Chemical Engineering Communications MATHEMATICAL MODELING

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MATHEMATICAL MODELING OF OSCILLATORY MHD COUETTE FLOW IN A ROTATING HIGHLY PERMEABLE MEDIUM PERMEATED BY AN OBLIQUE MAGNETIC FIELD

O. Anwar Béga; S. K. Ghoshb; M. Naraharic a Biomechanics and Engineering Magnetohydrodynamics Research, Mechanical Engineering Program, Department of Engineering and Mathematics, Sheffield Hallam University, Sheffield, UK b Magnetohydrodynamics Research Program, Mathematics Department, Narajole Raj College, Narajole, West Bengal, India c Fluid Dynamics Research, Fundamental and Applied Sciences Department, Universiti Teknologi Petronas, Malaysia Online publication date: 05 November 2010 To cite this Article Bég, O. Anwar , Ghosh, S. K. and Narahari, M.(2011) 'MATHEMATICAL MODELING OF

OSCILLATORY MHD COUETTE FLOW IN A ROTATING HIGHLY PERMEABLE MEDIUM PERMEATED BY AN OBLIQUE MAGNETIC FIELD', Chemical Engineering Communications, 198: 2, 235 — 254 To link to this Article: DOI: 10.1080/00986445.2010.500165 URL: http://dx.doi.org/10.1080/00986445.2010.500165

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Chem. Eng. Comm., 198:235–254, 2011 Copyright # Taylor & Francis Group, LLC ISSN: 0098-6445 print=1563-5201 online DOI: 10.1080/00986445.2010.500165

Mathematical Modeling of Oscillatory MHD Couette Flow in a Rotating Highly Permeable Medium Permeated by an Oblique Magnetic Field O. ANWAR BE´G,1 S. K. GHOSH,2 AND M. NARAHARI3 Downloaded By: [Inst of Tech PETRONAS Sdn Bhd] At: 04:47 6 November 2010

1

Biomechanics and Engineering Magnetohydrodynamics Research, Mechanical Engineering Program, Department of Engineering and Mathematics, Sheffield Hallam University, Sheffield, UK 2 Magnetohydrodynamics Research Program, Mathematics Department, Narajole Raj College, Narajole, West Bengal, India 3 Fluid Dynamics Research, Fundamental and Applied Sciences Department, Universiti Teknologi Petronas, Malaysia We study theoretically the incompressible, viscous, oscillatory hydromagnetic Couette flow in a horizontal fluid-saturated highly permeable porous medium parallel-plate channel rotating about an axis perpendicular to the plane of the plates under the action of a uniform magnetic field, B0, inclined at an angle h to the axis of rotation. The flow is generated by the non-torsional oscillation of the lower plate of the channel. The reduced unsteady momentum equations are nondimensionalized with appropriate variables. Exact solutions under specified boundary conditions are obtained using the Laplace transform method (LTM). The flow regime is found to be controlled by a rotational parameter (K2), which is the reciprocal of the Ekman number (Ek), the square of the Hartmann magnetohydrodynamic number (M2), a porous medium permeability parameter (Kp), which is the inverse of the Darcy number (Da), oscillation frequency (x), dimensionless time (T), and magnetic field inclination (h). The influence of these parameters on the primary (u1) and secondary (v1) velocity field is presented graphically and studied in detail. Asymptotic behavior of the solutions is also examined for several cases of the square of the Hartmann number, rotation parameter, and oscillation angular frequency. The existence of modified Hartmann boundary layers is also identified. The present study has important applications in MHD (magnetohydrodynamic) energy generator flows, chemical engineering magnetic materials processing, conducting blood flows, and process fluid dynamics. Keywords Asymptotic solutions; Channel; Chemical engineering; Darcy number; Hartmann number; Laplace transform method (LTM); Magnetohydrodynamics (MHD); Oblique magnetic field; Oscillatory; Porous medium; Rotation; Secondary flow

Address correspondence to O. Anwar Be´g, Biomechanics and Engineering Magnetohydrodynamics Research, Mechanical Engineering Program, Department of Engineering and Mathematics, Sheaf Building, Sheffield Hallam University, Sheffield, S1 1WB, UK. E-mail: [email protected]; [email protected]

