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Research Article

MHD free convection in a vertical slit micro‑channel with super‑hydrophobic slip and temperature jump: non‑linear Boussinesq approximation approach

Basant Kumar Jha1 · Bello Jibril Gwandu1,2

© Springer Nature Switzerland AG 2019

AbstractAn exact solution is obtained for natural convection flow of electrically conducting fluid in a vertical slit micro-channel with super-hydrophobic slip and temperature jump using non-linear Boussinesq approximation approach. The inner surface of one of the micro-channel surfaces is structurally changed so that it exhibited super-hydrophobic velocity slip and jump in temperature. A transverse magnetic field is applied to the flow direction. A unified solution is derived for two different physical situations. Case I, represents a physical situation when super-hydrophobic surface is heated and no-slip surface is unheated while case II represents a physical situation when no-slip surface is heated and super-hydrophobic surface is unheated. The velocity slip lowers the Nusselt number in Case I and raises it in Case II while the temperature jump co-efficient does the opposite. Velocity and flow rate are higher in case of non-linear density variation with temperature in comparison to linear density variation with temperature.

Keywords MHD · Non-linear Boussinesq approximation · Super-hydrophobic slip · Temperature jump

Mathematics Subject Classification 76R10 · 76W05 · 80A20 · 97M50

List of symbolsu Dimensional upward velocity of the flow (m/s)y Dimensional Cartesian coordinate along the width

of the channel (m)g Acceleration due to gravity (m/s2)T Dimensional temperature of the fluid (K)T0 Dimensional temperature of the surroundings (K)Th Dimensional temperature higher than T0 (K)B⃗ Magnetic field vector (T)B0 Uniform applied magnetic field (T)Y Dimensionless Cartesian coordinate along the

width of the channel (–)U Dimensionless upward velocity (–)L Channel width (m)M Magnetic parameter (–)Q Dimensionless volume flow rate (–)

Nu Nusselt number (–)N Non-linear density variation with temperature

(NDT) parameter (–)V⃗ Velocity vector field (m/s)

Greek letters�0 The first degree thermal expansion coefficient of

the fluid (1/K)�1 The second degree thermal expansion coefficient

of the fluid (1/K2)� Dimensionless temperature jump (–)� ′ Dimensional temperature jump (m)� Kinematic viscosity of the fluid (m2/s)� Density of the fluid (kg/m3)� Electrical conductivity of the fluid (A2 s3/kg m3)� Dimensionless temperature (–)�b Bulk temperature (–)

Received: 30 January 2019 / Accepted: 14 May 2019 / Published online: 20 May 2019

* Bello Jibril Gwandu, mbj_gwn@yahoo.com; Basant Kumar Jha, basant777@yahoo.co.uk | 1Department of Mathematics, Ahmadu Bello University, Zaria, Kaduna State, Nigeria. 2Department of Mathematics, Federal University Birnin Kebbi, P.M.B. 1157, Birnin Kebbi, Kebbi State, Nigeria.

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� Dimensionless slip length (–)�′ Dimensional slip length (m)

1 Introduction

In both vertical and horizontal orientation, natural convec-tion phenomenon has been studied for decades. This is due to its natural occurrence and extensive real life appli-cations in the immediate environment. Various conditions and aspects were considered by different researchers depending on their immediate needs. Chen and Weng [1], Khadrawi et al. [2] and Chen and Weng [3] studied natu-ral convection involving heating effects through vertical micro-channels while Maynes et al. [4] and Davies et al. [5] investigated a similar convection in horizontal micro-channels with super-hydrophobic inner surfaces. Each of the two studies emphasized on the orientation of the micro-structures leading to the super-hydrophobicity. Super-hydrophobic slip and temperature jump were, also, considered. All the above studies are linear convec-tion problems. Super-hydrophobicity was not involved in [1–3]. This was covered by [4, 5]. All the papers described non-magnetic micro-fluidic flows with enough details. However, their applications become limited where mag-netism or thermal non-linearity is involved.

