Parametric modulation in the Taylor-Couette ferrofluid flow

A parametric instability of the Taylor-Couette ferrofluid flow excited by a periodically oscillating magnetic field, has been investigated numerically. The Floquet analysis has been employed. It has been found that the modulation of the applied magnetic field affects the stability of the basic flow. The instability response has been found to be synchronous with respect to the frequency of periodically oscillating magnetic field.     

A purely elastic transition in Taylor-Couette flow

Experimental evidence of a non-inertial, cellular instability in the Taylor-Couette flow of a viscoelastic fluid is presented. A linear stability analysis for an Oldroyd-B fluid, which is successful in describing many features of the experimental fluid, predicts the critical Deborah number,De _c , at which the instability is observed. The dependence ofDe _c on the value of the dimensionless gap between the cylinders is also determined.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Components of wall shear rate in wavy Taylor-Couette flow

The time-resolved axial and azimuthal components of the wall shear rate were measured as function of Reynolds number by a three-segment electrodiffusion probe flush mounted in the inner wall of the outer fixed cylinder. The geometry was characterized by a radius ratio of 0.8 and an aspect ratio of 44. The axial distribution of the wall shear rate components was obtained by sweeping the vortices along the probe using a slow axial flow. The wavelength and phase celerity of azimuthal waves, axial wavelength of vortices and their drifting velocity were calculated from the limiting diffusion currents measured by three simple electrodiffusion probes.     

Bifurcation of the stationary Ekman flow into a stable periodic flow

The two cases of stationary Ekman boundary layer flow of an incompressible fluid near i) a plane boundary and ii) a free surface with constant shear are considered. It is proven that a stable secondary flow in the form of traveling waves bifurcates from the stationary flow at a certain Reynolds number, and that the stationary flow is unstable above this number. The values of the critical Reynolds number and of the numbers that characterize the traveling wave are computed and compared with experimental values.     

High-intensity turbulence measurements in a Taylor-Couette flow reactor

Turbulence-intensity measurements were made in a Taylor-Couette flow reactor consisting of two counter-rotating concentric cylinders designed for the purpose of studying turbulent premixed-flame propagation. In the annulus separating the two cylinders, a nearly homogeneous turbulent flow is generated. The intensities of turbulent velocity fluctuations in the annulus in both axial and circumferential directions were measured by using laser-Doppler velocimetry for a wide range of cylinder rotation rates, corresponding to low through high (120 cm/s) intensities relative to typical laminar flame speeds for lean methane-air mixtures. The experimental measurements indicate a linear relation between turbulence intensities and average cylinder surface speed and demonstrate the usefulness of the Taylor-Couette apparatus for studies of premixed-flame propagation in high-intensity turbulent flow.     

Turbulent Taylor-Couette vortex flow between large radius ratio concentric cylinders

Turbulent Taylor vortices between two concentric cylinders have been studied at a very high radius ratio of 0.985, equivalent to that found in relatively small underwater thruster units (typically with gaps of 2 mm). In order to study the flow at this radius ratio, a 1.42-m diameter experimental apparatus (with a rotating inner cylinder and a stationary outer cylinder) was constructed possessing a gap of 10 mm. Consequently, air bubbles could be visualised translating in water. A method was developed for identifying Taylor vortex properties from filtered digital images of the air bubbles and summing intensities to produce bubble density distributions. Whereas individual instantaneous images can be misleading, averaged bubble density distributions make it possible to identify vortex separation sizes and the positions of vortex outflow boundaries.     

Unsteady flow in the Ekman layer of an elastico-viscous liquid

The transient flow in the Ekman layer of an elastico-viscous liquid near a flat plate is discussed. Initially the fluid and the plate were rotating together and the plate then suddently starts moving with a uniform velocity in its own plane relative to the rotating frame of reference. It is shown that the ultimate steady state is reached through decay of inertial oscillations whose frequency decreases with increase in the elastic parameter.     

A numerical study on nonlinear dynamics of three-dimensional time-depended viscoelastic Taylor-Couette flow

In this paper, three-dimensional viscoelastic Taylor-Couette instability between concentric rotating cylinders is studied numerically. The aim is to investigate and provide additional insight about the formation of time-dependent secondary flows in viscoelastic fluids between rotating cylinders. Here, the Giesekus model is used as the constitutive equation. The governing equations are solved using the finite volume method (FVM) and the PISO algorithm is employed for pressure correction. The effects of elasticity number, viscosity ratio, and mobility factor on various instability modes (especially high order ones) are investigated numerically and the origin of Taylor-Couette instability in Giesekus fluids is studied using the order of magnitude technique. The created instability is simulated for large values of fluid elasticity and high orders of nonlinearity. Also, the effect of elastic properties of fluid on the time-dependent secondary flows such as wave family and traveling wave and also on the critical conditions are studied in detail.     

