Effect of Coriolis force on the thermosolutal convection threshold in a rotating annular Hele-Shaw cell

O R I G I N A L

Effect of Coriolis force on the thermosolutal convection threshold in a rotating annular Hele-Shaw cell

K. Souhar^• S. Aniss

Received: 14 June 2010 / Accepted: 27 June 2011 / Published online: 16 July 2011 ÓSpringer-Verlag 2011

Abstract In this study, the linear stability analysis is used to determine the onset of thermosolutal convection in fluids confined in rotating annular Hele-Shaw cell. The fluid layer is submitted to radial gradients of temperature and con- centration. The effects of both Coriolis force and curvature parameter on the stationary and oscillatory convection are investigated when the Prandtl number is of the order of unity or larger than unity.

List of symbols

C^* Dimensional concentration C Dimensionless concentration

C_b Dimensional concentration of the basic state e Thickness of the cell

E Eckman number

g Gravity

k Unit vector upward

Le Lewis number

m Azimuthal wave number P^* Dimensional pressure

P_b Dimensional pressure of the basic state P Dimensionless pressure

Pr Prandtl number R₁ Inner radii of the cell R₂ Outer radii of the cell Ra Thermal Rayleigh number Rs Solutal Rayleigh number t^* Dimensional time

t Dimensionless time

T^* Dimensional temperature

T Dimensionless temperature

T_b Dimensional temperature of the basic state u^*,v^*,w^* Dimensional velocity components

u,v,w Dimensionless velocity components r^*,h, z^* Dimensional spatial coordinates V^* Dimensional velocity vector Greek symbols

a Solute coefficient of volume expansion b Thermal coefficient of volume expansion d Curvature parameter

l Dynamic viscosity m Kinematic viscosity e Aspect ratio of the cell q Density

j_s Solute diffusivity j Thermal diffusivity r Grow rate of perturbations X Angular velocity

1 Introduction

There has been a steady interest in free convection driven by centrifugal body force in rotating homogeneous fluid. This force may generate free convection in the same manner as the gravity force causes natural convection. Thermal con- vection driven by centrifugal force, in the absence of gravity, in the case of a fluid filled gap between two co-rotating coaxial cylinders kept at different temperatures was studied in [1–3]. In these works, different models have been extensively used for understanding thermal atmospheric instabilities. This configuration in which the temperature gradient resulting from the conditions imposed on the boundaries is collinear with centrifugal body force was also K. SouharS. Aniss (&)

Faculty of Sciences Ain-Chock, Laboratory of Mechanics, University Hassan II, BP 5366, Maaˆrif, Casablanca, Morocco e-mail: s.aniss@fsac.ac.ma

DOI 10.1007/s00231-011-0849-x

studied in the case of a rotating fluid saturated porous layer placed at an arbitrary positive distance from the rotation axis in the presence of gravity [4]. In this study, the effect of gravity and centrifugal driven convection was investigated analytically neglecting Coriolis force, the effect of Coriolis force was discussed in [5]. Tagare [6] has investigated the onset of instabilities in rotating magnetoconvection for viscous fluid and for inviscid fluid has been investigated by Tagare and Rameshwar [7]. Ramezani et al. [8] have ana- lyzed the linear stability problem of thermal convection in a homogeneous Newtonian liquid confined in a horizontal annular Hele-Shaw cell subject to a constant rotation and submitted to a centrifugal gradient of temperature. In this study, the analysis was focused on the effect of Coriolis force and curvature parameter on the stationary convective threshold. It was shown that the Coriolis force and curvature parameter of the cell have a stabilizing effect and the wave number remains independent of these physical and geo- metrical parameters. Recently, Govender have investigated the Coriolis effect on the stability of centrifugally driven convection in a rotating anisotropic porous layer subject to gravity [9]. He has estimated the critical Rayleigh number for the onset of convection in the presence of thermal and mechanical anisotropy. Malashetty et al. [10] have studied the effect of rotation on the onset of convection in an anisotropic porous layer using a thermal non-equilibrium model.