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Introduction Oscillatory hydromagnetic flows have received considerable attention in the scientific literature owing to fundamental applications of such flows in aeronautics (Soundalgekar and Takhar, 1977), biomechanics (Rao and Deshikachar, 1986), astrophysics (Singh, 1984), electroplating flows in chemical materials processing (Yonemochi and Agoyaki, 2000), Czochralski growth melt control with magnetic fields in chemical engineering (Munakata and Tanasawa, 1990), and magnetohydrodynamic (MHD) energy generators (Tanaka, 1985). Planar and rotating Couette hydromagnetic flows have in particular been extensively studied. Gupta and Arora (1974) studied analytically the MHD Couette flow between two parallel, nonconducting, infinite planes, one oscillating and the other stationary, for small magnetic Prandtl number. Kulshrestha and Puri (1983) studied the wave structures in oscillatory hydromagnetic plane Couette flow under a pressure gradient. Rao (1984) investigated transient hydromagnetic flow between two torsionally oscillating eccentric disks, showing that a uniform rotation of the central region arises in the steady flow but is absent for the oscillatory flow case. Mai et al. (1999) studied both theoretically and experimentally the axial oscillation of a magnetically levitated nonmagnetic liquid column in a magnetic fluid in the presence of a nonuniform weak magnetic field. Singh (2000) presented exact solutions for magnetohydrodynamic oscillatory Ekman boundary layer flow in a rotating parallel-plate channel with one plate at rest and the other oscillating in its own plane, under a transverse magnetic field. Singh and Sharma (2001) studied hydromagnetic Couette flow and heat transfer in a horizontal channel with transverse sinusoidal injection at the stationary plate and constant suction at the other plate in uniform motion. The above studies were all confined to purely fluid regimes and scenarios wherein the applied magnetic field is transverse to the flow plane or aligned with the flow. However, in many industrial applications, the regime may be a porous medium and the magnetic field may be inclined, i.e., oblique to the plane of the flow. Inclination of the magnetic field may have a significant effect on velocity distributions in, for example, MHD generator systems. Studies considering inclined magnetic field effects on either steady or transient hydromagnetic flows have been presented by Ghosh (2001) and Ghosh and Pop (2002, 2006). Very recently Ghosh et al. (2010) considered the effects of Maxwell displacement currents on unsteady rotating hydromagnetic channel flow. Guria et al. (In press) also recently investigated the oscillatory hydromagnetic Couette channel flow in a rotating system under an inclined magnetic field, for the case where the upper plate is held at rest and the lower plate oscillates non-torsionally. They obtained exact solutions and also discussed asymptotic behavior of the solutions, showing that a thin boundary layer is generated close to the lower oscillating plate and that boundary layer thickness is enhanced with greater magnetic field inclination. Oscillatory flows in porous media are also of great interest in materials processing and astrophysical flows. Generally, the flows in such regimes are viscous-dominated and associated with low Reynolds numbers. Singh and Verma (1995) studied the viscous flow in a highly porous medium when the free stream velocity oscillates in time about a nonzero constant mean. They employed a periodic permeability and derived expressions for the transient velocity, transverse velocity component, and skin friction using the series expansion method, showing that separation is suppressed in the porous medium even with larger frequency of the free stream oscillations. Graham and Higdon (2002) used the finite element method to study oscillatory forcing flow