Ekman vortices and the centrifugal instability in counter-rotating cylindrical Couette flow

The finite length of a Taylor–Couette cell introduces endwall effects that interact with the centrifugal instability. We investigate the interaction between the endwall Ekman boundary layers and the vortical structures in a finite-length cavity with counter-rotating cylinders via direct numerical simulation using a three-dimensional spectral method. To analyze the nature of the interaction between the vortices and the endwall layers we consider four endwall boundary conditions: fixed endwalls, endwalls rotating with the outer cylinder, endwalls rotating with the inner cylinder, and stress-free endwalls. The vortical structure of the flow depends on the endwall conditions. The waviness of the vortices is suppressed only very near the endwall, primarily due to zero axial velocity at the endwall rather than viscous effects. In spite of their waviness and random behavior, the vortices generally stay inside of the v=0 isosurface by adjusting quickly to the radial transport of azimuthal momentum. The thickness and strength of the Ekman layer at the endwall match with that predicted from a simple theoretical approach.     

New experimental methods for turbulent spots and turbulent spirals in the Taylor-Couette flow

 The flow between concentrically counter-rotating circular cylinders is investigated experimentally with respect to the appearance of turbulent spots, combining two new methods, based on digital image processing. The simultaneous visualization of the transition to turbulence in the whole flow field in the gap between the two cylinders leads to a qualitative understanding of the phenomenon of turbulent spots. Quantitative results about spiral turbulence are obtained from measurements based on a time-resolved technique introducing a special method of image processing for long sequences of video frames. Variations of the gap width between the cylinders and the investigation of different boundary conditions at the end plates of the rotating cylinders allow conclusions concerning the importance of locally defined parameters of the flow field. Received: 4 December 1995/Accepted: 15 May 1997     

Experimental study on radial temperature gradient effect of a Taylor-Couette flow with axial wall slits

The flow between two concentric cylinders with the inner cylinder rotating and an imposed radial temperature gradient was studied using a digital particle image velocimetry method. The flow transition process under both a positive and negative temperature gradient with four different models of a stationary outer cylinder without and with differing numbers of slits (6, 9 and 18) was studied. The results showed that the buoyant force due to the temperature gradient clearly generated a helical flow when the rotating Reynolds number was small. For the plain and 6-slit models, the transition to a turbulent Taylor vortex flow was not affected by the temperature gradient considered in this study. In addition, the transition process of a larger number of slits (9-, 18-slit models) was accelerated due to the slit wall. As the temperature gradient became larger, the critical Reynolds number of the transition process decreased.     

Poiseuille flow and drop circulation in microchannels

Microfluidics aims to control precisely the transport of fluids and suspended particles or drops. Two characteristics of such transport in rectangular microchannels are addressed here, as a function of the cross-sectional aspect ratio. First, we highlight a convenient expression for the ratio of the centerline to bulk flow velocities, which is relevant for controlling the flow of suspended or flow-focused objects. Then, using the theory of Nadim and Stone, the droplet circulation fountain-flow pattern in such channels is evaluated explicitly, and implications for interfacial mobility measurements are discussed. For example, when the interface is retarded, part of the fountain reverses direction, thus alleviating stagnation, promoting mixing, and reducing interfacial concentration gradients.     

Modulated Taylor-Couette Flow: Onset of Spiral Modes

The linear stability of the flow between concentric cylinders, with the inner cylinder rotating at a constant angular velocity and the outer cylinder with an angular velocity varying harmonically about a zero mean, is addressed. The bifurcations of the base state are analyzed using Floquet theory, paying particular attention to non-axisymmetric bifurcations which are dominant in significant regions of parameter space. In these regions the spiral modes of the unforced system become parametrically excited and dominant. This is typical behavior of parametrically forced extended systems, where some modes are stabilized, but others are simultaneously excited. The flow structure of the bifurcated states are examined in detail, paying particular attention to the dynamic implications of their symmetries, and in particular how and when subsequent period doublings are inhibited. Received 5 November 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work was supported by NSF Grants INT-9732637 and CTS-9908599 (U.S.A.) and MCYT Grants PB97-0685 and BFM2001-2350 (Spain). Communicated by H.J.S. Fernando     