A non-linear study was done by Partha et al. [6]. This involved thermal diffusivity, non-Darcy porosity and solutal dispersion under the action of a magnetic field act-ing transverse to the embedded vertical plate. Similar non-linear convective studies were carried out by Partha [7], Kameswaran et al. [8] and Hayat et al. [9]. Momentum and thermal slips in the presence of an isothermal wall heat-ing were considered by Biswal et al. [10] and Maynes et al. [11]. While [10] dealt with a vertical channel, [11] investi-gated a horizontal channel. In all these above work, no-slip and no temperature jump conditions are used. Buonomo and Manca [12] investigated a micro-channel slip flow of a vertical orientation when the channel walls are heated by a constant heat flux. Other constant heat flux studies include Maynes et al. [13] and Maynes and Crockett [14]. Both [13] and [14] studied super-hydrophobic slip flow in horizontal micro-channels. The two studies differed only in the orientation of the micro-ribs and micro-cavities caus-ing the super-hydrophobicity.

In another development, both isothermal and iso-flux heating modes are considered for natural convection in a vertical slit micro-channel by Wang and Ng [15]. Jha and Gwandu [16] extended the work of Wang and Ng [15] by considering electrically conducting fluid in the presence of transverse magnetic field. Bhatti and Zeeshan [17] ana-lysed the hydrodynamics and heat transfer of dust-parti-cle-containing fluid with variable viscosity in a rectangular

horizontal channel. Their major findings include: (1) the velocity profile diminishes as a result of the influence of particle volume fraction (2) the temperature profile is a decreasing function of volume fraction. The trend are reversed for the Eckert and Prandtl numbers. Bhatti and Lu [18] studied the dynamics of elastic water waves moving through icy rectangular channels obtainable at the polar regions of the earth. They used the Poincare–Lighthill–Kuo (PLK) method to solve the non-linear differential equa-tions. The colliding solitary waves are observed to undergo phase shifts. However, they regained their original shapes and positions after the collision.

A similar work was discussed by the same authors [19] based on the model formulated by Plotnikov and Toland [20] concerning a thin sheet of ice floating on an infinite ocean surface. Another study on hydro-elasticity was con-ducted by Seth et al. [21] for free convection flow with partial slip. The study incorporated both heat and mass transfer in the presence of magnetic field. Bhatti et al. [22] incorporated suction and injection into a magneto-hydro-dynamic (MHD) flow past a stretching sheet. The resulting differential equations are solved using a combination of two numerical methods, namely, successive lineariza-tion method (SLM) and Chebyshev’s spectral collocation method (SCM). They found that the combined method is more efficient than using the individual methods and it converges easily to the exact solution.

In this study, an alternate isothermal wall heating is applied to a vertical super-hydrophobic micro-channel during a magneto-hydrodynamic (MHD) flow of an elec-trically conducting fluid. It is assumed that the capillarity pressure is not exceeded. This allows the fluid to rise up the channel (buoyancy effects) without overflowing into the micro-cavities on the super-hydrophobic surface. This traps vapour into the cavities, thereby, leading to a mixed kind of interface: On the micro-ridge, there is a solid–liquid interface; at the micro-groove, there are solid–gas and gas–liquid interfaces next to each other. This gives rise to a partial velocity slip and causes a par-tial thermal insulation to guarantee a temperature jump. For economic and technical reasons, only one plate sur-face is made super-hydrophobic (extremely difficult to wet). The study will have applications relevant to hydrau-lic micro-devices, heat dissipation in micro-electronics, electrolytic cells maintenance, anti-wetting micro-tech-nology, etc.

2 Mathematical analysis

Consider the fully developed steady natural convection flow of viscous, incompressible, electrically conducting fluid in a vertical slit micro-channel formed by two vertical