When a fluid has two diffusive components, heat and salt for instance, convection depends on the solute strati- fication as well as the thermal stratification. For a hori- zontal fluid layer, in which both a vertical temperature gradient and a vertical concentration gradient are imposed, the buoyancy due to the temperature gradient relative to the viscous dissipation is measured by the thermal Rayleigh number, whereas the buoyancy due to the concentration is measured by the solutal Rayleigh number. Nield [11] and Baines and Gill [12] have used a linearized density with respect to temperature and concentration and have ana- lyzed the linear stability of the conduction state. They have revealed that the primary convection sets in for a thermal Rayleigh number above a critical value that is a function of the solutal Rayleigh number. The onset of convection is oscillatory for a solutal Rayleigh number above a positive threshold, and steady for a solutal Rayleigh number below the threshold. For both types of onset, the critical thermal Rayleigh number increases as the solutal Rayleigh number increases. That is, the buoyancy due to the negative con- centration gradient in the upward direction stabilizes the conduction state, and the positive gradient destabilizes it.

From a practical point of view, the double diffusive mixed convection in rotating system is very important in many engineering applications and several studies were reported on the effect of rotation in double diffusive

convection. For instance, Tagare et al. [13] have investi- gated linear and non linear properties of thermohaline convection in rotating fluids and discussed the stability regions of travelling and standing waves. Sharma et al. [14]

have studied the effect of rotation on thermosolutal con- vection in ferromagnetic fluid saturating a porous medium layer heated and soluted from below in the presence of a uniform vertical magnetic field. It has been shown in [14]

that the oscillatory modes exist due to the presence of stable solute gradient and rotation. Recently, using both linear and non-linear stability analyses Malashetty and Rajashekhar [15] have analyzed the double diffusive con- vection in a fluid-saturated rotating porous layer heated from below and cooled from above when the fluid and solid phases are not in local thermal equilibrium. In this study, the Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation and it has been shown that there is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary.

In this study, we further extend the results reported by Ramezani et al.[8] to the configuration of a binary mix- tures confined in a rotating and horizontal annular Hele- Shaw cell. The inner and outer boundaries of the cell are submitted to constant temperatures and concentrations.

Generally, in the real mixtures the concentration and thermal diffusive modes are usually coupled through the Soret and Dufour effects. Here, we restrict our analysis to the linear thermohaline (thermosolutal) instability in which both effects are absent. Hence, the gradients of temperature and concentration are imposed and are collinear to the centrifugal force. The objective is to investigate the com- bined effect of Coriolis force and curvature parameter on the stationary and oscillatory convection.

2 Formulation

Consider a Newtonian fluid confined in an annular Hele- Shaw cell (Fig.1). Denote byR₁andR₂the inner and outer radii of the cell, respectively, and by eits thickness. The cell is subject to a constant angular velocity X¼Xk around its vertical symmetry axis. The fluid is bounded vertically by two thermally insulating walls located at z¼ ^e₂. The inner circular boundary is kept at a constant temperatureT₁with solute concentrationC₁, and the outer one is kept at a constant temperature T₂ with solute con- centration C₂. We denote by e¼e=ðR2R1Þ 1 the aspect ratio of the cell. The curvature parameter is defined byd¼ ðR2R1Þ=R1:In this study we neglect the gravity assuming thatgX²ðR2R1Þ. The governing equations

in the Boussinesq approximation are written in the relative frame (O,e_r,e_h,k) and are

r:V¼0 ð1Þ

qDV

Dt ¼ rPþlDVqX^ ðX^OMÞ 2qX^V ð2Þ DT

Dt ¼jDT ð3Þ

DC

Dt ¼jsDC ð4Þ

The pressure, temperature and concentration of the basic motionless state are given by

rP_bþqX^ ðX^OMÞ ¼0 ð5Þ

T_bT1

T2T1

¼ ln _R^r

2R1

ln ^R_R²

1

ln _R^R1

2R1

ln ^R_R²

1

ð6Þ

C_bC1

C2C1

¼ ln _R^r

2R1

ln ^R_R²

1

ln _R^R1

2R1

ln ^R_R²

1

ð7Þ

To introduce a perturbation parameter involving only the aspect ratio of the cell, e, we perform a dimensional analysis by means of an appropriate choice of scales used in convection problems in Hele-Shaw cell [16,17]. Thus, the time is scaled by ^R2R_j ¹, the vertical coordinate z^* is scaled bye, the dimensionless radial coordinate is defined byr¼_R^rR1