The asymmetric Ekman decay of cyclonic and anticyclonic vortices

Recent studies have shown differences in the behaviour of cyclonic and anticyclonic quasi-two-dimensional vortices in laboratory experiments in a rotating fluid. In this paper, the role of dissipative effects due to bottom topography is investigated as a possible cause for the asymmetry in the spin-down of both types of vortices. The basic mechanism of Ekman friction in 2D mathematical models is the presence of a linear damping term in the vorticity equation, which produces the flow decay. Here, an extended 2D formulation including nonlinear Ekman corrections is considered. The aim is to show that nonlinear Ekman effects are responsible for the different decay of cyclonic and anticyclonic vortices, while the conventional formulation (only containing the linear friction term) predicts a symmetric decay for both cases.In order to illustrate the role of nonlinear Ekman effects, axisymmetric vortices are simulated numerically. The relatively simple structure of such vortices allows a better understanding of their evolution. The main difference in the spin-down process of cyclones and anticyclones is the decay rate, which is faster for cyclonic motion. Furthermore, it is shown that the basic mechanism for such a difference is the outward advection of fluid in cyclones and inward in anticyclones, both effects due to Ekman pumping and suction, respectively. The results derived here intend to provide a physical interpretation which could be applied for more general, non-axisymmetric structures.     

Normal flow boundary conditions in 3D circulation models

The problem of enforcing normal transport conditions on 3D velocity fields is considered in the context of ‘wave equation’ finite element models. A procedure for strong enforcement of the transport constraint is given. The procedure is identical for Neumann (transport known) and Dirichlet (pressure known) problems, which are treated reversibly. All local mass and force balance relations are retained in the FEM system. A global mass conservation property is proven for the general 3D, discrete-time case. Examples demonstrate the quality of the solutions and the practicality of the approach. © 1997 John Wiley & Sons, Ltd.     

Flow visualization of the elastic Taylor-Couette instability in Boger fluids

Flow visualization is performed on an elastically-dominated instability in several similar Boger fluids in Taylor-Couette flow. The onset and evolution of secondary flow are observed over a range of shear rates using reflective mica platelet seeding. Sequences of ambiently and sheet-illuminated images were digitally processed. Rotation of the inner cylinder was ramped from rest to its final value over a time on the order of a polymer relaxation time. Dilute solutions of high molecular weight polyisobutylene in oligomeric polybutene manifest a flow transition at a Deborah number, De _s = _s 1.5 with a Taylor number of 0.00022 in a cell with dimensionless gap ratio = 0.0963. At this transition, simple azimuthal shearing is replaced by steady, roughly square, axisymmetric counter-rotating vortices grossly similar to the well-known Taylor vortex flow that is observed at De _s = 0, Ta = 3612. At De _s = 3.75, Ta = 0.0014, an axisymmetric oscillatory secondary flow develops initially but is replaced by the steady vortices. At De _s = 7.5, Ta = 0.0054, the oscillatory and vortex flow coexist and possess an irregular cellular cross-section. A wide span of growth rates is observed: the ratio of onset to polymer relaxation time ranges from 170000 at De _s = 1.5 to O(10) at De _s > 5. The role of inertia was explored through changing the solvent viscosity. A transition similar to the one that occurs at De _s = 3.75, Ta = 0.0014, from the base azimuthal shearing flow to axisymmetric vortices, was also observed with a much lower viscosity fluid at De _s = 3.3, Ta = 74.     

Stability of Large Ekman Boundary Layers in Rotating Fluids

The aim of this paper is to investigate the stability of Ekman boundary layers for rotating fluids when the Ekman number and the Rossby number go to zero. More precisely, we prove that spectral stability implies linear and nonlinear stabilities of approximate solutions. In particular, we replace the smallness condition obtained with energy methods in [5] by a weaker spectral condition which is sharp.     

Consideration of longitudinal diffusion in flow within a channel

The stationary problem of convective diffusion in a channel with absorbent walls is considered. It is assumed that a Poiseuille flow exists. Two methods are employed in the solution, the method of separation of variables, and the method of expansion in eigenfunctions of the corresponding problem with piston profile (expansion method). It is established by comparison with independently obtained solutions for high Peclet number that for the first eigenfunctions and eigenvalues the expansion method gives satisfactory results over the entire Peclet-number range. For approximate calculation of subsequent eigenfunctions and eigenvalues a modification of the smooth asymptotic expansion method is used. The results are used to calculate matter flow density on the wall, to evaluate the length of the entrance region, and to obtain an analytical expression for the limiting Nusselt number in terms of the Peclet number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 63–73, November–December, 1973.     

Creeping flow past and within a permeable spheroid

Flow past and within an isolated permeable spheroid directed along its axis of symmetry is studied. The flow velocity field is solved using the Stokes creeping flow equations governing the fluid motion outside the spheroid, and the Darcy equation within the spheroid. Expressions for the hydrodynamic resistance experienced by oblate and prolate spheroids are derived and analyzed. The limiting cases of permeable circular disks and elongated rods are examined. It is shown that the spheroid’s resistance varies significantly with its aspect ratio and permeability, expressed via the Brinkman parameter.