Centrifugal instability of pulsed Taylor-Couette flow in a Maxwell fluid

DOI 10.1140/epje/i2016-16082-9 Regular Article

T HE E UROPEAN

P ^HYSICAL J ^OURNAL E

Centrifugal instability of pulsed Taylor-Couette flow in a Maxwell fluid

Mehdi Riahi^1,a, Sa¨ıd Aniss¹, Mohamed Ouazzani Touhami¹, and Salah Skali Lami²

1 University Hassan II, Ain-Chock Faculty of Sciences, Laboratory of Mechanics, B.P.5366 Mˆaarif, Casablanca, Morocco

2 Lemta-UMR CNRS 7563-Ensem, 2, avenue de la Forˆet de Haye, BP 160, Vandoeuvre-les-Nancy, 54504, France Received 21 June 2016 and Received in final form 27 July 2016

Published online: 31 August 2016 – cEDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2016 Abstract. Centrifugal instability of a pulsed flow in a viscoelastic fluid confined in a Taylor-Couette system is investigated. Both cylinders are subject to an out-of-phase modulation of rotation with equal modulation amplitude and modulation frequency. The fluid is assumed to obey a linear Maxwell fluid with a relaxation time and a constant viscosity. Attention is focused on the linear stability analysis and on the effect of Deborah and frequency numbers on the critical values of the Taylor and wave numbers. Using Floquet theory, we show that in the limit of low frequency, the Deborah number has no effect on the stability of the basic state which tends to the classical configuration of steady circular Couette flow. When the frequency number increases, the stability of the basic flow is enhanced and the Deborah number has a destabilizing effect which is strongly pronounced in the high-frequency limit. In this frequency limit, the critical parameters tend to constant values independently of the frequency number. These numerical results are in good agreement with the asymptotic solutions obtained in the limit of low and high frequencies.

Moreover, a correlation between the rheological proprieties of the fluid in a rheometric experience, especially the loss and storage modulus, and this hydrodynamical instability behavior is presented.

1 Introduction

Hydrodynamical systems subjected to time periodic mod- ulation have received great interest during the past sev- eral years due to their importance in many natural and industrial processes. This modulation gives the possibility of controlling the onset of instability. In some industrial applications, many polymer processing operations are lim- ited at the onset where instabilities occur . Therefore, it may be desirable to operate at control parameters smaller than the critical ones at which the instability occurs and thus to have no instability.

Since the last century, pioneering investigations con- cerning the modulation of the boundary conditions in the Taylor-Couette problem have been conducted, for a New- tonian fluid, both experimentally and theoretically in dif- ferent configurations in which the angular velocities of the inner and outer cylinders are, respectively,Ω1+1cos(ωt) andΩ₂+₂cos(ωt) [1–6]. The configurations correspond- ing to a zero mean modulation (Ω₁ = Ω₂ = 0) in out of phase (=1/2=−1) have been revisited experimen- taly and theoretically in the narrow gap approximation by Aouidefet al.[8,9] and Tennakoonet al.[10]. The numer- ical results obtained in this case reveal a good agreement with the experimental observations at high and moderate

a e-mail:mehdi riahi@hotmail.fr

frequencies, but show some disagreement at low frequen- cies.

Many practical processes as well fundamental experi- mental studies of instability involve non-Newtonian, vis- coelastic liquids. In an industrial setting, this instability is detrimental to the quality of the final product, and avoiding it involves imposing limitations on throughput, or modifying the flow apparatus. It is therefore of practical importance to understand this instability, and if possible, come up with methods to delay their onset. In this pa- per, we extend the stability analysis of the pulsating flow of a Newtonian fluid in Taylor-Couette geometry to that of a viscoelastic Maxwell model, in the case of a narrow gap approximation and when the cylinders are counter- oscillating with the same amplitude. We consider a linear Maxwell fluid that is relatively simple to apply, but which has a relatively limited range of applications. This model is used in analyses of small deformations of plastics and also in some real fluid flow problems. Some specific examples of liquids that often show simple Maxwell-like behavior are the associative polymers such as hydrophobic ethoxylated urethane (HEUR) [11, 12], and aqueous surfactant solu- tions containing thread-like micelles [13–18]. The simple Maxwell model describes a viscoelastic fluid with a single relaxation time and constant viscosity. In an unsteady con- figuration and for deformations occurring at time scales shorter than the relaxation time, the viscoelastic fluid

behaves like a purely elastic solid, while for time scales longer than the relaxation time, the viscoelastic liquid be- haves like a simple Newtonian fluid. Recently, Bandopad- hyayet al.[19] investigated the interactions between fluid rheology and the interfacial electrokinetic phenomenon pertaining to time periodic electro-osmotic flows of lin- ear Maxwell fluids. A recent study by Tomar et al. [20]

dealt with the instability and dynamics of thin viscoelas- tic Maxwell liquid as a limiting case of a linear Jeffery fluid.

