The onset of Taylor-like vortices in the flow induced by an impulsively started rotating cylinder

The onset of instability in the flow by an impulsively started rotating cylinder is analyzed under linear theory. It is well-known that at the critical Taylor number T_c=1695 the secondary flow in form of Taylor vortices sets in under the narrow-gap approximation. Here the dimensionless critical time _c to mark the onset of instability for TT_c is presented as a function of the Taylor number T. Available experimental data of water indicate that deviation of the velocity profiles from the primary flow occurs starting from a certain time 4_c. It seems evident that during _c4_c the secondary flow is very weak and the primary state of time-dependent annular Couette flow is maintained.     

Hydrodynamic models as applied to the investigation of magnetic plasma stability

In the present paper magnetohydrodynamic models are employed to investigate the stability of an inhomogeneous magnetic plasma with respect to perturbations in which the electric field may be regarded as a potential field (rot E 0). A hydrodynamic model, actually an extension of the well-known Chew-Goldberg er-Low model [1], is used to investigate motions transverse to a strong magnetic field in a collisionless plasma. The total viscous stress tensor is given; this includes, together with magnetic viscosity, the so-called inertial viscosity.Ordinary two-fluid hydrodynamics is used in the case of strong collisions=. It is shown that the collisional viscosity leads to flute-type instability in the case when, collisions being neglected, the flute mode is stabilized by a finite Larmor radius. A treatment is also given of the case when epithermal high-frequency oscillations (not leading immediately to anomalous diffusion) cause instability in the low-frequency (drift) oscillations in a manner similar to the collisional electron viscosity, leading to anomalous diffusion.Notation f particle distribution function - E electric field component - H₀ magnetic field - density - V particle velocity - e charge - m, M electron and ion mass - _i, _e ion and electron cyclotron frequencies - viscous stress tensor - P pressure - r_i Larmor radius - P pressure tensor - t time - frequency - T temperature - collision frequency - collision time - j current density - _i, _e ion and electron drift frequencies - k_x, k_y, k_z wave-vector components - n₀ particle density - g acceleration due to gravity. The authors are grateful to A. A. Galeev for valuable discussion.     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

The mirror relations and nonlinear viscoelasticity of polymer melts

An attempt is made to incorporate into a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type the two mirror relations of Gleissle, as well as his relation between the steady-state first normal-stress difference and the shear viscosity curve. It is shown that the three relations can hold separately within this constitutive model, but not simultaneously, because they require a different nonlinear strain measure, namelyS ₁₂ () = – a ( – 1) (a = 0 for 1,a = 1 for 1) for the mirroring of the viscosities,S ₁₂ () = – a (–k ²/) (a = 0 for k, a = 1 for k) for the mirroring of the first normal-stress coefficients, and for the third relation. Here denotes the shear strain and erf the error function. Experimental data on melts of a low-density polyethylene, a high-density polyethylene and a polypropylene show that the mirror relations are passable approximations, but that the third relation meets reality surprisingly close if the right value ofk is used.     

Testing viscometric predictions of the internal viscosity model,using dilute viscous theta solutions

A stress-symmetrized internal viscosity (I.V.) model for flexible polymer chains, proposed by Bazua and Williams, is scrutinized for its theoretical predictions of complex viscosity ^* () = – i and non-Newtonian viscosity (), where is frequency and is shear stress. Parameters varied are the number of submolecules,N (i.e., molecular weightM = NM _s ); the hydrodynamic interaction,h ^*; and/f, where andf are the I.V. and friction coefficients of the submolecule. Detailed examination is made of the eigenvalues _p (N, h ^*) and how they can be estimated by various approximations, and property predictions are made for these approximations.Comparisons are made with data from our preceding companion paper, representing intrinsic properties [], [], [] in very viscous theta solutions, so that theoretical foundations of the model are fulfilled. It is found that [ ()] data can be predicted well, but that [ ()] data cannot be matched at high. The latter deficiency is attributed in part to unrealistic predictions of coil deformation in shear.     