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parallel plates under the effect of transverse magnetic field. The x-axis is parallel to the gravitational acceleration, g, but in the opposite direction while the y-axis is orthogo-nal to the vertical parallel plates. A magnetic field of uni-form strength B⃗ = (0, B0, 0) is assumed to be applied in the direction perpendicular to the direction of flow. The flow velocity field is of the form V⃗ = (u, 0, 0) . It is assumed that

the magnetic Reynolds number is very small, which cor-responds to negligibly induced magnetic field compared to the externally applied one. The plates are heated asym-metrically with one plate maintained at a temperature T0 while the other plate at a temperature Th where Th > T0. Due to this temperature gradient between the plates, nat-ural convection flow occurs in the channel. The geometry of the system under consideration in this present study is shown schematically in Fig. 1. In the present work, natural convection is due to non-linear density variation with tem-perature (NDT). The study is divided into two cases. Case I is when the super-hydrophobic surface is heated. Case II is when the no-slip surface is heated. The micro-structures on the super-hydrophobic surface resulted into partial thermal and hydrodynamic slips. Incorporating the non-linear Boussinesq approximation into the Navier-Stoke equations used in Jha and Gwandu [16], the dimensional governing equations for the phenomenon are as follows:

(1)�d2u

dy2+ g�0(T − T0) + g�1(T − T0)

2 −�B2

0u

�= 0

(2)d2T

dy2= 0

Equation (1) is the momentum equation showing non-linear buoyancy under the action of Lorentz force. Equa-tion (2) is a no-source and no-sink heat energy equation for the system.

T = T (y) is the fluid temperature at any point between the plates.

The dimensional boundary conditions are:

(3)T (y) = T0 + A(Th − T0) + � �

dT

dy

u(y) = ��du

dy

⎫⎪⎬⎪⎭y = 0,

T (y) = T0 + B(Th − T0)

u(y) = 0

�y = L

g

No-slip surfaceSuper-hydro-phobic surface

( )u y

( )T y

y L0y

0B y

Fig. 1 Geometric configuration of the flow

Case I Super-hydrophobic surface heated (A = 1, B = 0).

Case II No slip surface heated (A = 0, B = 1).

In Eq. (3), � ′ and �′ are super-hydrophobic temperature jump coefficient and super-hydrophobic velocity slip length respectively.

The dimensionless variables and parameters are:

U0 = g�0L2(Th − T0)∕� is the characteristic velocity of

the flow.The dimensionless equations and boundary conditions

are given by Eqs. (5)–(7).

Case I Super-hydrophobic surface heated (A = 1, B = 0).

Case II No slip surface heated (A = 0, B = 1).

. Solving Eq.  (6) and incorporating the solution into Eq. (5) lead to the following:

(4)

(Y , �, �) = (y, ��, � �)∕L;U = u∕U0; � = (T − T0)∕(Th − T0);

M = B0L√�∕��;N = �1(Th − T0)∕�0,

�

(5)d2U

dY2+ � + N�2 −M2U = 0

(6)d2�

dY2= 0

(7)�(Y) = A + �

d�

dY

U(Y) = �dU

dY

⎫⎪⎬⎪⎭Y = 0,

�(Y) = B

U(Y) = 0

�Y = 1

(8)�(Y) = C1 + C2Y

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The skin frictions at the channel surfaces are given by Eqs. (10) and (11). Equation (10) is for the super-hydropho-bic surface, while Eq. (11) is for the no-slip surface.

The mass flow rate and bulk temperature are, respec-tively, given by:

These led to the following expressions:

The constants Ci (i = 1 to 4) and kj (j = 1 to 11) are defined in the “Appendix”.

The rate of heat transfer expressed in the form of Nus-selt numbers are given by:

Equation (15) is for the super-hydrophobic surface and Eq. (16) is for the no-slip surface.

3 Results and discussion

After the exact solutions were obtained by hand calcu-lations and the graphs were produced by the means of user-written programs using MATLAB (Version R2010a) software, a number of observations were made. When the super-hydrophobic plate is heated, high values of tem-perature jump coefficient lower the temperature of the

(9)

U(Y) = C3Cosh(MY) + C4Sinh(MY) −(k1Y

2 + k2Y)

M2+ k6

(10)�0 =dU

dY

||||Y=0 = MC4 + k5

(11)

�1 =dU

dY

||||Y=1 = M[C3Sinh(M) + C4Cosh(M)

]−[(2k1 + k2)∕M

2]

(12)Q = ∫1

0

U(Y)dY ; �b =

{∫

1

0

U(Y)�(Y)dY

}/Q;

(13)

Q =1

M

[C3Sinh(M) + C4[Cosh(M) − 1]