Following the recent study carried out by Riahi et al.[21, 22] related to pulsed Taylor-Couette flow in differ- ent configurations, we are interested in the present work in a zero mean modulation of both inner and outer cylinders in an out-of phase case. In this context, we investigate the effect of frequency modulation and elasticity characterized by Deborah number on the threshold of instability. A dis- cussion related to the Newtonian case is also presented since new branches in the stability diagrams are obtained, which have not been reported nor in the theoretical studies carried out by Aouidefet al.[8,9], nor in the experimental observations obtained by Tennakoonet al.[10]. A compar- ison is made between the results obtained for viscoelastic liquid and those for Newtonian liquid in order to discern the effect of viscoelasticity on the dynamics of the flow.

This paper is organized as follows. In sect. 2, we deter- mine the basic pulsed flow generated by the modulation in out of phase. After that, we perform in sect. 3 a linear stability analysis in which asymptotic critical parameters of instability in the limit of low and high frequency are de- termined. In sect. 4, we present the numerical approach.

The numerical results concerning the effect of elasticity and frequency modulation as well as a comparison with the asymptotic ones are discussed in sect. 5. Section 6 is devoted to conclusion.

2 Base flow

Consider a time periodic flow in an incompressible vis- coelastic fluid filling the annulus space between two in- finitely long cylinders of radii R₁ and R₂ = R₁ +d, with d being the gap width (fig. 1). The angular veloc- ity of the inner and the outer cylinder is, respectively, Ω1 =Ωo cos(ω t^∗) andΩ2 =−Ωo cos(ω t^∗), where Ωo

and ω denote, respectively, the amplitude and the fre- quency of the modulated rotation. The governing equa- tions are the conservation equations for momentum and mass

ρ ∂V^∗

∂t^∗ +V^∗· ∇V^∗

=−∇P^∗+∇ ·τ, (1)

∇ ·V^∗= 0, (2)

whereV^∗is the velocity vector,τis the extra stress tensor and P^∗ is the pressure. The fluid is assumed to obey a linear Maxwell model and can be represented by a purely viscous damper and a purely elastic spring connected in series

τ +λ∂τ

∂t^∗ =μD. (3)

Fig. 1. Sketch of the modulated Taylor-Couette geometry.

We denote by D the rate of strain tensor defined by D=∇V^∗+∇^tV^∗. We designate by ρthe density,μ the dynamic viscosity and λ the relaxation time. Under the linear model that we consider in this study, we assume that the normal stresses are negligible compared to shear stress. However, it is noteworthy that the normal forces play a significant role in developing purely elastic insta- bilities of curved streamlines. These instabilities discussed in refs. [23–25] can occur even in the absence of inertia.

Although the Maxwell model is empirical and its range of validity is limited due to its simplicity, it is used in this study to investigate the effect of elasticity on the thresh- old of instability. Also, it is often argued that studying the linear Maxwell model allows to understand how the relax- ation time of the fluid affects instability, as opposed to the existence of normal stress differences, shear-thinning, and other effects that arise from various nonlinear terms in the constitutive models. The linearly viscoelastic fluids in gen- eral do not predict normal stress differences that can arise just if one consider a nonlinear rheological behavior, see for example the upper convected Maxwell and Oldroyd- B fluids. On the other hand, and from an experimental point of view, the normal stress behavior of viscoelastic surfactant solutions in a strain-controlled rheometer was investigated by Peter Fischer [26] who claimed that in the linear regime one observes the Maxwellian behavior of the viscoelastic solutions while no normal stresses were ob- served in this flow regime.

In dimensional cylindrical-polar coordinates (r^∗, θ, z^∗), the velocity components are given, respectively, in the radial, azimuthal and axial direction by (U^∗, V^∗, W^∗).

