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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 971456, 13 pages
http://dx.doi.org/10.1155/2011/971456
Research Article

Effect of Rotation and Magnetic Field through Porous Medium on Peristaltic Transport of a Jeffrey Fluid in Tube

1Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt

Received 8 June 2011; Accepted 25 July 2011

Academic Editor: Angelo Luongo

Copyright © 2011 S. R. Mahmoud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the analysis of peristaltic motion of a Jeffrey fluid in a tube with sinusoidal wave travelling down its wall. The effect of rotation, porous medium, and magnetic field on peristaltic transport of a Jeffrey fluid in tube is studied. The fluid is electrically conducting in the presence of rotation and a uniform magnetic field. An analytic solution is carried out for long wavelength, axial pressure gradient, and low Reynolds number considerations. The results for pressure rise and frictional force per wavelength were obtained, evaluated numerically, and discussed briefly.

1. Introduction

The dynamics of the fluid transport by peristaltic motion of the confining walls has received a careful study in the literature. The need for peristaltic pumping may arise in circumstances where it is desirable to avoid using any internal moving parts such as pistons in a pumping process. The peristalsis is also well known to the physiologists to be one of the major mechanisms of fluid transport in a biological system and appears in urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, the movement of spermatozoa in the ductus efferentes of the male reproductive tract and the ovum in the female fallopian tube, the locomotion of some worms, transport of lymph in the lymphatic vessels, and vasomotion of small blood vessels such as arterioles, venules, and capillaries. Technical roller and finger pumps also operate according to this rule. The behavior of most of the physiological fluids is known to be non-Newtonian. Several models have been proposed to explain the non-Newtonian behavior of fluids.

Mahmoud et al. [13] investigated effect of the rotation on the radial vibrations in a nonhomogeneous orthotropic hollow cylinder and effect of the rotation on wave motion through cylindrical bore in a micropolar porous cubic crystal, and he investigated effect of the rotation on the radial vibrations in a nonhomogeneous orthotropic hollow cylinder. Abd-Alla et al. [47] investigated effect of the rotation on a nonhomogeneous infinite cylinder of orthotropic material, influences of rotation, magnetic field, initial stress and gravity on rayleigh waves in a homogeneous orthotropic elastic half space, and magneto-thermoelastic problem in rotating nonhomogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, and they studied effect of the rotation on propagation of thermoelastic waves in a nonhomogeneous infinite cylinder of isotropic material. Mahmoud [8] studied effect of rotation on generalized magneto-thermoelastic Rayleigh waves in a granular medium under influence of gravity field and initial stress. Afifi et al. [911] investigated effect of magnetic field and wall properties on peristaltic motion of micropolar fluid in circular cylindrical tubes and interaction of peristaltic flow with pulsatile magnetofluid through a porous medium, and they studied aspects of a magnetofluid with suspended particles. Various attempts [1214] are made to solve the extremely complex equations of motion of non-Newtonian fluids. The good number of recent investigations [1523] on the peristalsis of non-Newtonian fluids has been presented with various perspectives, in channels or tubes. Most of the analytic studies are asymptotic expansions with small Reynolds number, wave number, and amplitude ratio as a perturbation parameter. Siddiqui et al. [24] examined the peristaltic motion of a magnetohydrodynamic Newtonian fluid in a tube by taking long wavelength approximation. More recently Hayat and Ali [22] studied the peristaltic motion of a third-order fluid in a tube under long wavelength and small Deborah number approximation. However, no attempt has been made to discuss the peristaltic motion of a magnetohydrodynamic (MHD) non-Newtonian fluid in a tube which holds for all values of non-Newtonian parameters. In the present analysis, such an attempt has been made. The liquid considered is of the Jeffrey type and is electrically conducting. This shows worthwhile the first attempt for MHD non-Newtonian flow in a tube for all values of the rheological parameters. The Jeffrey model is relatively simpler linear model using time derivatives instead of convected derivatives, for example, what the Oldroyd-B model does; it represents a rheology different from the Newtonian. Although more sophisticated viscoelastic models than the Jeffrey model exist, in a first study of the MHD peristaltic motion of a non-Newtonian fluid in circular cylindrical tube, the choice of Jeffrey fluid model is motivated by the following.

