DOI 10.1140/epjp/i2015-15253-7 Regular Article
P HYSICAL J OURNAL P LUS
Pulsed Taylor-Couette flow in a viscoelastic fluid under inner cylinder modulation
Mehdi Riahi1,a, Sa¨ıd Aniss1, Mohamed Ouazzani Touhami1, and Salah Skali Lami2
1 Universit´e Hassan II - Casablanca, Faculty of Sciences A¨ın Chock, 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: 24 July 2015 / Revised: 25 October 2015
Published online: 18 December 2015 – cSociet`a Italiana di Fisica / Springer-Verlag 2015
Abstract. The influence of elasticity on pulsed Taylor-Couette flow in a linear Maxwell fluid is investigated.
We consider the case, in which the inner cylinder is oscillating with a periodic angular velocity,Ω0cos(ωt), and the outer cylinder is fixed. Attention is focused on the linear stability analysis which is solved using the Floquet theory and a technique of converting a boundary value problem to an initial value problem.
Results obtained in this framework show that, in the high-frequency limit, the Deborah number has a destabilizing effect and the critical Taylor and wave numbers tend toward constant values independently of the frequency number. However, in the low-frequency limit, the Maxwell fluid behaves as a Newtonien one and the Deborah number has no effect on the stability of the basic state which tends to the classical configuration of steady circular Couette flow. These numerical results are in good agreement with the asymptotic analysis performed in the limit of low and high frequencies.
1 Introduction
The temporal periodic modulation of the boundary conditions in hydrodynamic systems is the subject of several works. This modulation gives the possibility of controlling the threshold of instability. In some applications, it may be desirable to operate at control parameter of instability higher than the critical one at which the instability occurs and yet have no instability. For example, in the Rayleigh-B´enard convection, the modulation of the temperatures imposed on the horizontal boundaries of a fluid layer can delay or accelerate the onset of convection depending on the amplitude and frequency of modulation and the threshold of convection is characterized by harmonic or subharmonic solutions [1–11].
Several investigations concerning the modulation of the Taylor-Couette flow have been conducted both experimen- tally and theoretically in different configurations in which the angular velocities of the inner and outer cylinders are, respectively,Ω1+1cos(ωt) andΩ2+2cos(ωt). The effect of this modulation on the thereshold of instability compared to the unmodulated case was discussed in [12–21]. In the case of pulsed flow in a Taylor-Couette geometry, where both cylinders rotate with the same angular velocity, two configurations corresponding to a zero mean modulation (Ω1=Ω2= 0) in phase (1=2= 1) or out of phase (1=−2) have been revisited theoretically in the narrow-gap approximation, respectively in [22, 23] and in [24]. Moreover, the first experimental evidence of these configurations was achieved in these references. The experimental results, corresponding to modulation in phase, have been in good agreement with the linear stability analysis and have shown that the flow is less unstable in the limit of low and high frequency while destabilization is maximum for an intermediate frequency. For modulation in out of phase, the numerical results compared well with the experimental observations at high and moderate frequencies but showed disagreement at low frequencies. The case in which the modulation concerns only the inner cylinder while the outer is at rest was first investigated in [13]. In this work a linear stability analysis was performed using finite differences and experiments for verification were achieved. The same configuration has been revisited in the small-gap approximation in [15, 23] and for a finite gap size in [16]. A stabilization of the base flow in the high-frequency limit was revealed while a destabilization occurred in the low frequencies.
The instabilities in flows of viscoelastic liquids are of fundamental importance to the understanding of the physics of complex fluids and of practical importance to materials processing and fluid characterization. For instance, instabilities
a e-mail:mehdi riahi@hotmail.fr
Fig. 1. The sketch of the modulated Taylor-Couette geometry.
in rheometry compromise the quality and extent to which viscoelastic material parameters may be measured. In this paper, 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 the narrow-gap approximation. 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) [25, 26], and aqueous surfactant solutions containing thread-like micelles [27–32]. Recently, viscoelastic fluids obeying a linearized Maxwell model were used to investigate the interactions between fluid rheology and interfacial electrokinetic phenomenon pertaining to time periodic electro-opsmotic flows [33].