Oscillatory Rotating Oblique MHD Flow

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through porous media, showing that for strong forcing in the form of sinusoidal oscillations, the mean flow rate is lowered to 40% of its unforced steady-state value. They also found that for a porous medium occupied by two fluids with disparate viscosities, oscillatory forcing can be successfully employed to suppress the flow rate of the less viscous fluid, with negligible effect on the more viscous fluid. As such this study showed that forcing can be employed as an oscillatory filter mechanism to separate two fluids of different viscosities, driving them in opposite directions in the porous medium, a feature of some importance in enhanced oil recovery processes. Several investigations of hydromagnetic oscillatory flow in electrically conducting saturated porous media have also appeared. Sengupta and Ray (1990) considered the oscillatory hydromagnetic Couette flow in a rotating porous medium channel for the case of a viscoelastic fluid. Dhulipala et al. (2008) obtained analytical solutions for viscous hydromagnetic flow in a porous medium near an oscillating infinite porous flat plate in the presence of a transverse magnetic field of uniform strength, fixed relative to the fluid, showing that the porous medium decelerates flow. Khan et al. (2006a) considered the magnetohydrodynamic flow of a generalized Oldroyd-B fluid in a circular pipe containing a porous regime using a modified Darcy law for simulating the porous medium. Khan et al. (2006b) further studied using fractional calculus, the effect of Hall currents, and oscillatory behavior on dynamics of an Oldroyd-B flow through a porous space. Hayat et al. (2007) considered the combined effects of rotation and Hall currents on oscillating hydromagnetic non-Newtonian flow in a Darcian porous medium. They considered flows induced by general periodic oscillations and elliptic harmonic oscillations of a plate. Khan et al. (2007) derived closed-form solutions for transient magnetohydrodynamic rheological flow in a porous medium channel, showing that the velocity profile is boosted with an increase of retardation time and permeability, but reduced with an increase in relaxation time and magnetic parameter. Khan et al. (2009a) studied the influence of variable suction and heat transfer on the oscillatory hydromagnetic flow in the presence of slip at the wall, utilizing a second-grade non-Newtonian model and a modified Darcian formulation. Khan et al. (2009b) further studied analytically the oscillatory rotating hydromagnetic rheological flow through a porous medium with the transient effects induced by plate oscillations, using the fractional calculus approach. The above studies have, however, not considered the influence of an oblique magnetic field. Be´g et al. (2009) were among the first researchers to investigate numerically the influence of an inclined magnetic field on porous medium hydromagnetic flow with Hall currents using the robust network simulation technique. In the present study we shall consider the primary and secondary velocity distributions in viscous, oscillatory hydromagnetic Couette flow in a horizontal, fluid-saturated porous medium parallel-plate channel rotating about an axis perpendicular to the plane of the plates under the action of a uniform oblique magnetic field, B0, inclined at an angle h to the axis of rotation. Exact solutions are developed using the Laplace transform method (LTM). Additionally, asymptotic solutions are presented. Such a study has thus far not appeared in the scientific literature and is of importance in rotating MHD energy generators, planetary hydromagnetics, etc.

Mathematical Model Consider the transient MHD Couette flow of a viscous, incompressible, electricallyconducting Newtonian fluid through a horizontal layer of saturated, homogenous,

O. A. Be´g et al.


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isotropic porous medium confined between two infinite parallel plates, a distance d apart. The Darcian model for porous media drag is employed, which is generally valid for low Reynolds number, viscous-dominated regimes. The channel plates rotate in unison with uniform angular velocity, X, about an axis perpendicular to the plates under the influence of a uniform magnetic field, B0, which is inclined at an angle h to the positive direction of the axis of rotation (z-axis). The regime is illustrated in Figure 1. The coordinate system is such that the x-axis is directed along the lower plate, the y-axis is perpendicular to it, and the z-axis is normal to the x-y plane. The flow is induced due to the non-torsional oscillation of the lower plate. Magnetic Reynolds number is small enough to discard magnetic induction effects. Since the plates are infinite along the x- and y-directions, all physical quantities will be functions of the independent variables z, t only. We neglect charge density and external electrical field effects, and polarization voltage is also negligible. Denoting (u,v) as the components of velocity along the x- and y-directions, the equations governing the flow in the regime, i.e., the primary and secondary momentum equations, may be shown to reduce to: @u @ 2 u rB20 n 2Xv ¼ n 2 ½cos2 h u u @t @z Kp1 q

ð1Þ

@v @ 2 v rB20 n þ 2Xu ¼ n 2 v v @t @z Kp1 q

ð2Þ

The corresponding initial and boundary conditions at the plates are: u¼v¼0 u ¼ U;

v¼0

u¼v¼0

for t 0

at z ¼ 0 ðlower plateÞ

at z ¼ d ðupper plateÞ

Figure 1. Physical model and coordinate system.