We assume that the base flow is azimuthal and axisymet- ric and then it is written asV^∗= (0, V^∗,0), whereP^∗and V^∗ areθindependent. Under these assumptions, a combi- nation of eqs. (1)-(2)-(3) leads to the following system:

ρ

1 +λ ∂

∂t^∗ −V^∗2 r^∗

=−

1 +λ ∂

∂t^∗ ∂P^∗

∂r^∗ , (4)

ρ

1 +λ ∂

∂t^∗ ∂V^∗

∂t^∗ =μ ∂²V^∗

∂r^∗2 + ∂

∂r^∗ V^∗

r^∗

, (5) 0 =−

1 +λ ∂

∂t^∗ ∂P^∗

∂z^∗ . (6)

We introduce the following dimensionless variables:

r^∗ =R1+d x, t= t^∗

d² ν

, VB = V^∗ R₁Ω_o,

PB = P^∗

ρR1dΩ²_o. (7)

Equation (6) shows that the pressure is independent of z. Assuming that the gap width d is small compared to the radius R₁ of the inner cylinder and using the small- gap approximation in which all terms of order d/R1 are neglected, the dimensionless azimuthal velocity satisfies eq. (5) which is now in the form

Γ∂²V_B

∂t² +∂V_B

∂t =∂²V_B

∂x² , (8)

with the boundary conditions

V_B(x= 0, t) = cos(σt) and V_B(x= 1, t) =−cos(σt).

(9) The parametersσ= (ωd²/ν) and Γ =λν/d² denote, respectively, the frequency number and the Deborah num- ber. σ is the ratio of the viscous diffusive time and the period of modulation, whereasΓ is the ratio of the relax- ation time and the viscous diffusive time.

The solution of eqs. (8) and (9) is

VB(x, t) =V1(x) cos(σt) +V2(x) sin(σt). (10) Here the functionsV₁ andV₂ are given by

V₁(x) =

cos(γβx) cosh(γξ(1−x))−cos(γβ(1−x)) cosh(γξx)

cosh(γξ)−cos(γβ) ,

(11) V2(x) =

sin(γβx) sinh(γξ(1−x))−sin(γβ(1−x)) sinh(γξx)

cosh(γξ)−cos(γβ) ,

(12) where γ =

σ/2, β = (√

1 +Γ²σ² +Γ σ)^1/2 and ξ = (√

1 +Γ²σ²−Γ σ)^1/2.

The pressure of the base flow, P_B(x, t), is obtained from the integration of the centrifugal force density that is determined by eq. (10) and which is written as

1 +Γ ∂

∂t

V_B=−

1 +Γ ∂

∂t ∂PB

∂x . (13) The parameter γ expresses the ratio of two lengths γ = d/δN, where δN =

2ν/ω is the thickness of the Stokes layer for a Newtonian fluid. The parametersβ and ξ ex- press also the ratio of two lengths β = ξ⁻¹ = δM/δN, where δM = (σ/2)^−1/2(σΓ +√

1 +σ²Γ²)^1/2 is the thick- ness of the Stokes layer for a linear Maxwell fluid [27].

3 Linear stability analysis

We assume that the base state is disturbed so that the velocity and the pressure fields in the perturbed state are written as the sum of the base flow variables and small perturbations,

u= (0, V_B,0) + (u(x, z, t), v(x, z, t), w(x, z, t)),

P =PB+p(x, z, t). (14)

Substituting the above expressions into eqs. (4)-(6) and then linearizing yields

M− ∂

∂t−Γ ∂²

∂t²

u+ 2T_a²

1 +Γ ∂

∂t

VBv=

1 +Γ ∂

∂t ∂p

∂x, (15)

M− ∂

∂t−Γ ∂²

∂t²

v=

1 +Γ ∂

∂t ∂VB

∂x u, (16)

M− ∂

∂t−Γ ∂²

∂t²

w=

1 +Γ ∂

∂t ∂p

∂z, (17)

∂u

∂x+∂w

∂z = 0, (18)

with the boundary conditions

u=v=w= 0 atx= 0,1, (19) where M = _∂x^∂22 + _∂z^∂22 and T_a is the Taylor number de- fined asT_a= (R₁Ω₀d/ν)(d/R₁)^1/2. Hereafter, we seek the solution of the system of equations (15)-(19) in terms of normal modes as

(u, v, w) = (ˆu(x, t),v(x, t),ˆ w(x, t),ˆ p(x, t)) exp(iqz),ˆ (20) where q is the axial wave number. We have focused in this paper only on the axisymmetric critical modes. This study is carried out in the case of a weak elasticity. Then, we consider that the first instability is axisymmetric as in the case of a Newtonian fluid.