Following the study carried out recently on the modulation in phase [34] where both cylinders are oscillating with the same amplitude and frequency, we are interested in the present work to a zero mean modulation of the inner cylinder and a fixed outer one. In this context, we investigate the effect of frequency modulation and elasticity characterized by the Deborah number on the threshold of instability.
This paper is organized as follows. In sect. 2, we determine the basic pulsed flow generated by the oscillation of the inner cylinder. 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 determined. In sect. 4, we present the numerical approach. The numerical results on the effect of elasticity and comparison of these results with the asymptotic ones are discussed in sect. 5. Section 6 is the conclusion.
2 Base flow
Consider an incompressible viscoelastic fluid filling the annulus space between two infinitely long cylinders of radiiR1 and R2 =R1+d, where d is the gap width (fig. 1). The outer cylinder is considered at rest while the inner one is oscillating with the angular velocity,Ω=Ω0cos(ωt∗),Ω0andωdenote, respectively, the amplitude and the frequency of the modulated rotation. The governing equations are the conservation equations for momentum and mass,
ρ ∂V∗
∂t∗ +V∗· ∇V∗
=−∇P∗+∇ ·τ, (1)
∇ ·V∗ = 0, (2)
where V∗ is the velocity vector, τ is the extra stress tensor andP∗ is the pressure. The fluid is assumed to obey a linear Maxwell model which can be represented by a purely viscous damper and a purely elastic spring connected in series,
τ+λ∂τ
∂t∗ =μD. (3)
We denote by D the rate of the strain tensor defined by D =∇V∗+∇tV∗. We designate by ρ the density, by μ the dynamic viscosity and byλthe 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 force plays a significant role in developing purely elastic instabilities of curved streamlines. These instabilities discussed in refs. [35–37] 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 threshold of instability.
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 axisymmetric and then it is written as V∗= (0, V∗,0), whereP∗ andV∗ areθ independent. Under these assumptions, a combination of the eqs. (1)–(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:
x= r∗−R1
d , t= t∗
d2 ν
, VB = V∗ R1Ω0
, PB= P∗
ρR1dΩ02. (7)
Equation (6) shows that the pressure is independent ofz. Assuming that the gap width dis small compared to the radiusR1 of the inner cylinder and using the small-gap approximation in which all terms of orderd/R1are neglected, the dimensionless azimuthal velocity satisfies eq. (5) which is now in the form
Γ∂2VB
∂t2 +∂VB
∂t =∂2VB
∂x2 (8)
with the boundary conditions
VB(x= 0, t) = cos(σt) and VB(x= 1, t) = 0. (9)
The parametersσ= (ωd2/ν) andΓ =λ ν/d2denote, respectively, the frequency number and the Deborah number.
The parameterσis the ratio of the viscous diffusive time and the period of modulation, whereasΓ is the ratio of the relaxation time and the viscous diffusive time. The pressure of the base flow,PB(x, t), is obtained from eq. (4) which
is written as
1 +Γ ∂
∂t
VB=−
1 +Γ ∂
∂t ∂PB
∂x . (10)
The solution of eqs. (8) and (9) is
VB(x, t) =V1(x) cos(σt) +V2(x) sin(σt). (11) Here, the functionsV1 andV2 are given by
V1(x) =cos(γβx) cosh(γξ(1−x)) cosh(γξ)−cos(γβ(1−x)) cosh(γξx) cos(γβ)
cosh2(γξ)−cos2(γβ) , (12)
V2(x) =sin(γβx) sinh(γξ(1−x)) cosh(γξ)−sin(γβ(1−x)) sinh(γξx) cos(γβ)
cosh2(γξ)−cos2(γβ) , (13)
whereγ=
σ/2,β = (σΓ +√
1 +σ2Γ2)1/2andξ= (−σΓ +√
1 +σ2Γ2)1/2. The parameter γexpresses the ratio of two lengths, γ=d/δN, whereδN =
2ν/ω is the thickness of the Stokes layer for a Newtonian fluid. The parameters β and ξexpress also the ratio of two lengths β =ξ−1 =δM/δN, where δM =d(σ/2)−1/2(σΓ+√
1 +σ2Γ2)1/2 is the thickness of the Stokes layer for a linear Maxwell fluid [38].