ð3aÞ ð3bÞ ð3cÞ


239

To render the model dimensionless, we introduce the following parameters: u v nt x0 d 2 u1 ¼ ; v1 ¼ ; s¼ 2; x¼ 2 ; U0 U0 d d 2 r d 1 K P ; Da ¼ 21 M 2 ¼ B20 d 2 ¼ ; Kp ¼ qn KP1 Da d


g¼

z ; d

K2 ¼

Xd 2 ; n

U ¼ U0 F ðsÞ; ð4Þ

where X denotes angular velocity, n is kinematic viscosity, q is fluid density, r is electrical conductivity, B0 is magnetic flux density, t is time, Kp1 is the permeability (hydraulic conductivity) of the porous regime between the plates, h is inclination of the applied magnetic field to the z-axis, g is nondimensional separation of the plates, u1 is the dimensionless primary velocity, v1 is the dimensionless secondary velocity, s is dimensionless time, x is dimensionless frequency of oscillation, K2 is the rotational parameter (inverse of the Ekman number, i.e., ratio of Coriolis to viscous forces), F(s) is the dimensionless oscillatory velocity function, M2 is the hydromagnetic body force parameter (square of the Hartmann number), Kp is dimensionless permeability function, and Da is the Darcy number. Implementing the variables (4) into Equations (1) and (2) we arrive at the following pair of dimensionless primary and secondary momenta equations for the regime: @u1 @ 2 u1 1 u1 2K 2 v1 ¼ M 2 u1 cos2 h Da @s @g2

ð5Þ

@v1 @ 2 v1 1 v1 þ 2K 2 u1 ¼ M 2 v1 2 Da @s @g

ð6Þ

The dimensionless boundary conditions at the plates now take the form: u1 ¼ v 1 ¼ 0 u1 ¼ F ðsÞ;

v1 ¼ 0

u1 ¼ v 1 ¼ 0

for s 0 and

0g1

at g ¼ 0 ðlower plateÞ

at g ¼ 1 ðupper plateÞ

for s > 0

for s > 0

ð7aÞ ð7bÞ ð7cÞ

Since the flow is induced due to the movement of the lower plate with uniform velocity F(s), the oscillating flow velocity can be assumed as F ðsÞ ¼ aeixs þ beixs

ð8Þ

where a and b are arbitrary constants.

Laplace Transform Method (LTM) Solutions Employing the LTM technique, Equations (5) and (6) are transformed to: su1 2K 2 v1 ¼

@ 2 u1 1 u M 2 cos2 h u1 2 Da 1 @g

ð9Þ

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sv1 þ 2K 2 u1 ¼

@ 2 v1 1 v M 2 v1 Da 1 @g2

ð10Þ

The boundary conditions at the plates also transform to: u1 ¼

a b þ ; s ix s þ ix

v1 ¼ 0

at g ¼ 0 ðlower plateÞ


u1 ¼ v1 ¼ 0 at g ¼ 1 ðupper plateÞ

ð11aÞ ð11bÞ

Since the flow is induced due to the movement of the lower plate with uniform velocity F(s) the oscillating flow velocity, F , can be assumed as defined in Equation (8). The closed-form solutions for Equations (9) and (10) subject to conditions (11a, b) may thus be expressed as: "