]−

(M2 + 6

)k1

3M4−

(k2 + 2k3)

2M2

(14)�b =1

Q

(k7 + k8 + k9 + k10 + k11

)

(15)Nu0 = −d�∕dY |Y=0

�b= −

C2

�b

(16)Nu1 =d�∕dY |Y=1

�b=

C2

�b

fluid across the width of the channel. This is because the partial thermal insulation at the solid–gas-liquid interface on the super-hydrophobic surface delays the passage of heat energy into or away from the fluid. The temperature, also, drops towards the no-slip plate. This is due to the natural flow of heat from a higher energy region to a lower one. The two trends are, however, reversed when the no-slip surface is heated. These are illustrated in Fig. 2. Fig-ures 3 and 4 indicate the velocity profiles when the super-hydrophobic side is heated. The emphasis in Fig. 3 is on the magnetic parameter. As it increases, the magnitude

Fig. 2 Effects of temperature jump coefficient on non-linear tem-perature variation within the micro-channel

Fig. 3 Effects of the magnetic parameter on the velocity when the super-hydrophobic plate is heated

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of the velocity decreases. This is due to attraction of the fluid molecules to the magnetic field. Figure 4 illustrates that non-linear temperature variation parameter, N, raises the magnitude of the velocity as it rises. This is because N magnifies the temperature, which leads to increase in the kinetic energy and consequently the velocity rises. When the no-slip surface is heated, the velocity profiles are depicted by Figs. 5 and 6. In Fig. 5, it is seen that the

magnetic parameter M reduces the upward velocity of the fluid, as in Fig. 3. Like in Fig. 4, the profile in Fig. 6 indicates that the higher the non-linearity factor, the higher the fluid flow velocity.

The bulk temperature of the fluid involves both the temperature rise and the flow velocity. Figure 7 describes the effects of velocity slip length and temperature jump co-efficient on the bulk temperature. The temperature jump co-efficient inhibits bulk temperature rise and the

Fig. 4 Effects of the NDT parameter on the velocity when the super-hydrophobic plate is heated

Fig. 5 Effects of the magnetic parameter on the velocity when the no-slip plate is heated

Fig. 6 Effects of the NDT parameter on the velocity when the no-slip plate is heated

Fig. 7 Effects of temperature jump coefficient on the bulk tempera-ture of the fluid

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velocity slip length favours increase in the bulk tempera-ture, when the super-hydrophobic side is heated. This is because the super-hydrophobicity causes a partial thermal insulation that retards the heating up of the fluid. The slip velocity, also, increases the kinetic energy and, hence, the temperature. When heating the no-slip side, the reverse is the case for the two parameters. This is because the ther-mal insulation at the opposite plate inhibits heat loss away from the fluid. In addition to that, lack of slip on the heated side reduces the bulk temperature rise. Figures 8 and 9 show that increase in the velocity slip co-efficient raises the bulk temperature when the super-hydrophobic side is heated and reduces it when the no-slip side is heated. It is observed that both the magnetic parameter, M, (in

Fig. 8) and non-linearity factor, N, (in Fig. 9) raise the bulk temperature as they increase, irrespective of which side is heated. This is because M slows down the fluid and the slower it moves the more heat it absorbs. Similarly, thermal non-linearity leads to more heat retention by the fluid in the micro-channel.

The variations of convective-to-conductive heat trans-fer ratio (Nusselt number), with respect to certain param-eters, are displayed in Figs. 10, 11 and 12. Figure 10 shows that the higher the hydrodynamic slip length, the lower the Nusselt number in Case I and the higher the Nusselt number in Case II. This is because some molecules slip off from the super-hydrophobic surface, thereby reducing the rate of heat transfer. This does not happen at the other