An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in concentric cylinder viscometers

An analytical solution is presented for the calculation of the flow field in a concentric cylinder viscometer of non-ideal Bingham-fluids, described by the Worrall-Tuliani rheological model. The obtained shear rate distribution is a function of the a priori unknown rheological parameters. It is shown that by applying an iterative procedure experimental data can be processed in order to obtain the proper shear rate correction and the four rheological parameters of the Worrall-Tuliani model as well as the yield surface radius. A comparison with Krieger's correction method is made. Rheometrical data for dense cohesive sediment suspensions have been reviewed in the light of this new method. For these suspensions velocity profiles over the gap are computed and the shear layer thicknesses were found to be comparable to visual observations. It can be concluded that at low rotation speeds the actually sheared layer is too narrow to fullfill the gap width requirement for granular suspensions and slip appears to be unavoidable, even when the material is sheared within itself. The only way to obtain meaningfull measurements in a concentric cylinder viscometer at low shear rates seems to be by increasing the radii of the viscometer. Some dimensioning criteria are presented.Notation A, B Integration constants - C Dimensionless rotation speed = µ/_y - c = 2µ - d = ₀ ²–2c_y - f() = (–₀)²+2c(–_y) - r Radius - r _b Bob radius - r _c Cup radius - r _y Yield radius - r ₀ Stationary surface radius - r Rotating Stationary radius - Y ₀ Shear rate parameter = /µ Greek letters Shear rate - = (r _y /r _b )²– 1 - µ Bingham viscosity - µ₀ Initial differential viscosity - µ µ₀-µ - Rotation speed - Angular velocity - Shear stress - _b Bob shear stress - _B Bingham stress - _y (True) yield stress - ₀ Stress parameter = _B +µ Y ₀ - _B - _y      

A general model for the effective viscosity of pseudoplastic and dilatant fluids

Summary A new and very general expression is proposed for correlation of data for the effective viscosity of pseudoplastic and dilatant fluids as a function of the shear stress. Most of the models which have been proposed previously are shown to be special cases of this expression. A straightforward procedure is outlined for evaluation of the arbitrary constants.

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Zusammenfassung Eine neue und sehr allgemeine Formel wird für die Korrelation der Werte der effektiven Viskosität von strukturviskosen und dilatanten Flüssigkeiten in Abhängigkeit von der Schubspannung vorgeschlagen. Die meisten schon früher vorgeschlagenen Methoden werden hier als Spezialfälle dieser Gleichung gezeigt. Ein einfaches Verfahren für die Auswertung der willkürlichen Konstanten wird beschrieben.

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Nomenclature b arbitrary constant inSisko model (eq. [5]) - n arbitrary exponent in eq. [1] - x independent variable - y(x) dependent variable - y ₀(x) limiting behavior of dependent variable asx 0 - y(x) limiting behavior of dependent variable asx - z original dependent variable - arbitrary constant inSisko model (eq. [5]) andBird-Sisko model (eq. [6]) - arbitrary exponent in eqs. [2] and [8] - effective viscosity = shear stress/rate of shear - _A effective viscosity at = _A - _B empirical constant in eqs. [2] and [8] - ₀ limiting value of effective viscosity as 0 - ₀() limiting behavior of effective viscosity as 0 - limiting value of effective viscosity as - () limiting behavior of effective viscosity as - rate of shear - arbitrary constant inBird-Sisko model (eq.[6]) - shear stress - _A arbitrary constant in eqs. [2] and [8] - ₀ shear stress at inBingham model - _1/2 shear stress at = ( ₀ + )/2 With 8 figures     

Nonlinear viscoelastic properties and change in entanglement structure of linear polymer

Simultaneous measurements of stress relaxation and differential dynamic modulus were made at 268 K over a time scale of 10 to 10⁴⁵ s for nearly monodisperse polybutadiene (M _w =2.2x10⁵, 1,2-structure 70%, M _e =3600) and also one having coarse cross-linking (M _c =29000). Static shear strain ranged from 0.1 to 2.0. In a long-time region (t> _k ), the relaxation modulus G (; t) could be expressed by the product G (0; t) h (y). The observed h() agreed well with the Doi-Edwards theory without use of IA approximation. Both the cured and uncured samples showed initial drop of the differential storage modulus G (), ; t) followed by gradual recovery, but did not attain the value before shearing G (, ; t) for the uncured sample showed smaller values than that for the cured one in the whole measured time scale at the higher strain, confirming the two origins of nonlinear viscoelasticity of well entangled polymer; induced chain anisotropy and induced decrement in entanglement density. G (, ; t) curves for the cured sample agreed well with the BKZ predictions. But the curves for the uncured sample agreed well with the BKZ prediction only at the time scale of t< _k . BKZ prediction showed significant upward deviations at t> _k . Such the differences are discussed in terms of the two origins.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

Heat transfer in laminar flow in a uniformly porous channel with an applied transverse magnetic field

