Stability of a pulsed Taylor-Couette flow in a viscoelastic fluid

(1)

321

Article

Nihon Reoroji Gakkaishi Vol.42, No.5, 321~327 (Journal of the Society of Rheology, Japan)

©2014 The Society of Rheology, Japan

1. INTRODUCTION

Extensive experimental and theoretical investigations have been conducted in the modulation of the boundary conditions in the Taylor-Couette problem

^1-10)

with a Newtonian fluid.

These works have considered the configuration in which the angular velocities of the inner and outer cylinders are respectively, W

1

+ e

1

cos(wt) and W

2

+ e

2

cos(wt). In this study, we are interested to the case considered by Carmi and Tustaniwskyj

⁶⁾

corresponding to a zero mean modulation, W

₁

= W

₂

= 0, in phase, e

₁

= e

₂

. This configuration has been revisited theoretically by Aouidef et al.

^11-12-14)

and Tennakoon et al.

¹³⁾

and for the first time experiments have been achieved by these authors. The experimental results, corresponding to the modulation in phase, considered with a narrow gap approximation, have been in good agreement with the linear stability analysis and shown that the threshold of instability is above that the one determined by Carmi and Tustaniwskyj.

⁶⁾

Furthermore, all results in references

^11-13)

, have shown that the flow is less unstable in the limit of low and high frequency while destabilization is maximum for an intermediate frequency.

In this paper, we consider the stability analysis of a pulsating flow of a viscoelastic fluid in the Taylor-Couette geometry.

This study is restricted to the small gap approximation and the purpose is to analyze, in the case of a zero mean modulation in phase, the effect of frequency modulation and elasticity characterized by Deborah number on the threshold of instability.

The paper is organized as follows. In section 2, we determine the basic pulsed flow generated by the modulation in phase. After that, we perform in section 3, a linear stability analysis in which asymptotic critical parameters of instability in the limit of low and high frequency are determined. In section 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 section 5.

Section 6 is the conclusion.

2. BASE FLOW

Consider an incompressible viscoelastic fluid filling the annulus between two infinitely long cylinders of radii R

1

and R

2

= R

1

+ d where d is the gap length (Fig. 1). The angular velocity of each cylinder is W

₀

cos( w t*), W

₀

and w denote respectively the amplitude and the frequency of the modulated rotation. The governing equations are the conservation equations for momentum and mass

Stability of a Pulsed Taylor-Couette Flow in a Viscoelastic Fluid

Mehdi R

iahi

1, Saïd a

niss^*,†

, Mohamed O

uazzani

T

Ouhami^*

, and Salah s

kali

l

ami^**

*

Université Hassan II, Faculty of Sciences Aïn-Chock, Laboratory of Mechanics, B.P.5366 Mâarif, Casablanca, Morocco

**

Lemta-UMR CNRS 7563-Ensem, 2, avenue de la Forêt de Haye, BP 160, Vandoeuvre-les-Nancy, 54504, France.

(Received : July 5, 2014)

The linear stability analysis of a pulsed flow in a linear Maxwell fluid confined in the Taylor-Couette system is investigated. Both cylinders are subjected to modulated rotation in phase with equal amplitude and frequency. 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 a stable configuration. The basic state is potentially unstable at an intermediate frequency and it becomes more unstable as Deborah number increases. At high frequency limit, the Deborah number has a strong destabilizing effect. These numerical results are in good agreement with the asymptotic solutions obtained in the limit of high and low frequencies.

Key Words: Linear stability / Oscillating flow / Taylor-Couette flow / Maxwell model / Floquet theory

† E-mail: s.aniss@etude.univcasa.ma

(2)

322 Nihon Reoroji Gakkaishi Vol.42 2014

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}^∗_∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}^∗_∗

� (4) 𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

^𝑉𝑉_𝑟𝑟_∗^∗

�� (5)

0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}_∗^∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