2R1, the velocity field Vðu;v;wÞis scaled by

j R₂R₁;_R ^j

2R₁;_R^ej

2R₁

;the pressure is scaled by^qmj_e2, and the

temperature and concentration are, respectively, scaled by T₂-T₁ and C₂-C₁. Hereafter, we denote by Ra¼

bR1X²ðR2R1Þ³ðT2T1Þ

mj e²d the thermal Rayleigh number, Rs¼

aR1X²ðR2R1Þ³ðC2C1Þ

mjs e²dthe solutal Rayleigh number,E¼_Xe^m2

the Eckman number,Le¼j=js the Lewis number, Pr¼^m_j the Prandtl number. Hence, the linear system of Eqs. (1–4) is written as

ou orþ d

1þdruþ d 1þdr

ov ohþow

oz ¼0 ð8Þ

e²Pr¹ou

ot ¼ oP

orþe²D2uþo²u oz²þ2

Ev þe² d

1þdr 2

2ov ohu

Ra1þdr d TþRs

Le 1þdr

d C ð9Þ

e²Pr¹ov

ot ¼ d 1þdr

oP

ohþe²D2vþo²v oz²2

Eu þe² d

1þdr ²

2ou orv

ð10Þ

e⁴Pr¹ow

ot ¼ oP

ozþe⁴D2wþe²o²w

oz² ð11Þ

e²oT

ot ¼ e² d 1þdr

1

ln 1ð þdÞuþe²D2Tþo²T

oz² ð12Þ e²oC

ot ¼ e² d 1þdr

1

ln 1ð þdÞuþ 1

Le e²D2Cþo²C oz²

ð13Þ whereD2¼_or^o22þ_1þdr^d _or^oþ_1þdr^d 2

o² oh².

3 Onset of thermosolutal convection-effect of the rotation

3.1 Hele-Shaw approximation

In this study we consider the formulation corresponding to Pr1 or Pr1. A first approximation is obtained from the system of Eqs. (8–13) by settingðe²¼0Þ. Remark that in this situation, the term ^oV_ot disappears. Let us denote by u_o,v_o,w_o,P_o,T_oandC_o, the solution of such approxima- tion. From Eq. (11), the pressure is independent ofz. Also, from Eqs. (12, 13) and using the adiabatic condition, the temperatureT_oand concentrationC_oare also independent of z. Thus, the system (8–13) becomes

ouo

or þ d

1þdruoþ d 1þdr

ovo

oh þowo

oz ¼0 ð14Þ

Fig. 1 Sketch of the liquid layer confined in rotating Hele-Shaw cell subjected to temperature and thermosolutal gradients

oPo

or Ra1þdr d T_oþRs

Le 1þdr

d C_oþ2

Ev_oþo²u_o oz² ¼0

ð15Þ d

1þdr oPo

oh 2

Eu_oþo²v_o

oz² ¼0 ð16Þ

oPo

oz ¼0 ð17Þ

Using the nonslip boundary conditions at the horizontal walls, u_o ¼v_o¼0 at z¼ ¹₂, Eqs. (15) and (16) are integrated, with respect to the variable z, to obtain the velocity field determined first in [8] without the solutal term. The averaged components of velocity can be written as

uo¼ d 1þdrM1

oPo

oh þM2

oPo

or þ1þdr

d RaTo1þdr d

Rs LeCo

ð18Þ

v_o¼ d

1þdrM₂oPo

oh M1

oPo

or þ1þdr

d RaTo1þdr d

Rs LeCo

ð19Þ

wo ¼0 ð20Þ

where M1ðcÞ ¼_8c¹2 1þ^sinðcÞcosðcÞþsinhðcÞcoshðcÞ 2cðcosh²ðcÞsin²ðcÞÞ

h i

, M2ðcÞ ¼

1 8c²

sinðcÞcosðcÞsinhðcÞcoshðcÞ 2cðcosh²ðcÞsin²ðcÞÞ

h i

;c¼₂^p1ffiffiffi_E:

The averaged continuity equation with respect to the variablezis now written as

D2Po¼ Ra 2Toþ1þdr d

oTo

or M₁ M2

oTo

oh

þRs

Le 2Coþ1þdr d

oCo

or M1

M2

oCo

oh

ð21Þ At the first order ðe²¼0Þ, Eqs. (14–17) cannot be coupled to the energy and concentration Eqs. (12, 13).