Eliminating the pressure and the third component of velocity, the linearized equations governing the behavior of the eigenfunctions ˆu, ˆv become

Δ− ∂

∂t−Γ ∂²

∂t²

Δˆu= 2q²T_a²

1 +Γ ∂

∂t

V_Bˆv, (21)

Δ− ∂

∂t−Γ ∂²

∂t²

ˆ v=

1 +Γ ∂

∂t ∂V_B

∂x u,ˆ (22) whereΔ= (_∂x^∂22 −q²). The boundary conditions are

ˆ

u= ˆv=∂uˆ

∂x = 0 atx= 0,1. (23) Before solving the system (21)-(23), we shall consider in the next section two asymptotic limiting cases correspond- ing to low and high frequencies.

3.1 Low-frequency behavior

In the limit of low-frequency regime, when σ 1, the spatial profilesV1(x) and V2(x) of the azimuthal velocity can be expanded in power ofσaccording to

V1(x) = 1−2x+ σ²

360x(x−1)(2x−1)

×(3x²−3x−1 + 60Γ) + h.o.t., (24) V₂(x) = σ

6x(2x−1)(x−1) + h.o.t. (25) The system of equations (21)-(22) can be simplified by taking into account the lower-order terms in the asymp- totic expansion of VB given in (24)-(25) and in the cor- responding expansion for its x-derivative. By introducing the new time variable η = σt, the system (21)-(22) be- comes

Δ−σ ∂

∂η −Γ σ² ∂²

∂η²

Δˆu= 2q²T_a²

1+Γ σ ∂

∂η

VBˆv, (26)

Δ−σ∂

∂η −Γ σ² ∂²

∂η²

ˆ v=

1 +Γ σ ∂

∂η ∂V_B

∂x u.ˆ (27) Using a quasi-static approach similar to that used in [8, 10] in which the time derivatives are neglected and at order σ, the system of equations (26)-(27) becomes

D²−q²2

ˆ

u= 2q²T_a²(1−2x) cos(η)ˆv, (28) D²−q²

ˆ

v= (−2) cos(η)ˆu. (29)

If we consider that cos(η) = 0, the set of ordinary differential equations (28)-(29) is solved numerically for an effective constant parameterC=T_a² cos(η)². The solution of this system with the boundary conditions (23) provides the following critical values:

Cc= 4665.66, qc= 4, (30) and then, the instantaneous Taylor number is minimum when cos(η) = 1 and its value is expected to be

Tac= 68.31, qc= 4. (31) In this frequency regime, the hydrodynamic behavior of the Maxwell model is similar to that of a Newtonian one.

Indeed, the terms of orderσin the asymptotic expansion of V_B and its x-derivative do not dependent on Deborah number. This result is expected since under low-frequency oscillations, i.e. long time periods, the time derivative component in the Maxwell rheological model becomes neg- ligible and the spring can be effectively removed from the model.

3.2 High-frequency behavior

In the limit of high-frequency regime, when σ 1, the choice of d as a length scale is not appropriate because

the instability is expected to occur in the Stokes bound- ary layer of size δ = dσ^−1/2(σΓ +√

1 +σ²Γ²)^1/2 [27].

Therefore it is convenient to make the following change of variables:

η=σt, (32)

x=σ^−1/2

σΓ+

1 +σ²Γ² 1/2

˜

x, (33)

q=σ^1/2

σΓ +

1 +σ²Γ² _−1/2

˜

q. (34)

A balance of the various terms in the second equation of the system (21)-(22) gives the relationship σ^1/2(σΓ +

√1 +σ²Γ²)^−1/2v˜∼u, which is then reported in the first˜ equation where the right- and the left-hand sides have the same magnitude if

Ta∼ σ^3/4 σΓ+√

1 +σ²Γ²3/4. (35) At this stage, one can notice that forΓ = 0 these laws correspond exactly to those obtained previously in [2,8,10]

for a Newtonian fluid

T_ac= 15.28γ^3/2 (36)

and the asymptotic behavior of the critical parametersTac

andqc related to a Maxwell fluid is written as Tac= Ta

Γ+√

σ⁻²+Γ²3/4, qc = q_c

Γ+√

σ⁻²+Γ²1/2, (37) which becomes whenσ→+∞

Tac= Ta

(2Γ)^3/4, qc = qc

(2Γ)^1/2. (38) Remark that in the high-frequency limit and in contrast to the Newtonian fluid case where Tac = 15.28γ^3/2, the critical parameters T_ac and q_c depend only on Deborah number which has a destabilizing effect. For a given value of Deborah number, we obtain

T_ac=C₁, q_c =C₂. (39) In this asymptotic case, the critical parameters tend to constant values which are independent of the frequency number.