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, VB,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+ 2Ta2
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)
whereM = ∂x∂22+∂z∂22 andTais the Taylor number defined asTa= (R1Ω0d/ν)(d/R1)1/2. The boundary conditions are
u=v=w= 0 at x= 0,1. (19)
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) whereqis the axial wave number. This number describes the axial periodicity of the perturbation. It is related to the wavelength by relationλ= 2π/q. The wavelength defines also the axial periodicity and corresponds to the size of two counter-rotating vortices.
Eliminating the pressure and the third component of velocity, the linearized equations governing the behavior of the eigenfunctions ˆuand ˆvbecome
Δ− ∂
∂t−Γ ∂2
∂t2
Δuˆ= 2q2Ta2
1 +Γ ∂
∂t
VBv,ˆ (21)
Δ− ∂
∂t−Γ ∂2
∂t2
ˆ v =
1 +Γ ∂
∂t ∂VB
∂x u,ˆ (22)
whereΔ= (∂x∂22−q2). The boundary conditions are ˆ
u= ˆv=∂uˆ
∂x = 0 at x= 0,1. (23)
Before solving the system (21)–(23). We shall consider in the next subsections two asymptotic limiting cases corre- sponding to low and high frequencies.
3.1 Low-frequency behavior
In the limit of the low-frequency regime, whenσ1, the spatial profiles of the azimuthal velocityV1(x) and V2(x) given by eqs. (12) and (13) can be expanded in power of σaccording to
V1(x) = 1−x+ σ2
360x(x−1)(x−2)(3x2−6x−4 + 60Γ) + h.o.t., (24) V2(x) = σ
6x(x−1)(x−2) + h.o.t. (25)
The system of equations (21), (22) can be simplified by taking into account the lower-order terms in the asymptotic expansion ofVB given in eqs. (24), (25) and in the corresponding expansion for its derivative. By introducing the new time variable η=σt, the system (21), (22) becomes
Δ−σ ∂
∂η −Γ σ2 ∂2
∂η2
Δuˆ= 2q2Ta2
1 +Γ σ ∂
∂η
VBv,ˆ (26)
Δ−σ ∂
∂η −Γ σ2 ∂2
∂η2
ˆ v=
1 +Γ σ ∂
∂η ∂VB
∂x u.ˆ (27)
Using a quasi-static approach similar to that used in [22–24] in which the time derivatives are neglected and at orderσ, the system of equations (26), (27) becomes
D2−q22
ˆ
u= 2q2Ta2(1−x) cos(η)ˆv, (28) D2−q2
ˆ
v= (−1) cos(η)ˆu, (29)
whereD= dxd. If we consider that cos(η)= 0 the set of ordinary differential equations (28), (29) is solved numerically for an effective constant parameter C =Ta2cos(η)2. The solution of this system with the boundary conditions (23) provides the following critical values:
Cc= 1707.01, qc= 3.12. (30)
and then, the instantaneous Taylor number is minimum when cos(η) = 1 and its value is expected to be
Tac= 41.31, qc= 3.12. (31)
These asymptotic values of the critical parameters agree with those of the unmodulated case studied in [39].
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 ofVB and itsx-derivative do not dependent on the 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 negligible and the spring can be effectively removed from the model.
3.2 High-frequency behavior
In the limit of the high-frequency regime, when σ 1, the choice of d as length scale is not appropriate because the instability is expected to occur in the Stokes boundary layer of sizeδM =d(σ/2)−1/2(σΓ+√
1 +σ2Γ2)1/2 [38].