Summary The steady laminar flow of an incompressible, viscous, and electrically conducting fluid between two parallel porous plates with equal permeability has been discussed by Terrill and Shrestha [6]. In this paper, using the solution of [6] for the velocity field, the heat transfer problems of (i) uniform wall temperature and (ii) uniform heat flux at wall are solved.For small suction Reynolds numbers we find that the Nusselt number, with increasing Reynolds number, increases for case (i) and decreases for (ii).Nomenclature stream function - 2h channel width - x, y distances measured parallel, perpendicular to the channel walls - U velocity of fluid in the x direction at x=0 - V constant velocity of suction at the wall - nondimensional distance, y/h - nondimensional distance, x/h - f() function defined in (1) - density - coefficient of kinematic viscosity - R suction Reynolds number, V h/ - Re channel Reynolds number, 4U h/ - B ₀ magnetic induction - electrical conductivity - M Hartmann number, B ₀ h(/)^1/2 - K constant defined in (3) - A constant defined in (5) - 4R/Re - q local heat flux per unit area at the wall - k thermal conductivity - T temperature of the fluid - X –1/ ln(1–) - C _p specific heat at constant pressure - j current density - Pr Prandtl number, C _p/k - P mass transfer Péclet number, R Pr - Pe mass transfer Péclet number, P/ - T ₀ temperature at x=0 - T _H() temperature in the fully developed region - T _h(X, ) temperature in the entrance region - Y _n () eigenfunctions, uniform wall temperature - _n eigenvalues - e() function defined by (24) - B _n 2/3 _n ² - A _n constants defined by (28) - a _2m constants defined by (30) - F _n () eigenfunctions, uniform wall heat flux - a _n , b _n , c _n , d _n , e _n constants defined by (45) and (48) - S a parameter, U ²/q - h ₁ heat transfer coefficient - T _m mean temperature - Nu Nusselt number - Nu _T Nusselt number, uniform wall temperature - Nu _q Nusselt number, uniform wall heat flux     

Zur analytischen Beschreibung des Füllstoffeinflusses auf das Fließverhalten von Kunststoff-Schmelzen

From literature some representative equations have been compiled describing the influence of filler on the viscosity of polymer melts. By application of these on the experimental results obtained from GF-SAN it was found that the relative viscosity _R , i.e. the ratio of the viscosities of the filled and unfilled melt, shows a pronounced dependence on the shear rate but not on the shear stress. Defining _R with constant and not with constant (as it is usually done), an analytical approach is possible independent of Further the influence of pressure, temperature and filler content on the zero-shear viscosity of filled polymer melts may be expressed by a modified Arrhenius equation.

     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

Some aspects of linear and nonlinear viscoelastic behaviour of polymer melts in shear

In linear viscoelastic investigations the frequency dependence of the phase shift between stress and strain appears to be very characteristic of the molecular structure of the material. This function is also a good approximation of the slope of the double logarithmic plot of the absolute value of the shear modulusG _d vs. the angular frequency. The product (G _d /) sin 2 comes very close to the relaxation spectrumH(), with = 1/, in all physical states of the material.The experimentally observed separability of time and strain effects in nonlinear viscoelasticity of highly viscous isotropic polymer fluids imposes restraints to the form of the constitutive equation. A single integral superposition equation of the Boltzmann type containing the product of a time function and a nonlinear strain function gives good results in describing experimental data in shear as well as in elongation. The molecular structure affects both functions in a different way. A universal definition of the nonlinear tensorial strain measure has not yet been developed. There are some indications that a definition on the basis of the principal stretch ratios may be fruitful.Invited paper, presented at the First Conference of European Rheologists at Graz (Austria), April 14–16, 1982.     

Experimental investigation of the creeping motion of a drop in a vertical tube

New experimental data regarding the motion of a drop along the axis of a vertical tube, filled with another highly viscous liquid, are obtained. The experiments are realised with sufficiently large drops for an internal circulation to develop and also for different pairs of fluids; the preponderant role of the gravity on the drop shape and consequently on its terminal velocity is pointed out. Moreover, by means of a visualization technique, details on the flow both inside and outside the drop are given.List of symbols g gravity acceleration - r distance from the drop center - R equivalent radius of the drop, i.e. the radius of the sphere having the same volume as the drop - R _EQ radius of the equatorial section of the drop - R _T tube radius - L _AX half length of the drop - U ₀ terminal velocity of the drop - P _s Poiseuille number= U _{0 e} /4 g R ² - Fr Fronde number = U ₀ ² _e /2 g R - Re Reynolds number = 2 U ₀ R _e / _e - E _o Eötvös number = 4g R ²/ - deformation parameter = _e U ₀/ - apparent density of the suspended liquid= | _i – _e | - _i viscosity of the suspended liquid - _e viscosity of the suspending liquid - drop-to-tube radius ratio = R/R _T - _EQ equatorial drop-to-tube radius ratio = R EQ/R _T - interfacial tension     