The thickness of the Stokes layer, for a Maxwellian fluid, is greater than that of a Newtonian fluid. Also it is well known that under high-frequency oscillations/short time periods, the time derivative component in the rhe- ological model dominate. Indeed, the dashpot resists changes in length and it can be approximated as a rigid rod that cannot be stretched from its original length.

Thus, only the spring connected in series to the dashpot will contribute to the total behavior.

4 Floquet analysis

The numerical approach used in this work was employed in the context of modulated flows instability [7,8,10]. The system of equations (21)-(23) is solved using the Floquet theory and then, the perturbed quantities are expanded in the form

(ˆu,v) = exp(μt)ˆ

n=+∞

n=−∞

(Un(x), Vn(x)) exp(i n σt). (40) The Floquet exponentμ=μo+iμ1is a complex num- ber. We aim to determine the marginal stability corre- sponding to harmonic solutions, μ₁ =μ_o= 0. Hereafter, the base flow is rewritten as

VB=F(x) exp(iσt) +F^∗(x) exp(−iσt), (41) where F(x) = ¹₂(V1(x)−iV2(x)) and the starred quan- tity is the complex conjugate. Introducing expressions (40) and (41) into the system (21)-(23), we get an infinite set of equations

D²−q²−inσ+Γ n²σ² D²−q² U_n=

2q²T_a²(1 +inΓ σ) (F V_n−1+F^∗V_n+1), (42) D²−q²−inσ+Γ n²σ² D²−q²

V_n= (1 +inΓ σ)

dF

dxU_n−1+dF^∗ dx U_n+1

, (43)

where D = _dx^d and the associated boundary conditions are

Un=Vn=DUn= 0, at x= 0, 1. (44) This system is transformed into a set of first-order or- dinary differential equations for the quantities,Un,DUn, (D²−q²)U_n, (D²−q²)DU_n,V_n, andDV_n. A set of 3 + 6N independent solutions satisfying the boundary conditions at x = 0 is constructed by a Runge-Kutta numerical scheme. A linear combination of these solutions satisfying the boundary conditions at the other extremex= 1 leads to a homogeneous algebraic system for the coefficients of the combination. A necessary condition for the existence of non-trivial solution is the vanishing of the determinant which can be formally written as

(σ, Γ, q, Ta) = 0. (45) For assigned values of the frequency number, σ, and Deborah number,Γ, the neutral curvesT_a(q) are obtained and then the critical parameters T_ac and q_c are deter- mined. The convergence of the numerical solutions de- pends greatly on the order N of the truncated Fourier series. This convergence is normally assumed when Tac

corresponding toN in the Fourier expansion is within 2%

of the one corresponding toN+ 1. The number of modes N retained in the system (42)-(43) depends strongly on the frequency value. For instance, N= 3 and N = 11 are the orders of the truncated of the Fourier series for high and low frequency, respectively.

Table 1. The critical Taylor and wave numbers for different values ofN forγ= 0.08 andΓ = 0 (Newtonian fluid).

γ N Tac qc

0.08

10 67.21 4

11 68.91 4

12 68.51 4

13 68.91 4

5 Numerical results and discussion

5.1 Newtonian fluid case (Γ=0)

The discussion on the Newtonian case allows us to com- pare our results with previous ones and to validate our numerical findings. In the limit of high frequency, there is non-discrepancy with the results in [8–10]. The evolu- tions of the numerical critical Taylor and wave numbers, Tac and qc, in figs. 4a and 6a agree with the asymptotic laws given by eq. (36) for Γ = 0: Tac = 15,28 γ^3/2 and q_c= 0,86γ[2,8,10]. A stabilization of the flow is revealed in this limit of high frequency.