Therefore, the energy and concentration Eqs. (12,13) are exploited at the ordere²by using the accurate expansions u¼u_oþe²u₁;T ¼T_oþe²T₁;C¼C_oþe²C₁ ð22Þ whereT₁andC₁verify the adiabatic conditions^oT_oz¹¼0 and

oC1

oz ¼0 at z¼ 1=2. Inserting expressions (22) into Eqs.

(12,13) and keeping only terms of order e², we obtain a system which is averaged with respect to the variablez

D2To Ra M2

ln 1ð þdÞTo¼ Rs M2

Le 1 ln 1ð þdÞCo

þ d 1þdr

1 ln 1ð þdÞ

d

1þdrM₁oPo

oh þM₂oPo

or

þoTo

ot ð23Þ

D2Coþ Rs M2

ln 1ð þdÞCo¼ Ra Le ln 1ð þdÞM2To

þ Le ln 1ð þdÞ

d 1þdr

d

1þdrM₁oPo

oh þM₂oPo

or

þLeoCo

ot ð24Þ In contrast to the original system (8–13), which needs eight boundary conditions, the system to the first order (21) and (23,24), resulting from the Hele-Shaw approximation, requires only the relevant six boundary conditions

d 1þdrM1

oPo

oh þM2

oPo

or ¼To¼Co ¼0 atr¼0;1 ð25Þ To perform a stability analysis, we seek the solutions of the system (21) and (23–24) in terms of normal mode as

Poðr;h;tÞ Toðr;h;tÞ Coðr;h;tÞ 0

@

1 A¼

fðrÞ gðrÞ hðrÞ 0

@ 1

Ae^{i mh}e^irt ð26Þ

wheremdenotes the azimuthal wave number which is an integer and r¼rRþirI (i²¼ 1). To simplify the boundary conditions (25), we introduce the change of variable KðrÞ ¼i m M1f þ^1þd_d ^rM2df

dr. We obtain the following system

DmðKÞ ¼ 1þdr d 2

M2 Rad²g dr²Rs

Le d²h dr²

51þdr

d M2 Radg drRs

Le dh dr

4M²₂þm²M²₁ M2

Ra gRs Leh

ð27Þ

DmðgÞ Ra M₂

lnð1þdÞg¼ Rs Le

1

ln 1ð þdÞM2h þ d

1þdr 2

1 lnð1þdÞK

þi mrIg ð28Þ D_mðhÞ þ Rs M2

lnð1þdÞh¼ Ra Le lnð1þdÞM₂g þ Le

lnð1þdÞ d 1þdr 2

K

þi mrILe h ð29Þ where Dm¼_dr^d22þ_1þdr^d _dr^d m²_1þdr^d 2

. This system is subject to the following simplified boundary conditions K¼g¼h¼0 atr¼0;1 ð30Þ 3.2 Numerical results

The numerical method consists to transform the system of Eqs. (27, 29) into a set of six differential equations. The

solution of this boundary value problem is sought as a superposition of six linearly independent solutions fol- lowing a method often used in stability problems [18].

Each independent solution which satisfies the boundary conditions at r=0 is constructed by a Runge–Kutta numerical scheme of fourth order. A linear combination of these solutions satisfying the boundary conditions at the other extreme, r=1, leads to a homogenous algebraic system for the coefficients of the combination. A necessary condition for the existence of nontrivial solutions is the vanishing of the determinant, which defines a characteristic equation, relating the cell curvature coefficient d, the Eckman number E, the thermal Rayleigh number Ra, the solutal Rayleigh numberRs, the Lewis numberLe, and the azimuthal wave numberm.