On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

Viscoelastic properties of semidilute poly(methyl methacrylate) solutions

Viscoelastic properties were examined for semidilute solutions of poly(methyl methacrylate) (PMMA) and polystyrene (PS) in chlorinated biphenyl. The number of entanglement per molecule, N, was evaluated from the plateau modulus, G _N . Two time constants, _s and ₁, respectively, characterizing the glass-to-rubber transition and terminal flow regions, were evaluated from the complex modulus and the relaxation modulus. A time constant _k supposedly characterizing the shrink of an extended chain, was evaluated from the relaxation modulus at finite strains. The ratios ₁/ _s and _k / _s were determined solely by N for each polymer species. The ratio ₁/ _s was proportional to N ^4.5 and N ^3.5 for PMMA and PS solutions, respectively. The ratio _k / _s was approximately proportional to N ^2.0 in accord with the prediction of the tube model theory, for either of the polymers. However, the values for PMMA were about four times as large as those for PS. The result is contrary to the expectation from the tube model theory that the viscoelasticity of a polymeric system, with given molecular weight and concentration, is determined if two material constants _s and G _N are known.     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Image processing of hydrogen bubble flow visualization for determination of turbulence statistics and bursting characteristics

A technique is described which employs automated image processing of hydrogen-bubble flow visualization pictures to establish local, instantaneous velocity profile information. Hydrogen bubble flow visualization sequences are recorded using a high-speed video system and then digitized, stored, and evaluated by a VAX 11/780 computer. Employing special smoothing and gradient detection algorithms, individual bubble-lines are computer identified, which allows local velocity profiles to be constructed using time-of-flight techniques. It is demonstrated how this techniques may be used to 1) determine local velocity behavior as a function of position and time, 2) evaluate time-averaged turbulence properties, and 3) correlate probe-type turbulent burst detection techniques with the corresponding visualization data.List of symbols Re Reynolds number based on momentum thickness, u / - t ⁺ nondimensional time tu ² / - T VITA variance averaging time period - u shear velocity = - u local instantaneous streamwise velocity,x-direction - u local fluctuating streamwise velocity,x-direction - u ⁺ nondimensional streamwise velocity, /u - local normal velocity,y-direction - w local spanwise velocity,z-direction - x ⁺ nondimensional coordinate in streamwise direction xu /v - y ⁺ nondimensional coordinate normal to wall, yu /v Greek momentum thickness, - kinematic viscosity - _w wall shear stress This paper was presented at the Ninth Symposium on Turbulence, University of Missouri-Rolla, October 1–3, 1984     

Quantitative analysis of flow visualizations in an unsteady channel flow

Quantitative results concerning the modulation of the ejection and bursting frequency in an unsteady channel flow obtained by flow visualizations are presented and compared with probe measurements. The frequency of the imposed velocity oscillations f covers a large range going from the quasi steady limit to the time mean bursting frequency in the corresponding steady flow. The imposed amplitudes of the velocity oscillations are 13% and 20% of the centerline velocity. The bursting process is identified by the intermittent lift up of the dye injected at the wall. Qualitative analysis of the flow visualizations show that the ejection activity at a given phase of the oscillation cycle is repetitive from one cycle to the other. The modulation amplitude of the ejection frequency f _e is sensitive to the imposed frequency. At low imposed frequency f _e is modulated as the wall shear stress, but the inner scaling does not hold when f ⁺ is high. Here, (+) corresponds to the quantities normalized with the inner variables, i.e. the friction velocity u and the viscosity . The grouping of the ejections into bursts show the coexistence of two categories of events which react differently to the forcing. The groups of ejections (Multiple Ejection Bursts) are governed by the modulation of the wall shear stress in the whole imposed frequency range. The solitary ejections (or the Single Ejection Bursts) have modulation amplitudes and phases which differ significantly from those of in the intermediate and high imposed frequency range. There is a good agreement between the flow visualization data and the probe measurements.