_𝑑𝑑2^𝑡𝑡^∗

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

_𝑅𝑅^𝑉𝑉^∗

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

_{𝜌𝜌 𝑅𝑅}^𝑃𝑃^∗

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕²^𝑉𝑉^𝐵𝐵

𝜕𝜕𝑡𝑡²

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

1

(𝑥𝑥) =

cos(𝛾𝛾𝛾𝛾𝑥𝑥)cosh�𝛾𝛾𝛾𝛾(1−𝑥𝑥)�+cos(𝛾𝛾𝛾𝛾(1−𝑥𝑥))cosh(𝛾𝛾𝛾𝛾𝑥𝑥)

cos(𝛾𝛾𝛾𝛾) + cosh(𝛾𝛾 𝛾𝛾)

(12)

𝑉𝑉

₂

(𝑥𝑥) =

sin(𝛾𝛾𝛾𝛾𝑥𝑥)sinh�𝛾𝛾𝛾𝛾(1−𝑥𝑥)�+sin(𝛾𝛾𝛾𝛾(1−𝑥𝑥))sinh(𝛾𝛾𝛾𝛾𝑥𝑥)

(13)

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}^∗_∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1) (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}^∗_∗

� (4)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}_∗^∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(2) where V

^*

is the velocity vector, t is the extra stress tensor and P

^*

is the pressure. There are many different models proposed for viscoelastic fluids. We describe the viscoelastic effect of the fluid using Maxwell’s model which can be represented by a purely viscous damper and a purely elastic spring connected in series

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}^∗_∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}^∗_∗

� (4) 𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}_∗^∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

1

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(3) We denote by D the rate of strain tensor defined by D =

∇V

^*

+ ∇

^t

V

^*

. We designate by r the density, m the dynamic viscosity and l 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, plays a significant role in developing purely elastic instabilities of curved Streamlines.

These instabilities discussed in references

^15-17)

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. Furthermore, assuming that the gap width d is small compared to the radius R

1

of the inner cylinder, we neglect curvature and we use the Cartesian coordinates instead of the radial coordinate.

In dimensional cylindrical-polar coordinates (r

^*

, q, 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 axisymetric and then it is written as V

^*

= (0, V

^*

, 0) where P

^*

and V

^*

are q independents.

Under these assumptions, a combination of the equations (1)- (2)-(3) leads to the following system

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4) 𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω0

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ωo2

(7)

Γ

𝜕𝜕𝑡𝑡²

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

1

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

2

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(4)

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5) 0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

cos(𝛾𝛾𝛾𝛾𝑥𝑥) cosh�𝛾𝛾𝛾𝛾(1−𝑥𝑥)�+cos(𝛾𝛾𝛾𝛾(1−𝑥𝑥)) cosh(𝛾𝛾𝛾𝛾𝑥𝑥)

(12)

𝑉𝑉

2

(𝑥𝑥) =

sin(𝛾𝛾𝛾𝛾𝑥𝑥) sinh�𝛾𝛾𝛾𝛾(1−𝑥𝑥)�+sin(𝛾𝛾𝛾𝛾(1−𝑥𝑥)) sinh(𝛾𝛾𝛾𝛾𝑥𝑥)

(13)

(5)

𝜕𝜕𝑡𝑡

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5) 0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(6) We introduce the following dimensionless variables

𝜕𝜕𝑡𝑡^∗

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4) 𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

^𝑉𝑉_𝑟𝑟^∗_∗

�� (5)

0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

^𝑡𝑡_𝑑𝑑2^∗

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

1

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(7) Equation (6) shows that the pressure is independent of z. Assuming that the gap width d is small compared to the radius R

1

of the inner cylinder and using the small-gap approximation in which all terms of order d/R

1

are neglected, the dimensionless azimuthal velocity satisfies equation (5) which is now in the form

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4) 𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω0

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ωo2

(7)

Γ

𝜕𝜕𝑡𝑡²

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

1

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

2

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(8) with the boundary conditions

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4) 𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω0

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ωo2

(7)

Γ

𝜕𝜕𝑡𝑡²

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

1

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

2

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(9) The parameters s = w d

²

/n and G = l n /d

²

denote respectively the frequency number and the Deborah number.