In spite of its relative simplicity, the Jeffrey model can indicate the changes of the rheology on the peristaltic flow even under the assumption of large wavelength, low Reynolds number, and small or large amplitude ratio. In Newtonian fluid, Mekheimer [25] studied the MHD peristaltic flow in a channel under the assumption of small wave number. Therefore, at least in an initial study, this motivates an analytic study of MHD peristaltic non-Newtonian tube flow that holds for all non-Newtonian parameters. By choosing the Jeffrey fluid modele it became possible to treat both the MHD Newtonian and non-Newtonian problems analytically under long wavelength and low Reynolds number consideration. Considering the blood as an MHD fluid, it may be possible to control blood pressure and its flow behavior by using an appropriate magnetic field. The influence of magnetic field may also be utilized as a blood pump for cardiac operations for blood flow in arterial stenosis or arteriosclerosis.

2. Formulation of the Problem

Consider the axisymmetric flow of a Jeffrey fluid in a uniform circular tube with a sinusoidal peristaltic wave of small amplitude travelling down its wall (see Figure 1). The geometry of wall surface is therefore described as𝑧,𝑡=𝑑+𝑎cos2𝜋𝜆𝑧𝑐𝑡.(2.1) Here 𝑎 is amplitudes of the waves, 𝜆 is the wavelength, 𝑑 is average radius of the undisturbed tube. The constitutive equations for an incompressible Jeffrey fluid are:𝐼0=𝑝𝐼+𝑆,𝜇𝑆=1+𝜆1̇𝛾+𝜆2,̈𝛾(2.2) where 𝐼0 and 𝑆 are Cauchy stress tensor and extra stress tensor, respectively, 𝑝 is the pressure, 𝐼 is the identity tensor, 𝜇 is dynamic viscosity, 𝜆1 is the ratio of relaxation to retardation times, 𝜆2 is the retardation time, ̇𝛾 is the shear rate, and dots over the quantities indicate differentiation with respect to time. In laboratory frame, the equations governing two-dimensional motion of an incompressible MHD Jeffrey fluid through a porous medium [24] are as follows:𝜕𝑈𝜕𝑅+𝜕𝑊𝜕𝑍+𝑈𝑅𝜌𝜕=0,𝑈𝜕𝑡+𝑈𝜕𝑈𝜕𝑅+𝑊𝜕𝑈𝜕𝑍+Ω2𝜕𝑈=𝑝𝜕𝑅+1𝑅𝜕𝜕𝑅𝑅𝜕𝑆𝑅𝑅𝜕𝑅𝑆𝑅𝑅𝑅2+𝜕2𝑆𝑅𝑍𝜕𝑍2𝜇𝜅0𝜌𝜕𝑈,𝑊𝜕𝑡+𝑈𝜕𝑊𝜕𝑅+𝑊𝜕𝑊𝜕𝑍𝜕=𝑝𝜕𝑍+1𝑅𝜕𝜕𝑅𝑅𝜕𝑆𝑅𝑍𝜕𝑅+𝜕2𝑆𝑍𝑍𝜕𝑍2𝜎𝐵20𝜇𝑊𝑘0𝑊+𝜌Ω2𝑊,(2.3) where 