In the case of stationary convection,rR=0 andrI=0, we present in Fig.2, for two values of the solutal Rayleigh numberRs, the variation of the critical Rayleigh number, Ra_c, versusE^-1. It turns out that the effect of buoyancy due to the concentration is important by comparing with the results in [8]. In Fig.3, the evolution of the stationary thermal critical Rayleigh number with respect to the solutal Rayleigh number for different values of Eckman’s number and for d =0.1 is presented. The results show that Ra_c increases with the inverse of Eckman’s number, and then the effect of Coriolis force on the onset of the thermosol- utal instability is stabilizing. Figures4and5show that this stabilizing effect of the Coriolis force increases with increasing the curvature parameter. Figure6illustrates the variation of the thermal critical Rayleigh number with respect to the solutal Rayleigh number forE=100 (small values of rotation) and ford =0.1, 0.5 and 0.9. Here, the

10^-1 10⁰ 10¹ 10² 10³ 10⁴

10¹ 10² 10³

E^-1 Rac

Rs= 0 RS=150 δ^{= 0.1}

Fig. 2 Evolution of the stationary critical thermal Rayleigh number, Ra_c, versus E^-1 for d = 0.1: Solid line R_s= 150 (presence of a concentration gradient),dashed line R_s= 0 (absence of concentration gradient, [5])

-500 0 500 1000

-1000 -500 0 500 1000 1500

Rs Rac

E=10² E=10^{- 1} E=10^{- 3}

Unstable

Stable δ^=0.1

Fig. 3 Stability diagram for stationary convection: Critical thermal Rayleigh number,Ra_c, versus the solutal Rayleigh number,R_s, ford= 0.1 and for different values of Eckman number

-500 0 500 1000

-1000 0 1000 2000

Rs Ra c

E=10² E=10^{- 1} E=10^{- 3}

Stable Unstable

δ^=0.5

Fig. 4 Stability diagram for stationary convection: Critical thermal Rayleigh number,Ra_c, versus the solutal Rayleigh number,R_s, ford= 0.5 and for different values of the Eckman number

-500 0 500 100

-1000 -500 0 500 1500 2500

Rs Ra c

E=10² E=10^-1 E=10^-3

Stable δ^=0.9

Unstable

Fig. 5 Stability diagram for stationary convection: Critical thermal Rayleigh number,Ra_c, versus the solutal Rayleigh number,R_s, ford= 0.9 and for different values of the Eckman number

weak stabilizing effect of the curvature on the thermosol- utal convection threshold is observed. We present in Figs.7 and 8, the stability diagram of the onset of ther- mosolutal convection for intermediate and for small values of the Eckman number. These curves show that the cur- vature has a stabilizing effect and this effect increases dramatically with the curvature parameter for small Eck- man number (high rotation). In Fig.9, we present the sta- tionary and oscillating stability regions for different values of the Eckman numberE. These curves show that upon a certain values of the solutal Rayleigh number Rs, the critical Rayleigh number Ra_c corresponds to oscillating convection. Here, we are in the presence of the oversta- bility. The regions corresponding to the oscillating con- vection narrows dramatically as the Eckman number

decreases. Note that the values of the solutal Rayleigh number corresponding to the threshold of oscillating ther- mosolutal convection increases as the Eckman number decreases.

3.3 Asymptotic study:E1 andd 1

In this case the averaged velocity field (18,19) is given by uo¼ 1

12 oPo

or þ1

dRaTo1 d

Rs LeCo

ð31Þ

vo¼ 1

12doPo

oh ð32Þ

with this approximation the averaged continuity, energy and concentration equations are

-500 0 500 1000

-1000 -500

0 500 1000

Rs Rac

δ^=0.1 δ^=0.5 δ^=0.9

Stable Unstable E=10²

Fig. 6 Stability diagram for stationary convection: Critical thermal Rayleigh number,Ra_c, versus the solutal Rayleigh number,R_s, forE= 100 and for different values of the curvature parameter

-1000 -500 0 500 1000

-500 0 500 1000 1500

Rs Ra c

δ^{= 0.1} δ= 0.5 δ= 0.9

Stable Unstable

E=0.1

Fig. 7 Stability diagram for stationary convection: Critical thermal Rayleigh number,Ra_c, versus the solutal Rayleigh number,R_s, forE= 0.1 and for different values of the curvature parameter

-1000 -500 0 500 1000

-1000 0 1000 2000 2500

Rs Rac

δ^=0.1 δ^=0.5 δ^=0.9 Unstable

Stable E=10^{- 3}

Fig. 8 Stability diagram for stationary convection: Critical thermal Rayleigh number,Ra_c, versus the solutal Rayleigh number,R_s, forE= 10^-3and for different values of the curvature parameter