Decreasing the value of the frequency number γ, the overall trend is toward a destabilization in agreement with [2, 3, 9, 10]. A break in the slope of the T_ac(γ) curve accompanied with a discontinuity in the curveq_c(γ) is ob- served. This topological feature results from the particular shape of the neutral curves ofTa(q). In general the neutral curves have several local minima and the location of the lower minimum when it exists corresponds to the different branches in the diagrams T_ac(γ). An example of the neu- tral curvesTa(q) for representative values of the frequency number is presented in fig. 2. Forγ= 3.6, two modes with different critical wave numbers, qc = 3.1 and qc = 6, are obtained giving rise to a discontinuity in the curveq_c(γ), see fig. 6a. These modes are nearly critical for the same value of the critical Taylor number: Tac= 166 which cor- responds to a change in the slope of the curveTac(γ) and to a discontinuity in the curve qc(γ) respectively seen in fig. 4a and fig. 6a. A second important point is that our nu- merical findings show that whenγ≤3 the critical Taylor number keeps a nearly constant valueTac≈100. These re- sults recover in the best way the experimental findings re- lated to the pulsed Taylor-Couette flow in the out-of-phase case [10]. It is worth noticing that our narrow gap results when 1< γ <2 differs from those obtained theoretically by Aouidef et al. [8, 9] and Tennakoon et al. [10]. They found that for the lowest values of γ considered the crit- ical Taylor number slightly increases above its minimum value T_ac^min ≈ 117 reached for γ = 2.5 in contradiction with the present results and experimental findings [10].

In the low-frequency regime, γ 1, the critical val- ues of Tac and qc versus the frequency were not deter- mined neither numerically nor experimentally in [8–10].

Our numerical results cover the limit when γ −→ 0 and reproduce quite well the asymptotic laws given by eq. (31) corresponding to the unmodulated case: Tac = 68.91 and qc = 4 as illustrated in fig. 4a and fig. 6a. The conver- gence criteria are always verified in this frequency limit.

0 1 2 3 4 5 6 7 8 9 100

200 300 400 500 600 700

q T

γ = 4.3

0 1 2 3 4 5 6 7 8 9

150 200 250 300

q T

γ = 3.6

0 1 2 3 4 5 6 7 8 9

140 160 180 200 220 240 260 280 300

q T

γ = 3.4

Fig. 2. Marginal stability curves for the Taylor numberversus qfor different values of the frequency numberγ.

In table 1, an example of the convergence of our numerical predictions is presented when γ = 0.08. As one can see, this convergence is rapidly obtained and only N = 11 is sufficient instead of N= 22 in refs. [8, 10].

5.2 Maxwellian fluid case (Γ=0)

In this section, we investigate numerically the effect of the frequency and Deborah numbers on the onset of instabil- ity. Figure 3 illustrates the marginal stability curves when γ = 1.9 for two values of Deborah number, Γ = 0 and Γ = 0.006. As one can see, the shape of these curves is affected by the Deborah number and the number of min- ima increases with this number. Also, one can note that the smaller value of these minima corresponding to the critical Taylor number decreases when the Deborah num- ber weakly increases. This first observation confirms the destabilizing effect of the Deborah number.

The variation of the critical Taylor numberversus the parameterγis reported in figs. 4 and 5 for different values of Deborah number. As observed in these figures, a stabi-

lization of the base flow is revealed in the high-frequency limit. In this limit, however, a change in the shape of the curves is revealed for the viscoelastic fluid. Indeed, in con- trast to the Newtonian fluid case,Γ = 0, where the criti- cal Taylor number increases infinitely with the frequency number,Tac= 15.28γ^3/2 [2, 8, 10], we note that the crit- ical Taylor number varies independently of the frequency number and tends to constant values: Tac = 863.32, T_ac= 223.03 and T_ac= 198.25 for Γ = 0.001,Γ = 0.006 and Γ = 0.008, respectively. Furthermore, in the high- frequency limit, the increase of Deborah number leads to a destabilization of the base flow. These numerical results are in good agreement with the asymptotic analysis pre- sented in sect. 3.2.

The effect of the Deborah number on the critical Tay- lor number is also shown in fig. 5 where results of fig. 4 are summarized. It turns out that in the low-frequency limit, when γ 1, the Deborah number has no effect on the critical Taylor number whose variation reproduces quite well the asymptotic law (eq. (31)),Tac= 68.91, for New- tonian fluid and steady rotation of the cylinders. As one can see, the Deborah number has a destabilizing effect

2 4 6 8 10 12 100

120 140 160 180 200 220

q Ta

γ = 1.9 ^{Γ = 0}

2 4 6 8 10 12

100 105 110 115 120 125 130 135 140 145 150

q Ta

γ = 1.9 ^{Γ = 0.006}

2 4 6 8 10 12

96 98 100 102 104 106 108 110 112 114

q Ta

γ = 1.9 ^{Γ = 0.01}

Fig. 3.Marginal stability curves forγ=

σ/2 = 1.9 for different values of the Deborah numberΓ (σ: dimensionless frequency).

observed for γ > 1 which becomes more pronounced as the frequency number increases.