𝑅, 𝑊 are the velocity components in the laboratory frame (𝑅, 𝑍), 𝜌 is the density, 𝑝 is the pressure, 𝜎 is the electrical conductivity of the fluid, 𝐵0 is a constant of magnetic field, 𝜇 is the kinematic viscosity, Ω is the rotation component, and 𝑘0 is the permeability of the porous medium, and we get [25]𝜇𝑆=1+𝜆11+𝜆2𝑈𝜕𝜕𝑅+𝑊𝜕𝜕𝑍̇𝛾,𝑆𝑅𝑅=2𝜇1+𝜆11+𝜆2𝑈𝜕𝜕𝑅+𝑊𝜕𝜕𝑍𝜕𝑈𝜕𝑅,𝑆𝑍𝑍=2𝜇1+𝜆11+𝜆2𝑈𝜕𝜕𝑅+𝑊𝜕𝜕𝑍𝜕𝑈𝜕𝑍,𝑆𝑅𝑍=𝜇1+𝜆11+𝜆2𝑈𝜕𝜕𝑅+𝑊𝜕𝜕𝑍𝜕𝑈𝜕𝑍𝜕𝑊𝜕𝑅.(2.4) We will carry out this investigation in a coordinate system moving with the wave speed in which the boundary shape is stationary. The coordinates and velocities in the laboratory frame (𝑅, 𝑍) and the wave frame (𝑥, 𝑦), are related by𝑟=𝑅𝑐𝑡,𝑧=𝑍,𝑢=𝑈𝑐,𝑤=𝑊,𝑝=𝑝𝑅,𝑡,(2.5) where 𝑢, 𝑤 are the velocity components in the wave frame (𝑟,𝑧). We introduce the following nondimensional variables and parameters for the flow:𝑟=𝑅𝑑1,𝑧=2𝜋𝑍𝜆,𝑢=𝑈𝑐𝛿,𝑤=𝑊𝑐,𝛿=2𝜋𝑑1𝜆,𝑡=2𝜋𝑐𝑡𝜆,𝑝=2𝜋𝑑21𝑝𝜎𝜇𝑐𝜆,𝐻=𝜇𝐵2o𝑑21,Re=𝜌𝑐𝑑1𝜇𝑑,𝑆=1𝜇𝑐Ω𝑆,2=Ω2𝜈𝑑21,𝑘𝑘=0𝑑21,1=1𝑑1𝑎,𝑎=1𝑑1,(2.6) where Re is the Reynolds number, 𝛿 is the dimensionless wave number, and 𝐻 is the magnetic parameter (Hartman number). Using nondimensional variables and parameters in (2.3), we get the following: 𝛿𝜕𝑢𝜕𝑟+𝛿𝜕𝑊𝑢𝜕𝑍+𝛿𝑟𝑢𝜕=0,Re𝛿𝜕𝜕𝑟+𝑤𝜕𝑧𝑢=𝜕𝑝𝜕𝑟+𝛿21𝑟𝜕𝑟𝜕𝑟𝜕𝑆𝑟𝑟𝑆𝜕𝑟𝑟𝑟𝑟2+𝛿2𝜕2𝑆𝑟𝑧𝜕𝑧21𝑘𝑢Ω2𝑢,Re𝛿3𝑢𝜕𝜕𝜕𝑟+𝑤𝜕𝑧𝑤=𝜕𝑝+1𝜕𝑧𝑟𝜕𝑟𝜕𝑟𝜕𝑆𝑟𝑧𝜕𝑟+𝛿2𝜕2𝑆𝑧𝑧𝜕𝑧2𝐻21𝑤𝑘𝑤+Ω2𝑤,(2.7) introducing the stream function 𝜓 as 𝛿𝑢(𝑟,𝑧)=𝑟𝜕𝜓1𝜕𝑧,𝑤(𝑟,𝑧)=𝑟𝜕𝜓𝜕𝑟.(2.8) We can write (2.7) as follows:𝛿2𝜕1𝜕𝑟𝑟𝜕𝜓𝜕𝜕𝑧𝛿1𝜕𝑧𝑟𝜕𝜓𝜕𝑟+𝛿2𝜕𝑟21𝑟𝜕𝜓𝛿𝜕𝑧=0,(2.9)Re𝛿𝑟𝜕𝜓𝜕𝜕𝑧1𝜕𝑟𝑟𝜕𝜓𝜕𝜕𝑟1𝜕𝑧𝑟𝜕𝜓𝜕𝑧=𝜕𝑝𝜕𝑟+𝛿21𝑟𝜕𝑟𝜕𝑟𝜕𝑆𝑟𝑟𝑆𝜕𝑟𝑟𝑟𝑟2+𝛿2𝜕2𝑆𝑟𝑧𝜕𝑧2𝛿𝑘1𝑟𝜕𝜓𝜕𝑧𝛿Ω21𝑟𝜕𝜓,𝜕𝑧(2.10)Re𝛿3𝛿𝑟𝜕𝜓𝜕𝜕𝑧+1𝜕𝑟𝑟𝜕𝜓𝜕𝜕𝑟1𝜕𝑧𝑟𝜕𝜓𝜕𝑟=𝜕𝑝+1𝜕𝑧𝑟𝜕𝑟𝜕𝑟𝜕𝑆𝑟𝑧𝜕𝑟+𝛿2𝜕2𝑆𝑧𝑧𝜕𝑧2+𝐻21𝑟𝜕𝜓+1𝜕𝑟𝑘1𝑟𝜕𝜓𝜕𝑟Ω21𝑟𝜕𝜓.𝜕𝑟(2.11) Eliminating pressure from (2.9), (2.11) by cross-differentiation, using the long wavelength (𝛿1) and low Reynolds number in (2.9)–(2.11), and neglecting 𝛿 and higher power, we obtain 𝜕𝑝𝜕𝑟=0,(2.12)𝜕𝑝=1𝜕𝑧𝑟𝜕𝑟𝜕𝑟1+𝜆11𝑟𝜕𝜓𝜕𝑟𝜔21𝑟𝜕𝜓𝜕𝑟,(2.13) where 𝜔2=1/𝑘Ω2+𝐻2,𝑆𝑟𝑟=2𝛿1+𝜆11+𝛿𝑐𝜆2𝑑1𝜕𝜓𝜕𝜕𝑧𝜕𝑟𝜕𝜓𝜕𝜕𝑟𝜕𝜕𝑧1𝜕𝑟𝑟𝜕𝜓,𝑆𝜕𝑧𝑧𝑧=2𝛿1+𝜆11+𝛿𝑐𝜆2𝑑1𝜕𝜓𝜕𝜕𝑧𝜕𝑟𝜕𝜓𝜕𝜕𝑟𝜕𝜕𝑧1𝜕𝑧𝑟𝜕𝜓.𝜕𝑟(2.14) From (2.12) we show that 𝑝𝑝(𝑟). Differentiating (2.13) with respect to 𝑟 we get𝜕=1𝜕𝑟𝑟𝜕𝑟𝜕𝜕𝑟1𝜕𝑟𝑟𝜕𝜓𝜕𝑟𝜒21𝑟𝜕𝜓𝜕𝑟=0,(2.15) where 𝜒2=(1+𝜆1)𝜔2.