-200 0 200 500 1000 1500

0 200 500 1000 1250 1500

Rs Rac

Unstable

Stable δ=0.1

Oscillating Unstability

E=100 E=10^-2 E=5 10^-3

Fig. 9 Stability diagram for stationary and oscillating convection:

Critical thermal Rayleigh number, Ra_c, versus the solutal Rayleigh number,R_s, ford= 0.1 and for different values of the Eckman number

D2Po¼ Ra d

oTo

or þ Rs dLe

oCo

or ð33Þ

D2ToþRa

12dTo¼ Rs

12dLeCo 1 12

oPo

or þoTo

ot ð34Þ

D2CoRs

12dCo¼ Ra Le

12d ToLe 12

oPo

or þLeoCo

ot ð35Þ

The system (33–35) is subject to the boundary conditions

oPo

or ¼T_o¼C_o¼0 atr¼0;1 ð36Þ The solution,P_o,T_oandC_ois written in normal modes Poðr;h;tÞ

Toðr;h;tÞ Coðr;h;tÞ 0

@

1 A¼

A1

A2

A3

0

@ 1

AsinðnprÞe^{i mh}e^rt ð37Þ

Substituting solution (37) into the system (33–35), we deduce the characteristic equation forr.

r²þ 1

12ðp²þd²m²Þ 12ð1þLeÞðp²þd²m²Þ þdm²ðRsLe RaÞ

rþ 1

12Leh12ðp²þd²m²Þ² þdm²ðRsRaÞ

¼0 ð38Þ

The neutral stability curve corresponds to r=irI. After, the real and imaginary parts of Eq. (38) are written as follows

r²_I þ 1

12Leh12ðp²þd²m²Þ²þdm²ðRsRaÞi

¼0 ð39Þ rI ð1þLeÞðp²þd²m²Þ þdm²ðRsRaÞ

¼0 ð40Þ For the stationary convection, r_I=0, we obtain the marginal stability condition

RaRs¼12dðp²þd²m²Þ²

d²m² ð41Þ

Thus, the critical parameters corresponding to the threshold of stationary convection are

mc¼p

d ð42Þ

Ra_cRs¼48dp² ð43Þ

In the case of oscillatory convection, rI 6¼0; the overstability is possible only if r²_I[0: Let us consider the overstability case form¼m_c¼^p_d:We obtain

RacRs\48dp² ð44Þ

The overstable marginal state, exists below the neutral stability line RaRs¼48dp² in the Ra-Rs plane.

Equating the two Eqs. (39) and (40), we have also the condition for realrI.

Ra[Rs Le

þ2ð1þLeÞ

Le d ð45Þ

Then from the inequations (44, 45), the overstable marginal state occurs below the neutral stability line Ra -Rs=48dp²and above the lineRa¼^Rs_Leþ^2ð1þLeÞ_Le din the Ra-Rsplane. We illustrate in Fig.10the numerical and analytical results, for d =0.1 and E =10³. These asymptotic results agree with the numerical one.

4 Conclusion

In this work, we have investigated the linear thermosolutal instability in a horizontal liquid layer confined in a rotating annular Hele-Shaw cell. The gradients of temperature and concentration are in the same direction that centrifugal force or in the opposite ones. We have analyzed the effects of both Coriolis force and curvature parameter on the sta- tionary and oscillatory instabilities when the Prandtl number is of the order of unity or larger than unity. We have shown that the critical thermal Rayleigh number increases in the stationary case with the inverse of the Eckman numberE^-1. Thus, the effect of the Coriolis force on the thermosolutal convection threshold is stabilizing. It turns out that the curvature parameter has also a stabilizing effect. For oscillating convection, we have shown that upon a certain value of the solutal Rayleigh number,Rs, the first instability is oscillating, and the overstability region nar- rows dramatically when Eckman number decreases.

Acknowledgments The authors greatly acknowledge financial support from the Centre National de la Recherche Scientifique (CNRS-France) and the Centre National pour la Recherche Scientif- ique et Technique (CNRST-Morocco).

-500 0 500 1000

-1000 -500 0 500 1000

Rs

Rac Unstable

Stable

Oscillating unstability Numeric al result

δ^=0.1 E=10³

Analytical result

Fig. 10 Stability diagram for stationary and oscillating convection:

Analytical and numerical results forE= 10³and ford= 0.1

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