Therefore, it is convenient to make the change of variables, x=σ−1/2
σΓ +
1 +σ2Γ2 1/2
˜
x, (32)
q=σ1/2
σΓ+
1 +σ2Γ2 −1/2
˜
q. (33)
However, the new time variable is the same as that defined in the asymptotic case of low frequency,η=σt. A balance of the various terms in the second equation of the system (21), (22) gives the relationshipσ1/2(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∼ σ3/4 σΓ+√
1 +σ2 Γ23/4. (34)
At this stage, one can notice that forΓ = 0 these laws correspond exactly to those obtained previously in [15,23,24]
for a Newtonian fluid,
Ta= 15.28γ3/2. (35)
From eq. (34) the asymptotic behavior of the critical parametersTacandqc is written as Tac= Ta
Γ +√
σ−2+Γ23/4, qc= qc
(Γ +√
σ−2+Γ2)1/2, (36)
which becomes, when σ→+∞,
Tac= Ta (2Γ)3/4, qc= qc
(2Γ)1/2. (37)
Note that in the high-frequency limit and in contrast to the Newtonian fluid case whereTac= 15.8γ3/2, the critical parametersTac andqc depend only on the Deborah number which has a destabilizing effect and for a given value of the Deborah number, 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 the one of a Newtonian fluid. Also, it is well known that under high-frequency oscillations/short time periods, the time derivative component in the rheological 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 Numerical approach
The numerical approach used in this work was employed in the context of modulated flows instability [22–24]. 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). (38)
The Floquet exponentμ=μ0+iμ1is a complex number. We aim to determine the marginal stability corresponding to harmonic solutions,μ1=μ0= 0. Hereafter, the base flow is rewritten as
VB =F(x) exp(iσt) +F∗(x) exp(−iσt), (39) where F(x) = 12(V1(x)−iV2(x)) and the starred quantity is the complex conjugate. Introducing expressions (38) and (39) into the system (21)–(23) we get an infinite set of equations,
D2−q2−inσ+Γ n2σ2 D2−q2
Un = 2q2Ta2(1 +inΓ σ) (F Vn−1+F∗Vn+1), (40) D2−q2−inσ+Γ n2σ2 D2−q2
Vn = (1 +inΓ σ) dF
dxUn−1+dF∗ dx Un+1
, (41)
where D=dxd and the associated boundary conditions are
Un=Vn=DUn= 0 at x= 0,1. (42)
This system is transformed into a set of first-order ordinary differential equations for the quantities, Un, DUn, (D2−q2)Un, (D2−q2)DUn, Vn, and DVn, (see the appendix). A set of 3 + 6N independent solutions satisfying the boundary conditions atx= 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 a nontrivial solution is the vanishing of the determinant which can be formally written as
(σ, Γ, q, Ta) = 0. (43)
For assigned values of the frequency number,σ, and of the Deborah number,Γ, the neutral curvesTa(q) are obtained and then the critical parameters Tac and qc are determined. The convergence of the numerical solutions depends greatly on the orderN of the truncated Fourier series. This convergence is normally assumed whenTac corresponding to N in the Fourier expansion is within 2% of the one corresponding toN+ 1. The number of modes N retained in the system (40), (41) depends strongly on the frequency value. For instance,N = 3 andN = 11 are the orders of the truncated of the Fourier series for high and low frequency, respectively.
1 2 3 4 5 6 7 75
80 85 90 95 100 105 110 115 120
q T
aγ = 1.7
1 2 3 4 5 6 7
70 75 80 85 90 95 100 105 110 115 120
q T
aγ = 1.5
1 2 3 4 5 6 7
60 70 80 90 100 110 120
q T
aγ = 1.4
Fig. 2. Marginal stability curves for representative values of the frequency number whenΓ = 0.
5 Numerical results and discussion
5.1 Newtonian fluid
We discuss first the Newtonien case when Γ = 0. Evolutions of the critical Taylor and wave numbers versus the frequency number are shown in figs. 3(a) and 5(a) As one can see, in the high-frequency limit, a stabilization of the flow is revealed and Tac(γ) andqc(γ) are in good agreement with the previous theoretical results obtained by Aouidef et al.[23] which reproduce quite well the asymptotic laws:Tac= 15.28γ3/2andqc = 0.86γ. Decreasing the value of the frequency numberγ, the overall trend is toward a destabilization in agreement with results obtained in [15, 16, 23]. A break in the slope of theTac(γ) curve accompanied by a discontinuity in the curveqc(γ) is obtained. 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, when it exists, corresponds to the different branches in the diagramsTac(γ).