# 2 2 2 2 2 2 M sin h þ iN Sinhr ð1 gÞ M sin h iN Sinhr ð1 gÞ 2 1 u1 ¼ F Sinhr2 Sinhr2 2iN 2 2iN 2 ð12Þ v1

¼

2K 2 iN 2

Sinhr1 ð1 gÞ Sinhr2 ð1 gÞ F Sinhr1 Sinhr2

ð13Þ

where: 1=2 2 2 M ð1 þ cos hÞ þ þ iN Da

ð14aÞ

1=2 1 2 2 2 2 M ð1 þ cos hÞ þ r2 ¼ s þ iN 2 Da

ð14bÞ

1=2 N 2 ¼ 16K 4 M 4 sin4 h

ð14cÞ

1 r1 ¼ s þ 2

2

2

Taking the inverse Laplace transforms of (12) and (13) we arrive at the following: aeixs sinhða2 ib2 Þð1 gÞ 2 2 2 u1 ¼ ðM sin h þ iN Þ sinhða2 ib2 Þ 2iN 2 sinhða þ ib1 Þð1 gÞ 1 2 2 2 ðM sin h iN Þ sinhða1 þ ib1 Þ beixs sinhða1 ib1 Þð1 gÞ 2 2 2 þ ðM sin h þ iN Þ sinhða1 ib1 Þ 2iN 2 sinhða ib 2 2 2 Þð1 gÞ 2 2 ðM sin h iN Þ sinhða2 ib2 Þ 1 X a b a b þ þ þ 2np sin npg es2 s es 1 s s s s s ix þ ix ix þ ix 2 2 1 1 n¼1

ð15Þ

Oscillatory Rotating Oblique MHD Flow sinhða1 þ ib1 Þð1 gÞ sinhða2 ib2 Þð1 gÞ sinhða1 þ ib1 Þ sinhða2 ib2 Þ 2 ixs 2bK e sinhða2 ib2 Þð1 gÞ sinhða1 ib1 Þð1 gÞ þ sinhða2 ib2 Þ sinhða1 ib1 Þ iN 2 1 X a b a b þ þ þ 2np sin npg es 1 s es 2 s s s s s ix þ ix ix þ ix 1 1 2 2 n¼1

241

2aK 2 eixs v1 ¼ iN 2

ð16Þ


where: 2( 3 )1=2 2 1=2

14 2 2 2 5 M 2 ð1 þ cos2 hÞ þ þ 2x þ N 2 þ M 2 ð1 þ cos2 hÞ þ a1 ¼ 2 Da Da ð17aÞ 2( 3 )1=2 2 1=2 14 2 2 5 M 2 ð1 þ cos2 hÞ þ þð2x þ N 2 Þ2 M 2 ð1 þ cos2 hÞ þ b1 ¼ 2 Da Da ð17bÞ 2( 3 )1=2 2 1=2 14 2 2 2 5 M 2 ð1 þ cos2 hÞ þ þð2x N 2 Þ þ M 2 ð1 þ cos2 hÞ þ a2 ¼ 2 Da Da ð17cÞ 2(

3 )1=2 2 1=2 1 2 2 5 M 2 ð1 þ cos2 hÞ þ þð2x N 2 Þ2 M 2 ð1 þ cos2 hÞ þ b2 ¼ 4 2 Da Da ð17dÞ 1 2 M 2 ð1 þ cos2 hÞ þ s1 ¼ n2 p2 þ þ iN 2 2 Da

1 s2 ¼ n p þ 2 2 2

2 2 M ð1 þ cos hÞ þ iN Da 2

2

ð17eÞ

ð17fÞ

The solutions (15) and (16) are valid for sin h < 4K 2 =M 2 . The and signs in (15) and (16) are in agreement with 2x > N2 and 2x < N2, respectively. For the special case of steady-state flow (s ! 1), the solutions (15) and (16) reduce to: aeixs sinhða2 ib2 Þð1 gÞ 2 2 2 u1 ¼ ðM sin h þ iN Þ sinhða2 ib2 Þ 2iN 2 sinhða1 þ ib1 Þð1 gÞ 2 2 2 ðM sin h iN Þ sinhða1 þ ib1 Þ

O. A. Be´g et al.

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beixs sinhða1 ib1 Þð1 gÞ 2 2 2 þ ðM sin h þ iN Þ sinhða1 ib1 Þ 2iN 2 sinhða2 ib2 Þð1 gÞ 2 2 2 ðM sin h iN Þ sinhða2 ib2 Þ


v1 ¼

ð18Þ

sinhða1 þ ib1 Þð1 gÞ sinhða2 ib2 Þð1 gÞ sinhða1 þ ib1 Þ sinhða2 ib2 Þ 2 ixs 2bK e sinhða2 ib2 Þð1 gÞ sinhða1 ib1 Þð1 gÞ þ ð19Þ sinhða2 ib2 Þ sinhða1 ib1 Þ iN 2 2aK 2 eixs iN 2