The critical wave number as a function of γ for dif- ferent values of the Deborah number is reported in fig. 6.

In the low-frequency limit, the critical wave number re- mains constantqc = 4. This value corresponds to that of the classical Taylor-Couette solution of a Newtonian fluid.

In the high-frequency limit, the same behavior as that of the critical Taylor number occurs. For Γ = 0, the criti- cal wave number increases versus the frequency number γ. However, we note that for Γ = 0.001,Γ = 0.006 and Γ = 0.008, the critical wave number becomes almost con- stant and takes, respectively, the valuesq_c= 16.2,q_c = 7 and qc = 6.5 independently of the frequency number. Fi- nally, one can notice that the critical wave number de- creases with increasing the Deborah number.

Also, it is obvious that the Deborah number affects the shape of the curvesTac(γ) andqc(γ), as seen in figs. 4 and 6. In particular, the intersecting points of the different branches in the stability diagramsTac(γ) and their corre- sponding discontinuities in the curvesq_c(γ). For example,

the intersection point which occurrs atγ= 3.6 in the curve Tac(γ) forΓ = 0 is tracking toward the right with the in- crease of the Deborah number (Γ = 0.001 → γ = 3.8;

Γ = 0.006→γ= 4.2 andΓ = 0.008→γ= 5.15).

To conclude, a correlation between the rheological pro- prieties, especially loss and storage modulus of a Maxwell fluid, and its hydrodynamic instability behavior seems very important since the configuration used in this work, counter-oscillating cylinders, is similar to that used in a rheometric experiment in the case of a frequency sweep. In the low-frequency limit, we have shown that the Maxwell fluid behaves as a Newtonian one. This result is correlated to the fact that the loss modulus, related to the viscous part of the model, is predominant in comparison with the storage modulus [28]. However, in the limit of high fre- quency, the storage modulus, related to the elastic part of the model, becomes predominant in comparison with the viscous modulus and tends to a constant value [28].

This behavior explains the fact that we obtain asymptotic critical parameters in this frequency range.

0 2 4 6 8 10 12 0

100 200 300 400 500 600 700

γ Tac

( a ) Equation 36

Equation 31 Γ = 0

0 2 4 6 8 10 12

50 100 150 200 250 300 350 400 450 500 550

γ T_ac

( b ) T_ac

Equation 31

Γ = 0.001 863.32

0 2 4 6 8 10 12

60 80 100 120 140 160 180 200 220 240 260

γ T_ac

223.03 Γ = 0.006

Tac ( c )

Equation 31

0 2 4 6 8 10 12

60 80 100 120 140 160 180 200 220

γ T_ac

T_ac 198.25

Equation 31

Γ = 0.008 ( d )

Fig. 4. Critical Taylor number versus the frequency number for different values of the Deborah number Γ: (a) Γ = 0, (b)Γ = 0.001, (c)Γ = 0.006, (d)Γ = 0.008.

In addition, the present paper shows experimental in- terests having a close relationship with the experimental rheology for low elasticity. Indeed, on the basis of the re- sults presented in this paper, it is proposed that instead of measuring elastic properties of a real viscoelastic fluid for verifying if its rheological behavior is well described by a linear Maxwell or not, one may equally resort to its instability behavior in pulsed Taylor-Couette flow as an ef- fective and simpler means for this purpose. The response of the linear Maxwell fluid is similar to that of a New- tonian fluid in the low-frequency regime. In other words, the critical amplitude giving rise to the instability corre- sponds to that of a Newtonian fluid in a steady rotation of the cylinders. Increasing the frequency oscillation of the cylinders is accompanied by an increase of this critical amplitude which is smaller than that of a Newtonian fluid and remains constant even if we increase the frequency oscillation.

The results that we have obtained concerning the effect of the elasticity of the Maxwell fluid through the Debo- rah number are similar to that of a previous work [29] in which the effects of a temperature modulation in phase on the convective instability of a viscoelastic layer were inves-

tigated. In this study [29], the Deborah number has the same effect on the critical parameters and it was shown that no effect was observed in the low-frequency limit whereas a destabilizing effect occurred when the modu- lation frequency increases.

5.3 Comparison to the case of co-oscillating cylinders and the case of oscillating inner cylinder and a fixed outer one

The case of an in-phase modulation of the cylinders and the case where the modulation concerns only the inner cylinder have been studied recently by Riahiet al.[21,22].