An example of the neutral curvesTa(q) for representative values of the frequency number is presented in fig. 2. For γ= 1.7, two modes with different wave numbers,qc= 2.4 andqc= 3.7, are obtained. Decreasing the frequency number γ untilγ= 1.5, these modes are nearly critical for the same value of the critical Taylor number,Tac≈75. This result contributes to a change of the shape of the curve slope,Tac(γ), and to a discontinuity in the curveqc(γ) in agreement with the theoretical results in [15].
A more destabilizing effect and other breaks in the slope of theTac(γ) curve are observed nearγ= 0.6 andγ= 0.3.
Note that in the low-frequency limit, whenγ→0, the curves Tac(γ) and qc(γ) reproduce quite well the asymptotic laws, Tac = 42.63 and qc = 3.12, corresponding to the unmodulated case [39]. The convergence criteria is always verified when γ→0. An example of this convergence is presented forγ= 0.08 in the table 1.
Table 1.Critical Taylor and wave numbers forγ= 0.08,Γ = 0 and for different values ofN.
γ N Tac qc
0.08
10 41.66 3.12
11 42.63 3.12
12 43.76 3.12
13 42.96 3.12
0 2 4 6 8 10 12
0 100 200 300 400 500 600 700
γ Tac
( a )
Tac = 15.28 γ3/2
Tac = 42. 63 Γ = 0
0 2 4 6 8 10 12
0 100 200 300 400 500 600 700
γ Tac
( b )
T ac
Tac = 42. 63 Γ = 0.001
866,25
0 2 4 6 8 10 12
0 50 100 150 200 250 300
γ Tac
( c )
Tac
Tac = 42. 63 Γ = 0.006
223,12
0 2 4 6 8 10 12
20 40 60 80 100 120 140 160 180 200 220
γ Tac
( d )
Tac
Tac = 42. 63 Γ = 0.01
158,12
Fig. 3. 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 = 0.01.
5.2 Viscoelastic fluid
We investigate the effect of the Deborah number on the onset of instability by determining the critical parameters, Taylor and wave numbers, versus the parameter γ = σ
2 where σ is the dimensionless frequency of modulation.
The variation of the critical Taylor number versus the parameter γ is reported in fig. 3 for different values of the Deborah number. As observed in this figure, a stabilization of the base flow is revealed in the high-frequency limit.
However, a change in the shape of the curves is revealed for the viscoelastic fluid in this frequency limit. Indeed, in contrast to a Newtonian fluid, Γ = 0, where the critical Taylor number increases infinitely with the frequency number,Tac = 15.28γ3/2 [15, 23, 24], we note that the critical Taylor number varies independently of the frequency number and reaches asymptotic values: Tac = 866.25, Tac = 223.12 andTac = 158.12 for Γ = 0.001,Γ = 0.006 and Γ = 0.01, respectively. Furthermore, the increase of the Deborah number leads to a destabilization of the base flow.
These numerical results are in good agreement with the asymptotic analysis, in the high-frequency limit, presented in subsect. 3.2.
0 2 4 6 8 10 0
50 100 150 200 250 300 350 400 450 500
γ T
acΓ = 0 Γ = 0.001 Γ = 0.006 Γ = 0.01
Fig. 4.Critical Taylor numberversus the frequency number for different values of the Deborah numberΓ.
Decreasing the value of the parameter,γ, leads to a destabilization of the basic state. Also, when the frequency number tends to zero, the critical Taylor number reproduces quite well the classical Taylor-Couette solution corre- sponding to a steady rotation of the inner cylinder in the case of a Newtonian fluid,Tac= 42,63 [39]. In this frequency limit, it turns out that the Deborah number has no effect on the critical Taylor number, whereas a remarkable desta- bilizing effect is observed beyondγ= 3 and this effect becomes more pronounced as the frequency number increases, see fig. 4.