971456.fig.001
Figure 1: Geometry of peristaltic motion on asymmetric channel through porous medium.

3. Rate of Volume Flow

The instantaneous volume flow rate in fixed coordinate system is given by𝑄𝑧,𝑡=2𝜋0𝑤𝑍,𝑡𝑅𝑑𝑅,(3.1) where is a function of 𝑍 and 𝑡. On substituting (2.5) into (3.1) and then integrating, one obtains 𝑄=𝑞+𝜋𝑐2,(3.2) where𝑞=2𝜋0𝑤𝑟𝑑𝑟(3.3) is the volume flow rate in the moving coordinate system and is independent of time. Here, is a function of 𝑧 alone. Using the dimensionless variables, we find𝐹=𝑞2𝜋𝑐𝑎21=0𝑤𝑟𝑑𝑟.(3.4) The time-mean flow over a period 𝑇=𝜆/𝑐 at a fixed Z-position is defined as1𝑄=𝑇𝑇0𝑄𝑑𝑡.(3.5) Using (3.2) into (3.5), 0<𝜖<1, we obtain𝑄=𝑞+𝜋𝑐𝑎2𝜖1+22.(3.6) Using dimensionless variables we write:𝑄2𝜋𝑐𝑎2=𝑞2𝜋𝑐𝑎2+12𝜖1+22.(3.7) Equation (3.6) becomes 1𝛽=𝐹+2𝜖1+22,𝑄𝛽=2𝜋𝑐𝜕2,𝐹=0𝜕𝜓𝜕𝑟𝑑𝑟=𝜓()𝜓(0),(3.8) where 𝛽 and 𝐹 are, respectively, the flow rates in the fixed and wave frames.