Following Yamamoto and Iwamura (1976) the porous medium is simulated via Darcy’s law as an assemblage of small identical spherical particles fixed in space. Angular frequency of oscillation in unsteady motion will have an important role in steady-state oscillation within a porous medium (s ! 1). In response to a steady-state oscillation, resonance exhibits a dynamo mechanism of significance in astronautical and astrophysical flow regimes. Microwave background radiation has been observed. For example, in connection with this phenomenon. Ghosh (2001) and Ghosh and Pop (2006) analyzed the steady-state case for xs ! 0 (in the absence of angular frequency of oscillation) showing that this is consistent 1=2 with x > 1=2½16K 4 M 4 sin4 h . They also indicated that for xs ¼ p=2 subject 1=2 to x > 1=2 cos xs½16K 4 M 4 sin4 h the angular frequency x > 0. The condition 1=2

x > 1=2 cos xs½16K 4 M 4 sin4 h is valid for turbulent dynamo mechanisms on a solar and also terrestrial context when xs ¼ p=2. Therefore, resonance exhibits a transition from laminar to turbulent flow with an irregular fluctuation of electrons to migrate from large eddy motions to smaller eddy motions with the growth of a magnetic field at the resonant level. Further details are described in Ghosh (2001) and Ghosh et al. (2006, 2010). It may further be demonstrated that when xs ¼ 0, the solutions (18) and (19) further reduce to the case for steady-state one-dimensional hydromagnetic flow (s ! 1). Our computations, discussed later, reveal that the primary flow does exist when xs ¼ 0 and p=2; however, secondary flow vanishes when xs ¼ 0. Thus, the steady-state approach in the current magnetohydrodynamic study may exercise a marked influence on the primary (main) flow for xs ¼ 0 and p=2. In this context, steady-state oscillations embody a physical meaning when xs ¼ p=2 under resonance.

Asymptotic Behavior of Solutions We now consider several asymptotic cases of the general solutions. Case I: M2 > (1), K2 O(1), x O(1) and a ¼ b ¼ 1/2 In this case, again rotational (Coriolis) forces are of the same order as viscous hydrodynamic forces. Magnetohydrodynamic drag is however much greater than the viscous force, corresponding to very strong magnetic field imposition. Expressions (18) and (19) for this situation reduce to: u1 ¼

1 a1 g 2 2 e M sin hfsinðxs b2 gÞ sinðxs b1 gÞg 2N 2 þ N 2 fcosðxs b2 gÞ þ cosðxs b1 gÞg

ð28Þ


v1 ¼

2K 2 a1 g e ½sinðxs b1 gÞ sinðxs b2 gÞ N2

245

ð29Þ


where 1 2 1=2 a1 ¼ a2 ¼ pffiffiffi M 2 ð1 þ cos2 hÞ þ Da 2

ð30aÞ

2x N 2 b1 ¼ b2 ¼ pffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 M 2 ð1 þ cos2 hÞ þ Da

ð30bÞ

It is evident from (28) and (29) that in this case, there exists a single-deck boundary layer of thickness O(1=a1). This may be identified as the modified Hartmann layer. Analyzing (28) it is apparent that the boundary layer in the primary flow is unaffected by K2 (i.e., rotation) but is dependent on Hartmann number squared (M2), and Darcy number (Da), under an oblique magnetic field. A phase lag of p=2 (when xs ¼ p=2) will arise in this boundary layer. Inspection of (29) shows that the boundary layer in the secondary flow is dependent on rotation (K2), Hartmann number squared (M2), Darcy number (Da), and frequency (x). Again a phase lag of p=2 (when xs ¼ p=2) will arise in this boundary layer. In the presence of a longitudinal magnetic field (h ¼ p=2) the boundary layer thickness will be affected by M2 and Da.

Laplace Transform Method Numerical Solutions and Discussion We now provide numerical evaluations of the generalized solutions given in (15) and (16). These are shown in Figures 2 to 13, where we have presented computations for the variation of primary and secondary velocity components (u1, v1) with all key

Figure 2. Primary velocities for K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ p=2.