The critical wave number as a function of γ for different values of the Deborah number is reported in fig. 5. In the low-frequency limit, the critical wave number remains constant,qc = 3.12. This value corresponds to that of the unmodulated Taylor-Couette flow in the case of a Newtonian fluid [39]. In the high-frequency limit, for Γ = 0, the critical wave number increases versus the frequency number γ. However, forΓ = 0.001, Γ = 0.006 and Γ = 0.01, the critical wave number tends to asymptotic valuesqc = 16.1,qc= 6.4 andqc= 5.6 independently of the frequency number. Also, one can notice that the critical wave number decreases with increasing Deborah number. Note that these numerical findings are in good agreement with the asymptotic analysis, in the high-frequency limit, presented in subsect. 3.2.
Moreover, it is obvious that Deborah number affects the shape of the curvesTac(γ) andqc(γ), as seen in figs. 3 and 5 concerning the intersecting points of the different branches in the stability diagramsTac(γ) and their corresponding discontinuities in the curvesqc(γ). For instance, the intersection point which occurred at γ= 1.5 in the curveTac(γ) for Γ = 0 is slightly tracking toward the right with the increase of the Deborah number (Γ = 0.001 → γ = 1.7;
Γ = 0.006→γ= 1.8 andΓ = 0.01→γ= 2).
The results obtained in this work concerning the effect of the elasticity through the Deborah number are similar to those of previous works [34, 40]. The linear stability of a pulsed flow of a viscoelastic fluid in the Taylor-Couette geometry when the outer and the inner cylinders are oscillating in phase with the same amplitude and frequency is investigated in [34], whereas the effect of a temperature modulation in phase on the convective instability of a viscoelastic layer is studied in [40]. In these studies, 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 modulation frequency increased. Although the physical mechanisms of instability resulting from the modulations of temperature and rotation are not similar, the results of both modulations on the effect of the Deborah number on the threshold of instability are similar in the low and high frequency of modulation. Moreover, even for a Newtonian fluid, similarity has been observed in the curves representing the critical parameters versus the frequency number in the case of modulations of rotation [22] and temperature [41] with equal amplitude and frequency. These findings are explained by the following considerations. First, the basic states in modulations of rotation and temperature are obtained, respectively, in terms of velocity and temperature, from diffusion equations of momentum and temperature.
In the thermal modulation in phase with equal amplitude and frequency, we have considered that the temperature at the horizontal walls of a fluid layer isT0cos(ωt), where T0 and ω represent, respectively, the amplitude and the dimensional frequency. Thus, since the dimensionless diffusion equations with their boundary conditions are similar, the base velocity in modulation of rotation and the base temperature in thermal modulation depend on space and on time in the same manner. Second, for both modulations, the conservation equations corresponding to the perturbed basic states lead to two reduced systems of equations which have expressions very close from a mathematical point of view.
0 2 4 6 8 10 12 2
3 4 5 6 7 8 9 10 11
γ q c
( a )
q c = 0.86 γ
q c = 3.12 Γ = 0
0 2 4 6 8 10 12
2 3 4 5 6 7 8 9 10 11
γ q c
( b )
q c
q c = 3.12 Γ = 0.001
16,1
0 2 4 6 8 10 12
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
γ q c
( c )
q c
q c = 3.12 Γ = 0.006
6.4
0 2 4 6 8 10 12
2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
γ q c
( d )
q c
q c = 3.12 Γ = 0.01
5,6
Fig. 5. Critical wave 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 = 0.01.
To conclude, a correlation between the rheological properties, especially of the loss modulus and of the storage modulus of a Maxwell fluid, and its hydrodynamic instability behavior seems very important since the configuration used in this work, oscillating inner cylinder and fixed outer one, is similar to that used in a rheometric experience 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 [42]. However, in the limit of high-frequency, 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 [42]. This behavior explains how to obtain asymptotic critical parameters in this frequency range.
6 Conclusion
We have examined the linear stability of a pulsed flow of a viscoelastic fluid in the Taylor-Couette geometry. The outer cylinder is assumed to be at rest while the inner one is modulated about a zero mean rotation. We have focused on the effects of elasticity and frequency on the critical parameters, Taylor and wave numbers. From a linear stability analysis and using Floquet theory, the numerical results show that at high frequencies the Deborah number has a strong destabilizing effect. The critical Taylor and wave numbers tend toward asymptotic values independently of the frequency number. In this frequency limit, only the spring will contribute to the total behavior. In the low-frequency limit, the Maxwell fluid behaves as a Newtonian one and the critical Taylor number corresponds to the unmodulated case. In this situation, the spring connected in series with a dashpot could be removed from the rheological model and only the dashpot contributes to the total behavior. The numerical observations are in good agreement with the asymptotic analysis performed in the limit of low and high frequency.
