of the method to non-swirling flow problems. Some of these problems are well covered in the literature whilst others are new. Swirling flows are considered in Part III. In particular, details of separation and vortex breakdown induced by swirl are discussed. All results are supplemented by a selection of contour plots to indicate the nature of the flows predicted.     

Note on an integral equation occurring in stagnation point flow

Zusammenfassung Die Differentialgleichung der Staupunktströmung wird nach einer Methode vonWeyl behandelt, und es wird gezeigt, dass das Weylsche Verfahren für elektronische Rechenautomaten besonders geeignet ist.     

Unsteady,laminar, incompressible flow through rectangular ducts

Résumé L'étude porte sur la théorie de lécoulement non permanent d'un fluide visqueux incompressible dans de canaux rectangulaires d'allongement divers, sous l'influence d'un gradient de pression arbitraire, dépendant du temps. Des solutions ont été obtenues dans 4 cas particuliers: 1. gradient de pression impulsif, 2. gradient de pression constant et établi brusquement, 3. gradient de pression en fonction harmonique du temps, 4. gradient de pression à une composante constante et une composante harmonique. On donne les répartitions de vitesse, les coefficients de frottement et la dissipation d'énergie par unité de longueur.

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This paper summarizes the second portion of the Ph. D. Thesis [1]²) byC. Fan.     

Efficient solver for swirling flow problems in ducts with variable radius

The paper presents preliminary results of a work in progress addressing the hydrodynamic stability of swirling flows problems in ducts with variable radius which imply mathematical modeling, dynamic and stability investigations. The proposed quasi-analytical method aims to obtain the velocity profiles with a low order approximation method of which the computation costs were neglijable and regain the central stagnation zone developed in the fluid. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Extreme instability of the horocycle flow

Let g be aC ³ negatively curved Riemannian metric on a compact connected orientable surfaceS. LetB be the collection of all metrics resulting from sufficiently small conformal changes of the metricg. (1) Then there is a constantA > 0 such that ifB then the \(\bar d\) distance between the horocycle flow? _t (Margulis parametrization) of (S, ?) and the rescaled horocycle flowh _ct of (S, g) is at leastA (?c > 0). No other dynamical system is known to have such extreme instability. (2) Fix ε > 0. Then there is anN > 0 so that if we are given samples {ξ} ₀ ^N {η} ₀ ^N which arose from the horocycle flows corresponding to two of the metrics?, g ∈B, then either the two samples are \(\bar d\) farther thanA/2 apart or the two surfaces are closer than ε. This holds even if these samples are slightly inaccurate.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

Numerical simulation of magnetohydrodynamic buoyancy-induced flow in a non-isothermally heated square enclosure

Numerical simulation of magnetohydrodynamic (MHD) buoyancy-induced heat transfer and fluid flow has been analyzed in a non-isothermally heated square enclosure using finite volume method. The bottom wall of enclosure were heated and cooled with a sinusoidal function and top wall was cooled isothermally. Vertical walls of the enclosure were adiabatic. Effects of Rayleigh number (Ra = 10⁴, 10⁵ and 10⁶), Hartman number (Ha = 0, 50 and 100) and amplitude of sinusoidal function (n = 0.25, 0.5 and 1) on temperature and flow fields were analyzed. It was observed that heat transfer was decreased with increasing Hartmann number and decreasing value of amplitude of sinusoidal function.     

The Design of axisymmetric ducts for incompressible flow with vorticity.

In this paper a numerical algorithm is described for solving the boundary value problem associated with axisymmetric, inviscid, incompressible, rotational (and irrotational) flow in order to obtain duct wall shapes from prescribed wall velocity distributions. The governing equations are formulated in terms of the stream function and the function as independent variables where for irrotational flow can be recognized as the velocity potential function, for rotational flow ceases being the velocity potential function but does remain orthogonal to the stream lines. A numerical method based on finite differences on a uniform mesh is employed. The technique described is capable of tackling the so–called inverse problem where the velocity wall distributions are prescribed from which the duct wall shape is calculated, as well as the direct problem where the velocity distribution on the duct walls are calculated from prescribed duct wall shapes. The two different cases as outlined in this paper are in fact boundary value problems with Neumann and Dirichlet boundary conditions respectively. Even though both approaches are discussed, only numerical results for the case of the Dirichlet boundary conditions are given. A downstream condition is prescribed such that cylindrical flow, that is flow which is independent of the axial coordinate, exists. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of fluid flow and heat transfer in coolant ducts of friction clutches

This paper deals with the numerical simulation of the fluid flow in coolant ducts as wall as the calculation of temperature field in fluid and solid of a friction clutch. The simulations are carried out by means of commercial software based on a finite volume method in a relative framework, where the fluid flow is not time dependent. The results in dependence of rotation speed and slip velocity show interesting secondary flow in the ducts. Different flow pattern have distinct influence on the temperature field and the heat transfer. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear instability of hypersonic flow over a cone

This study investigates the nonlinear stability of hypersonicviscous flow over a sharp slender cone. The attached shock andthe effects of curvature are taken into account. Asymptoticmethods are used for large Reynolds number and large Mach numberto examine the viscous modes of instability, which may be describedby a triple-deck structure. A weakly nonlinear analysis is carriedout allowing an equation for the amplitude of disturbances tobe derived. The coefficients of the terms in the amplitude equationare evaluated for axisymmetric and non-axisymmetric disturbances.Thus, the effects of the shock and curvature on the nonlinearstability of the flow may be deduced.     