Fig. 8 Effects of the magnetic parameter on the bulk temperature of the fluid

Fig. 9 Effects of the NDT parameter on the bulk temperature of the fluid

Fig. 10 Effects of temperature jump co-efficient on the Nusselt number

Fig. 11 Effects of the magnetic parameter on the Nusselt number

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surface. It, also, demonstrates that the higher the thermal slip length, the bigger the Nusselt number in Case I and the smaller the Nusselt number in Case II. This is because the thermal insulation on the super-hydrophobic side implies that convection is higher than conduction, while the reverse is the case on the other side. Figures 11 and 12 confirm the Nusselt number variations with the hydro-dynamic slip length as portrayed by Fig. 10. In addition to that, Fig. 11 reveals that the stronger the magnetic effect, the lower the heat transfer ratio in both cases. This is because there is more heat absorption by the fluid in both cases, which means that there is more convection. Similarly, Fig. 12 adds that the bigger the non-linearity parameter, the smaller the Nusselt number in the heating

Fig. 12 Effects of the NDT parameter on the Nusselt number

Fig. 13 Effects of the magnetic parameter on the flow rate

Fig. 14 Effects of the NDT parameter on the flow rate

Fig. 15 Effects of the Magnetic Parameter on the Skin Friction

Fig. 16 Effects of the NDT parameter on the Skin Friction

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of each surface. This is because there is more conduction than convection at that surface.

Figure 13 exposes that heating the no-slip surface gives higher flow rate than heating the super-hydrophobic surface at unit non-linearity and thermal creep length of 0.5. This is because lack of thermal insulation causes a faster conversion of heat energy to kinetic energy. It, also, shows that the flow rate is enhanced by longer velocity slip lengths and lower magnetic field strength in both cases. This is due to their effects on the velocity. The velocity affects the flow rate directly. As seen in Fig. 14, the bigger the non-linearity factor, the higher the flow rate in both cases. There is more heat due to temperature non-linearity. So, there exists a higher flow rate. In Fig. 15, it could be seen that the longer the velocity slip lengths, the lower the skin friction (for the heated surface) in both cases, and higher skin friction exists in the first case than in the second case. More velocity slip means less friction effect. Apart from that, the magnetic parameter reduces the skin friction in the first case and increases it in the second case. Since there is no velocity slip, there will be more friction. Figure 16 illustrates that the NDT parameter increases the skin friction in case I and reduces it in case II. This is because of high velocity and surface roughness on the super-hydrophobic side (high friction) and high velocity with little or no roughness (low friction).

In order to validate the present study, it was compared with two previous studies [15, 16]. [15] is a linear non-magnetic Boussinesq buoyancy approximation of a natural convection in a vertical parallel plate micro-channel, while [16] is a magnetic aspect of the linear Boussinesq approxi-mation of a vertical free convection in a rectangular micro-channel. Both studies have the same super-hydrophobic conditions with the present study. The variables and

parameters considered for comparison are itemised in Table 1. Throughout the work, the values of variables and parameters are arbitrarily chosen large enough to show clear effects of their variations.

4 Conclusions

Based on the results obtained and discussions ensued above, the following conclusions, regarding the present research, were reached:

(1) An increase in the temperature jump coefficient causes a decline in the temperature when the super-hydro-phobic surface is heated and raises the temperature when the no-slip surface is heated

(2) No matter which surface is heated, an increase in the magnetic parameter, M, decreases the fluid flow velocity, and an increment in the NDT parameter, N, increases the velocity.

(3) An increase in the temperature jump co-efficient decreases the bulk temperature when the super-hydrophobic surface is heated and increases it when the no-slip surface is heated.

(4) A rise in the velocity slip coefficient raises the bulk temperature when the super-hydrophobic surface is heated and brings it down when the no-slip surface is heated.

(5) An increment in either M or N causes an increment in the bulk temperature, irrespective of which sur-face is heated.

(6) The patterns of change in Nusselt number are the exact reverse of the changes in the bulk tempera-ture.

Table 1 Comparison of values of certain parameters with some previous studies

a These are the correct numerical values obtained in [16]

Variables and parameters

Present work� = � = 1,

M = 0.01,

N = 0, Y = 0.5.

Wang and Ng [15]� = � = 1,

Y = 0.5.

Present work� = � = 1,

M = 1,N = 0,

Y = 0.5.

Jha and Gwandu [16]a

� = � = 1,

M = 1,

Y = 0.5.