Appendix A.
The system of equations (40), (41) is transformed into a set of first-order ordinary differential equations for the quantities, Un,DUn, (D2−q2)Un, (D2−q2)DUn,Vn, andDVn such as
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
U1+p=Un, U2+p=DUn, U3+p= (D2−q2)Un, U4+p=D(D2−q2)Un, U5+p=Vn,
U6+p=DVn
(A.1)
withp= 6(N+K), whereN is the truncation order or the Fourrier series and−N≤K≤N. Thus, the system (40), (41) becomes
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
U1+p=U2+p,
U2+p=q2U1+p+U3+p, U3+p=U4+p,
U4+p= 2q2Ta2(1 +inΓ σ) (F U−1+p+F∗U11+p) + (q2+inσ−Γ n2σ2)U3+P, U5+p=U6+p,
U6+p= (1 +inΓ σ) dF
dxU−5+p+dF∗ dx U7+p
+ (q2+inσ−Γ n2σ2)U5+p.
(A.2)
The associated boundary conditions are
U1+p=U2+p=U5+p= 0, at x= 0,1. (A.3)
The solution of the obtained boundary value problem (A.2) is sought as a superposition of 6(2N + 1) linearly independent solutions Ui+pj wherej= 1,2,3, . . . ,2(N+ 1). Thus,
Ui+p=
j=2(N+1)
j=1
Cj+pUi+pj+p. (A.4)
Imposing the following initials values,
Ui+pj+p(0) =δij =
0, i=j,
1, i=j, (A.5)
a set of 3 + 6N independent solutions satisfying the boundary conditions at x= 0 is constructed by a Runge-Kutta numerical scheme
⎧⎪
⎪⎨
⎪⎪
⎩
U1+p(0) =C1+pU1+p1+p(0) = 0 =⇒C1+p= 0, U2+p(0) =C2+pU2+p2+p(0) = 0 =⇒C2+p= 0, U3+p(0) =C5+pU5+p5+p(0) = 0 =⇒C5+p= 0.
(A.6)
Finally, the solution of the system (A.2) is
Ui+p=
l=2N
l=1
C3+6lUi+p3+6l+C4+6lUi+p4+6l+C6+6lUi+p6+6l
. (A.7)
A linear combination of these solutions satisfying the boundary conditions at the other extremex= 1,
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
U1+p(1) =
l=2N
l=1
C3+6lU1+p3+6l(1) +C4+6lU1+p4+6l(1) +C6+6lU1+p6+6l(1)
= 0,
U2+p(1) =
l=2N
l=1
C3+6lU2+p3+6l(1) +C4+6lU2+p4+6l(1) +C6+6lU2+p6+6l(1)
= 0,
U3+p(1) =
l=2N
l=1
C3+6lU5+p3+6l(1) +C4+6lU5+p4+6l(1) +C6+6lU5+p6+6l(1)
= 0,
(A.8)
leads to a homogeneous algebraic system for the coefficients of the combination. A necessary condition for the existence of nontrivial solution is the vanishing of the determinant
U13 U14 U16 · · · U13+12N U14+12N U16+12N U23 U24 U26 · · · U23+12N U24+12N U26+12N U53 U54 U56 · · · U53+12N U54+12N U56+12N
· · · · · · · · ·
· · · · · · · · ·
· · · · · · · · ·
U1+12N3 U1+12N4 U1+12N6 · · · U1+12N3+12N U1+12N4+12N U1+12N6+12N U2+12N3 U2+12N4 U2+12N6 · · · U2+12N3+12N U2+12N4+12N U2+12N6+12N U5+12N3 U5+12N4 U5+12N6 · · · U5+12N3+12N U5+12N4+12N U5+12N6+12N
= 0, (A.9)
which can be formally written as
(σ, Γ, q, Ta) = 0.
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