s is the ratio of the viscous diffusive time and the period of modulation whereas G is the ratio of the relaxation time and the viscous diffusive time. The pressure of the base flow, P

_B

(x,t), is obtained from equation (4) which is written as

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

𝐵𝐵

=

1Ω₀

, 𝑃𝑃

𝐵𝐵

=

1𝑑𝑑Ωo2

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(10) The solution of equations (8) and (9) is

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

^𝜕𝜕_{𝜕𝜕𝑡𝑡}²^𝑉𝑉₂^𝐵𝐵

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(11) Here, the functions V

1

and V

2

are given by

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4) 𝜌𝜌 �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

_𝐵𝐵

=

1Ω₀

, 𝑃𝑃

_𝐵𝐵

=

1𝑑𝑑Ω_o²

(7)

Γ

𝜕𝜕𝑡𝑡²

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

_𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

1

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(12)

𝜌𝜌 �

^{𝜕𝜕𝑽𝑽}_{𝜕𝜕𝑡𝑡}_∗^∗

+ 𝑽𝑽

^∗

∙ ∇𝑽𝑽

^∗

� = − ∇𝑃𝑃

^∗

+ ∇ ∙ 𝝉𝝉 (1)

∇ ∙ 𝑽𝑽

^∗

= 0 (2)

𝝉𝝉 + λ

_{𝜕𝜕𝑡𝑡}^{𝜕𝜕𝝉𝝉}_∗

= 𝜇𝜇 𝑫𝑫 (3)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �−

^𝑉𝑉_𝑟𝑟^∗2_∗

� = − �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑟𝑟}_∗^∗

� (4)

𝜌𝜌 �1 + λ

^𝜕𝜕

𝜕𝜕𝑡𝑡^∗

�

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}_∗^∗

= 𝜇𝜇 �

^𝜕𝜕_{𝜕𝜕𝑟𝑟}²^𝑉𝑉_∗2^∗

+

_{𝜕𝜕𝑟𝑟}^𝜕𝜕_∗

�

�� (5)

0 = − �1 + λ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕_∗

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑧𝑧}^∗_∗

� (6)

𝑥𝑥 =

^𝑟𝑟^∗^{− 𝑅𝑅}_𝑑𝑑 ¹

, 𝑡𝑡 =

𝜈𝜈

, 𝑉𝑉

𝐵𝐵

=

1Ω₀

, 𝑃𝑃

𝐵𝐵

=

1𝑑𝑑Ωo2

(7)

Γ

𝜕𝜕𝑡𝑡²

+

^{𝜕𝜕𝑉𝑉}_{𝜕𝜕𝑡𝑡}^𝐵𝐵

=

^𝜕𝜕_{𝜕𝜕𝑥𝑥}²^𝑉𝑉₂^𝐵𝐵

(8)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 0, 𝑡𝑡) = 𝑉𝑉

_𝐵𝐵

(𝑥𝑥 = 1, 𝑡𝑡) = cos(𝜎𝜎𝑡𝑡) (9)

�1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� 𝑉𝑉

𝐵𝐵

= − �1 + Γ

_{𝜕𝜕𝑡𝑡}^𝜕𝜕

� �

^{𝜕𝜕𝑃𝑃}_{𝜕𝜕𝑥𝑥}^𝐵𝐵

� (10)

𝑉𝑉

_𝐵𝐵

(𝑥𝑥, 𝑡𝑡) = 𝑉𝑉

₁

(𝑥𝑥) cos(𝜎𝜎𝑡𝑡) + 𝑉𝑉

₂

(𝑥𝑥, 𝑡𝑡) sin(𝜎𝜎𝑡𝑡) (11)

𝑉𝑉

₁

(𝑥𝑥) =

(12)

𝑉𝑉

₂

(𝑥𝑥) =

(13)

(13) where g = , b = ( sG+ )

^1/2

and

x = ( - sG+ )

^1/2

.

The parameter g expresses the ratio of two lengths g = d/d

N

where d

N

= is the thickness of the stokes layer for a Newtonian fluid. The parameters b and x express also the ratio Fig. 1. The Sketch of the modulated Taylor-Couette geometry.

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