We note that represents the dimensionless form of the surface of the peristaltic wall:𝑎(𝑧)=1+𝜖cos2𝜋𝑧,𝜖=1𝑑1.(3.9) Choosing the zero value of the streamline along the central line (𝑤=0), we have 𝜓(0)=0. Then the shape of the wave at the boundary is the streamline with value 𝜓()=𝐹 in wave frame, the boundary conditions in terms of stream 𝜕𝜓=0,1𝜕𝑟𝑟𝜕𝜓1𝜕𝑟=0,at𝑟=0,𝜓=𝐹,𝑟𝜕𝜓𝜕𝑟=1,at𝑟=.(3.10)

4. Method of Solution

Integration of (2.15) along with boundary conditions (3.10) gives𝜕1𝜕𝑟𝑟𝜕𝜓𝜒𝜕𝑟2𝑟𝑟𝜓=2𝑐1,(4.1) where 𝑐1 is an arbitrary function of 𝑧. Equation (4.1) after using the transformation𝜙1=𝜓𝑟+𝑟2𝜒2𝑐1(4.2) can be reduced into the following modified Bessel equation:𝑟2𝜕2𝜙1𝜕𝑟2+𝑟𝜕𝜙1𝜒𝜕𝑟2𝑟2𝜙+11=0,(4.3) whose solution along with (4.2) and boundary conditions (3.10) is given below:𝑟𝜓=𝑟𝐹𝐼0(𝜒)+𝑟𝐼1(𝜒)2𝐹+2𝐼1(𝜒𝑟)2𝜒𝐼2(𝜒),(4.4) where 𝐼0, 𝐼1, and 𝐼2 are the modified Bessel function of order zero, one, and two, respectively.

Substitution of (4.4) into (2.8) and (2.13) yields the following expressions for axial velocity (𝑤) and axial pressure gradient:𝑤=2𝐹𝜒𝐼0(𝜒)+2𝜒𝐼1(𝜒)2𝜒2𝑓+2𝐼0(𝜒𝑟)2𝜒𝐼2,(𝜒)𝑑𝑝=𝑑𝑧𝜒22𝐹+2𝐼0(𝜒𝑟)2𝐼2.(𝜒)(4.5) The expressions for pressure rise (Δ𝑃𝜆) and frictional force (𝐹𝜆) per wavelength are, respectively, given by Δ𝑝𝜆=20𝑑𝑝𝐹𝑑𝑧𝑑𝑧,𝜆=02𝜋2𝑑𝑝𝑑𝑧𝑑𝑧.(4.6)

5. Results and Discussion

To investigate the effects of rotation (Ω), magnetic parameter (𝑀), material parameter (𝜆1), permeability of the porous medium (𝑘), and mean flux (𝐹), we plotted Figures 26.

fig2
Figure 2: Show the stress distributions for tube 𝑎=0.3, 𝑏=0.4, and 𝑑=1.1.
fig3
Figure 3: Show the velocity distributions for 𝑎=0.3, 𝑏=0.4, 𝑑=1.1, and 𝜆=0.5.
fig4
Figure 4: Show the Graph of pressure rise versus 𝐹 in symmetric channels for 𝑎=0.3, 𝑏=0.4, and 𝑑=1.1.
fig5
Figure 5: Plot showing 𝐹𝜆 for 𝐹[1,1] for changing rotation (Ω), (a), Hartman number 𝑀 (b), and material parameter 𝜆1 (c).
fig6
Figure 6: Plot showing variation of the pressure gradient (d𝑝/d𝑧) within wavelength for various values of rotation (Ω) (a) magnetic parameter (𝑀) (b), and material parameter (𝜆1) (c).

The stress distribution (𝑆𝑟𝑟), (𝑆𝑟𝑧), and (𝑆𝑧𝑧) in tube for different values of the rotation (Ω) is presented in Figures 2(a), 2(b), and 2(c), respectively. We notice that the stress is in oscillatory behaviour, which may be due to peristalsis. The absolute value of stress distribution (𝑆𝑟𝑟), (𝑆𝑟𝑧), and (𝑆𝑧𝑧) increases at first with increasing the rotation (Ω), and then it decreases with increasing the rotation (Ω) when large values of 𝑟 have been taken into account. It is observed that the absolute values of the stress are larger in case of a Jeffrey fluid when compared with Newtonian fluid.