Here, we compare results of these studies to those of the present work dedicated to out-of-phase modulation. We report in fig. 7 the evolution of the critical Taylor number versus the frequency number for the three cases of modu- lation and for two values of Deborah number, Γ = 0.006 and Γ = 0.01. As one can see, a common behavior is ob- served in the high-frequency limit where the critical Taylor number tends to the same constant value independently of the frequency number, T_ac ∼ 223 for Γ = 0.006 and

0 2 4 6 8 10 12 100

200 300 400 500 600 700

γ T

Γ = 0

Γ = 0.001 Γ = 0.006 Γ = 0.008

Tac

T_ac 198,25 863.32 Tac

223.03 Tac = 68.91

Fig. 5.Critical Taylor numberversus the frequency number for different values of the Deborah numberΓ.

0 2 4 6 8 10 12

3 4 5 6 7 8 9 10

γ q c

( a )

Equation 31 Equation 35 Γ = 0

0 2 4 6 8 10 12

3 4 5 6 7 8 9 10

γ q _c

( b ) q

c

Equation 31

Γ = 0.001 16.2

0 2 4 6 8 10 12

2 3 4 5 6 7 8

γ q c

( c ) Γ = 0.006

q c

Equation 31

7

0 2 4 6 8 10 12

3 4 5 6 7 8 9 10

γ

q _c q 6.5

c Γ = 0.008

Equation 31

( d )

Fig. 6. Critical wave numberversus the frequency number for different values of the Deborah number Γ: (a) Γ = 0, (b) Γ = 0.001, (c)Γ = 0.006, (d)Γ = 0.008.

10⁰ 10¹ 10²

γ T ac

Tac = 41.63 T_ac = 68.91 T_ac = 68.91

T_ac ~ 223

inner cylinder modulation out−of phase modulation in−phase modulation

Γ = 0.006

(a)

10⁰ 10¹

10²

γ

Tac = 68.91

Tac ~ 160

T ac

T_ac = 41.63 inner cylinder modulation out−of phase modulation in−phase modulation

Γ = 0.01

(b)

Fig. 7. Critical Taylor number versus the frequency number for different cases of modulation and for different values of Deborah numberΓ: (a)Γ = 0.006, (b)Γ = 0.01.

Tac ∼160 for Γ = 0.01. In this frequency range, the in- stability occurs near the oscillating cylinders in layers of thickness δ = dσ^−1/2(σΓ +√

1 +σ²Γ²)^1/2. This behav- ior was pointed out by Aouidefet al. [8] for the case of a Newtonian fluid.

Differences are observed when the frequency number decreases, particularly whenγ <7. We find that the curve related the out-of-phase modulation is located between the upper curve corresponding to the case of in-phase modula- tion and the lower one corresponding to the inner cylinder modulation case. It turns out that the most unstable case is the inner cylinder modulation. A particular feature re- lated to the in-phase modulation is observed at the inter- mediate frequencies where the flow is potentially unstable.

For this case of modulation, a stabilization of the flow is revealed in the low-frequency limit in contrast to the other cases of modulation where the critical Taylor num- ber tends towards the classical solutions corresponding to the unmodulated cases.

6 Conclusion

We have examined the linear stability of a pulsed flow in a Maxwell fluid in the Taylor-Couette geometry when the outer and the inner cylinders are oscillating out of phase with the same amplitude and frequency. We have focused on the effects of elasticity and frequency on the critical parameters, Taylor and wave numbers. The numerical re- sults show that at high frequencies, and in contrast to a Newtonian fluid, the critical Taylor and wave numbers are independent of the frequency number when the Debo- rah number increases and the critical parameters tend to asymptotic values depending only on the Deborah num- ber. In this frequency limit, we have shown that the fluid elasticity has a strong destabilizing effect and only the spring will contribute to the total behavior. Decreasing the frequency number to intermediate frequency, the over- all trend of the pulsed flow is toward destabilization and the effect of elasticity is always destabilizing. In the low- frequency limit, the Maxwell fluid behaves as a Newtonian one and the critical Taylor number is given byT_ac= 68.91 which corresponds to the unmodulated case. In this situ- ation, the spring connected in series with a dashpot could be removed from the rheological model and only the dash- pot will contribute to the total behavior. These numerical observations are in good agreement with the asymptotic analysis carried out in the limit of low and high frequency.

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