246

O. A. Be´g et al.

Figure 3. Primary velocities for K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ 0.0.

parameters in the model. Although clearly oscillatory flow will be exhibited in distributions plotted against xs we have opted here to present spatial velocity distributions at a fixed time in the flow. All data are provided in the figure captions. Figures 2 and 3 present the primary velocity distributions (u1) with various Hartmann numbers squared magnetic field (M2) with K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, and h ¼ p=4 for xs ¼ p=2 (transient) and xs ¼ 0 (steady state, s ! 1), respectively. These plots correspond to a 45 orientation of the applied magnetic field to the rotational (z-) axis for high permeability media with strong rotation. There is a substantial difference in the two graphs. In Figure 2, clearly for all M2 the profiles are inversely parabolic and consistently negative, indicating that primary flow reversal

Figure 4. Primary velocities for K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, M2 ¼ 5.0, xs ¼ 0.0.



247

Figure 5. Primary velocities for K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, M2 ¼ 5.0, xs ¼ p=2.

is always present across the channel. Magnetic field therefore exerts an inhibiting effect on primary flow. With larger M2 the magnetohydrodynamic drag is greater and the flow is more markedly decelerated, i.e., u1 values are more negative. With lower M2 values the opposite effect is observed and values are least negative. We also note that there is a nonzero primary velocity at the lower late (g ¼ 0) owing to the oscillating nature of the plate, whereas at the upper stationary plate the velocity vanishes. The maximum effect of magnetic field is witnessed in the lower half-space of the channel (0 < g < 0.5) at g 0.3. Conversely in Figure 3, the velocity profiles are always positive, showing that in the steady state the flow is significantly

Figure 6. Primary velocities for K2 ¼ 2.0, M2 ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ p=2.


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Figure 7. Primary velocities for K2 ¼ 2.0, M2 ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ 0.0.

stabilized and back flow totally eliminated. In the steady-state case therefore the maximum primary velocity is always associated with a smooth descent to the minimum at the lower plate. With the weakest magnetic field (M2 ¼ 5), the primary velocity decay across the channel is approximately linear. With increasing M2 values the flow is indeed decelerated and profiles become increasingly curved. Values of the primary velocity in Figure 3 are also several orders of magnitude larger than the transient state (Figure 2). In Figures 4 and 5, the effect of magnetic field orientation (h) on primary velocity distributions (u1) for xs ¼ 0 (steady state, s ! 1) and xs ¼ p=2 (transient), respectively. The orientational effect of the magnetic field arises only in the primary

Figure 8. Primary velocities for Da ¼ 5.0, M2 ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ 0.0.



249

Figure 9. Primary velocities for Da ¼ 5.0, M2 ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ p=2.

flow equation (5) in the term M2u1cos2h. Clearly, the magnetohydrodynamic drag will be minimized for constant M2, when cos h is minimized, i.e., h ¼ p=2 (and cos p=2 ¼ 0 so that the magnetic field is along the positive x-axis, i.e., aligned to the primary flow) and maximized when cosh is maximized, i.e., h ¼ 0 (and cos 0 ¼ 1 so that the magnetic field is along the positive z-axis and transverse to the primary flow). As h increases from 0 through p=6, p=4, p=3, to p=2, the hydromagnetic drag force, M2u1cos2h, for the steady-state case (Figure 4) is continuously reduced, which serves to accelerate the primary flow. The profiles in Figure 4 are clearly in accordance with this where we observe that the primary flow velocity is largest for h ¼ p=2, and lowest for h ¼ 0. In all these profiles the values are positive, showing again that in the steady state back flow does not occur across the channel span.

Figure 10. Secondary velocities for K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, h ¼ p=4 xs ¼ p=2.


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Figure 11. Secondary velocities for K2 ¼ 2.0, Da ¼ 5.0, x ¼ 0.2, M2 ¼ 5.0, xs ¼ p=2.