Case I: the super-hydrophobic surface is heated U 0.0729 0.0729 0.0583 0.0583 �0 0.0833 0.0833 0.0677 0.0677 Q 0.0625 0.0625 0.0502 0.0502 �b 0.3111 0.3111 0.3120 0.3120a

 Nu0 1.6071 1.6071 1.6026 1.6026a

Case II: the no-slip surface is heated U 0.1771 0.1771 0.1435 0.1435 �1 − 0.5834 − 0.5833 − 0.5000 − 0.5000 Q 0.1459 0.1458 0.1179 0.1179 �b 0.7048 0.7048 0.7069 0.7069 a

 Nu1 0.7095 0.7095 0.7073 0.7073 a

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(7) The bigger the magnetic parameter, M, the lower the flow rate, Q, whichever surface is heated.

(8) The larger the non-linearity parameter, N, the higher the flow rate, Q, regardless of the nature of heated surface.

(9) An increment in M causes a decrease in the skin fric-tion when the super-hydrophobic surface is heated and makes it rise when the no-slip surface is heated.

(10) A rise in the velocity slip length decreases the skin friction regardless of which surface is heated.

(11) The skin friction at the super-hydrophobic surface when it is heated, is higher than that at the no slip surface, when the no-slip surface is heated.

(12) A rise in the non-linearity parameter, N, causes a rise in the skin friction when the super-hydrophobic sur-face is heated, and a fall in skin friction when the no-slip surface is heated.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest

Appendix

C1 =A + �B

1 + �; C2 =

B − A

1 + �; C3 = �MC4 +

(2k1 + k3M

2)

M4−

�k2

M2; C4 = k12(k13 + k14);

k1 = −N(A2 − 2AB + B2)

(1 + �)2; k2 = −

[2N(AB − A2 − AB� + B2�

)+ (B − A − A� + B�)]

(1 + �)2;

k3 = −[N(A2 + 2AB� + B2�2) + (A + A� + B� + B�2)]

(1 + �)2; k4 = −

k1

M2; k5 = −

k2

M2;

k6 = −(2k1 + k3M

2)

M4; k7 =

C1C3

M[Sinh(M)]; k8 =

C1C4

M[Cosh(M) − 1];

k9 =C2C3

M2[MSinh(M) − Cosh(M) + 1]; k10 =

C2C4

M2[MCosh(M) − Sinh(M)];

k11 =1

12[4(C1k4 + C2k5) + 6(C1k5 + C2k6) + 3C2k4 + 12C1k6]; k12 =

1

[�MCosh(M) + Sinh(M)];

k13 = k6[Cosh(M) − 1]; k14 =k1 + k2[1 + �Cosh(M)]

M2.

4. Maynes D, Jeffs K, Woolford B, Webb BW (2007) Laminar flow in a micro-channel with hydrophobic surface patterned micro-ribs oriented parallel to the flow direction. Phys Fluids 19:093603

5. Davies J, Maynes D, Webb BW, Woolford B (2006) Laminar flow in a micro-channel with super-hydrophobic walls exhibiting transverse ribs. Phys Fluids 18:087110

6. Partha MK, Murthy PVSN, Sekhar GPR (2006) Soret and dufour effects in a non-Darcy porous medium. J Heat Transfer 128:605–610

7. Partha MK (2010) Non-linear convection in a non-Darcy porous medium. Appl Math Mech (English Edition) 31(5):565–574

8. Kameswaran PK, Sibanda P, Partha MK, Murthy PVSN (2014) Thermophoretic and non-linear convection in non-Darcy porous medium. J Heat Transfer 136:042601

9. Hayat T, Bashir G, Waqas M, Alsaedi A (2016) MHD 2D flow of Williamson nano-fluid over a non-linear variable thicked surface with melting heat transfer. J Mol Liq 223:836–844

10. Biswal L, Som SK, Chakraborty S (2007) Effects of entrance region transport processes on free convection slip flow in ver-tical micro-channels with isothermally heated walls. Int J Heat Mass Transf 50:1248–1254

11. Maynes D, Webb BW, Davies J (2008) Thermal transport in a micro-channel exhibiting ultra-hydrophobic micro-ribs main-tained at constant temperature. J Heat Transfer 130:022402

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