Two-fluid spatial instability with velocity defect in the basic flow

The initial state of liquid atomization by a fast gas stream is considered by viscous linear spatial stability analysis for parallel two-fluid flow. The unbounded basic velocity profile is characterized by boundary layers near the interface and different asymptotic velocities. By computing generalized spatial branches we identify conditions for absolute instability when a velocity deficit is introduced to account for the conditions near the nozzle. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydrodynamical instability initiation in two-phase stratified flow using Spectral Method

In this article, the hydrodynamical instability initiation criterion in two-phase stratified flow in a horizontal duct is examined. The nonlinear two mass and two momentum conservation equations are used for numerical simulation using the two-phase two-fluid model. The model is solved using the Finite Volume and Spectral Methods, respectively. This paper is the first to utilize the Spectral Method for the simulation of two-phase flow problems. Using the Spectral Method, we show that the numerical error and CPU time decreases noticeably relative to the Finite Volume Method. The well established Kelvin–Helmholtz (K–H) instability is selected for the test case and comparison. The results taken from each set of computer codes developed in this paper are highly compatible with the theoretical and experimental results of previous researchers who used alternative numerical methods. The results obtained from the Spectral Method in comparison with the results of other well known codes exhibit greater consistency with prior analytical results, but with much smaller computer calculation time. The step taken in the present study shows a positive progress in two-phase two-fluid model numerical solution with hydrostatic assumption. It is recommended the research to be continued with two-phase two-fluid model but with hydrodynamical assumption.     

The instability of the flow in a suddenly blocked pipe

This paper considers the decay of Poiseuille flow within a suddenlyblocked pipe. For small to moderate times the flow is shownto consist of an inviscid core flow coupled with a boundarylayer at the pipe wall. A small-time asymptotic solution isdeveloped and it is shown that this solution is valid for timesup to the point at which the boundary layer fills the wholepipe. A small-time composite solution is used to initiate anumerical marching procedure which overcomes the small-timesingularity that arises in the flow and so allows us to describethe ultimate decay of the flow within a blocked pipe. The stabilityof this flow is then considered using both a quasi-steady approximationand a transient-growth analysis based upon marching solutionsof the linearized Navier–Stokes equations. Our transientstability analysis predicts a critical Reynolds number, fortransition to turbulence, in the range 970 < Re < 1370.     

Helical modes of instability in swirling flow between concentric cylinders

Zusammenfassung Die Stabilität einer reibungsfreien und schraubenartigen Strömung zwischen zwei konzentrischen Zylindern wird untersucht. Schranken für die Anfachungsgrösse und Fortpflanzungsgeschwindigkeit von nicht-rotationssymmetrischen Störungen sind abgeleitet und genaue Werte dieser Grössen für die Spezialfälle von Couette-Strömung zwischen einem ruhenden äusseren und einem drehenden inneren Zylinder mit zusätzlicher Axial-Geschwindigkeitskomponente, die lienare oder parabolische Funktionen der Radialrichtung sind, werden erhalten. Es ergibt sich, dass der Zusatz einer linear abhängenden axialen Geschwindigkeit destabilisierend wirkt, weil der Zusatz einer parabolischen Komponente stabilisierend ist. Die Maximalanfachungsgrössen befinden sich bei schraubenförmigen Störungen mit endlicher Steigung.     

Stability of plane parallel shear flow in a rotating layer heated from below

The discussion of stability of plane parallel shear flow in an infinite rotating layer heated from below requires a mathematical analysis of this problem in dependence on four parameters. These are the Reynolds- and Rayleigh-number, controlling the strength of the shear flow and the heating power, respectively, the Prandtl-number, which measures the relative influence of viscosity and thermal conductivity, and the rotation rate of the layer. After discussing some physical background, possible applications and laboratory experiments two major problems are addressed: i) To find out the cases where unconditional (global) stability up to criticality takes place. In these situations theory makes the clearest predictions and coincidence between experiments and mathematical theory can be expected. ii) To prove that the (monotonic) energy-stability limit is assumed by 2-dimensional (with respect to the spatial variables) perturbations. The solution of this variational problem shows that in certain situations the critical perturbations are 2-dimensional. In these situations, at least, the stability problem is completely solved. Received May 21, 1996 / Accepted December 17, 1996     

Note on heat transfer in laminar,fully developed pipe flow with axial conduction

Zusammenfassung Der Wärmeübergang bei laminarer Rohrströmung mit konstanter Wandtemperatur wurde analytisch untersucht. Das Resultat ist eine Darstellung der Nusseltschen Zahl als Funktion von der Pecletschen Zahl.