The effects of the rotation (Ω) and magnetic parameter (𝑀) on the velocity is plotted in Figure 3. Figure 3 shows that influence of the rotation (Ω) and magnetic parameter (𝑀) on the velocity increases with the increase of magnetic parameter (𝑀), and it decreases with the increase of the rotation (Ω).

Figure 4 shows the variation of Δ𝑃𝜆 with flow rate 𝐹 for values of rotation (Ω), magnetic parameter (𝑀), and material parameter (𝜆1) for tube. We observe that the peristaltic pumping rate increases with increase of magnetic parameter (𝑀) and material parameter (𝜆1), and it decreases with the increase of the rotation (Ω). The phenomenon of trapping is another interesting topic in peristaltic transport. The formation of an internally circulating bolus of the fluid by closed streamlines is called trapping, and this trapped bolus pushed ahead along the peristaltic wave.

Figure 5 shows the variation of 𝐹𝜆 with flow rate 𝐹 for values of rotation (Ω), magnetic parameter (𝑀), and material parameter (𝜆1) for tube. Figures 5(a), 5(b), and 5(c) display the influence of (Ω), magnetic parameter (𝑀), and material parameter (𝜆1), respectively, for tube on 𝐹𝜆. Figure 5(a) refers to the case when 𝐹=0.2. Here it is noted that 𝐹𝜆 increases with decrease of rotation (Ω) when 0.6𝐹0.2 and it increases with the increase of the rotation (Ω) when 0.2<𝐹. Figure 5(a) refers to the case when Ω=1.2, here, it is noted that 𝐹𝜆 is negative and positive when 0.6𝐹0.6 and 0.6<𝐹, respectively. When Ω=0.8. Here it is noted that 𝐹𝜆 is negative and positive when 0.6𝐹0.83 and 0.83<𝐹, respectively. When Ω=0.4, 𝐹𝜆 is negative for 0.6𝐹1.4 and positive for 1.4<F. Also, when Ω=0.0, 𝐹𝜆 is negative for 0.6𝐹3.0 and positive for 3.0<𝐹. Figure 5(b) refers to the case when 𝐹=0.8. Here it is noted that 𝐹𝜆 increases with decrease of magnetic parameter (𝑀) when 1.5𝐹0.8, and it increases with the increase of the magnetic parameter (𝑀) when 0.8<𝐹. Figure 5(c) refers to the case when 𝐹=0.8. Here it is noted that 𝐹𝜆 increases with decrease of material parameter (𝜆1) when 1𝐹0.48, and it increases with the increase of the magnetic parameter (𝑀) when 0.48<𝐹.

Figure 6 shows the distributions of the pressure gradient within a wavelength for various values of the rotation (Ω), magnetic parameter (𝑀) and material parameter (𝜆1). The effects of magnetic parameter (𝑀), on the pressure gradient (d𝑝/d𝑧) within a wavelength are plotted in Figure 2(b). It is noticed that magnetic parameter (𝑀) and material parameter (𝜆1) increase the maximum amplitude of (d𝑝/d𝑧) when compared to the case with zero magnetic parameter and zero material parameter (𝜆1).

6. Conclusion

The influence of the rotation (Ω), magnetic parameter (𝑀), and material parameter (𝜆1) on the peristaltic flow of a Jeffrey fluid in tube has been analyzed. The analytical expressions are constructed for axial velocity, 𝐹𝜆, and pressure gradient. Numerical investigation is plotted and discussed. The main findings can be summarized as follows. (i)The axial velocity for the MHD fluid is less when compared with hydrodynamic fluid in the central part of the tube. (ii)The magnitude of (d𝑝/d𝑧) and 𝐹𝜆 increases with increase of magnetic parameter (𝑀) and material parameter (𝜆1) and it increases with decrease of the rotation (Ω).(iii)The size of trapped bolus is smaller in Jeffrey fluid when compared with that of Newtonian fluid (𝜆1=0). (iv)The magnitudes of (d𝑝/d𝑧), Δ𝑃𝜆, and 𝐹𝜆 for Newtonian fluid are smaller than that of Jeffrey fluid.(v)For large values of magnetic parameter (𝑀) and material parameter (𝜆1), the magnitudes of Δ𝑃𝜆 and Δ𝑃𝜆 increase with decrease of the rotation (Ω).

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