Conversely, back flow is present for all inclinations of the magnetic field for the transient case (Figure 5). With increasing inclination (weaker Lorentz drag) the back flow is intensified. For the x-direction aligned magnetic field case, i.e., h ¼ 0, back flow is weakest, almost negligible. This indicates that back flow is inhibited with lower inclinations of the applied magnetic field but exacerbated with larger inclinations. Again, the primary velocities are several orders of magnitude lower for the transient case (Figure 5) than for the steady-state case (Figure 4). Figures 6 and 7 depict the response of primary velocity distributions (u1) to various Darcy numbers (Da) for the transient (xs ¼ p=2) and steady-state (xs ¼ 0 (steady state, s ! 1)) cases, respectively. With an increase in Da the Darcian porous 1 media resistance term, Da u1 , is clearly reduced in the primary flow momentum

Figure 12. Secondary velocities for K2 ¼ 2.0, M2 ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ p=2.



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Figure 13. Secondary velocities for Da ¼ 5.0, M2 ¼ 5.0, x ¼ 0.2, h ¼ p=4, xs ¼ p=2.

conservation Equation (5). As such the resistance of the porous media particles to the primary flow will be lowered, causing flow acceleration. In Figure 6 although once again back flow is present in the channel, the magnitudes of the velocity increased with an increase in Da, i.e., permeability increasing accelerates the flow. Similarly, in Figure 7 (steady state) an increase in Da is also observed to strongly accelerate the primary flow across the channel width. Velocity magnitudes in the steady-state case are clearly significantly greater than in the transient case. In both cases the lowest primary velocity magnitudes accompany the least permeable scenario, i.e., Da ¼ 0.025. The presence of a porous medium therefore exerts a marked regulatory effect on the primary flow and can be exploited as a useful control mechanism in hybrid MHD generator systems. Figures 8 and 9 illustrate the spatial primary velocity profiles (u1) for selected values of the inverse Ekman number (K2) for the steady state (xs ¼ 0 (steady state, s ! 1)) and transient (xs ¼ p=2) scenarios, respectively. For the former case (Figure 8) back flow is observed only for very strong rotation, i.e., for K2 ¼ 5. This arises at some distance from the lower plate and is sustained to the upper plate. Very strong Coriolis forces therefore reverse the primary flow entirely in the upper half-space of the channel (0.5 < g < 1) since viscous forces are swamped. In the lower channel half-space closer to the lower plate back flow does not arise although there is a sharp descent in primary flow from the lower plate. For K2 ¼ 1, where rotational and viscous forces are exactly balanced in the regime, a more gradual decay in primary velocity, u1, is witnessed from the maximum value at the lower plate to the least value at the upper plate. For K2 ¼ 3, the primary flow is clearly accelerated and in fact a velocity overshoot surfaces at g 0.15. Back flow is, however, never induced for this value of K2 anywhere in the regime. The presence of strong back flow with K2 ¼ 5 indicates that there exists a critical rotational parameter value between 3 and 5 where flow reversal is initiated. Such behavior would be instrumental in optimizing primary flow performance in rotating MHD generators in working operations since while increasing the rotational velocity (and therefore Coriolis force) serves to boost primary flow, further increase is noticeably counterproductive.


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For the transient case another interesting trend is observed from inspection of Figure 9. While flow reversal is evident for all values of K2, as this parameter is increased from 1 to 3, back flow is clearly enhanced; however, with subsequent increase in K2 to 5, back flow is reduced somewhat, although it remains more intense than for the K2 ¼ 1.0 case. While the magnitudes of primary velocity are very small compared with the steady-state case (Figure 8), our computations clearly indicate that a transition in flow behavior arises at some value of K2 between 3 and 5. In Figures 10–12 the secondary velocity distributions are presented across the channel for only the transient case (xs ¼ p=2), for the effects of Hartmann number squared (M2), magnetic field inclination (h), Darcy number (Da), and rotational parameter (K2). In all these graphs back flow never arises. Figure 10 shows that the magnetohydrodynamic drag force in Equation (6), M2v1, clearly reduces secondary velocity as the M2 parameter rises. The peaks in the profiles also migrate increasingly towards the lower plate as M2 increases. Magnitudes of the secondary velocity are, however (as with primary velocity for the transient case), very small. With an increase in magnetic field inclination, h, secondary velocity, v1, as shown in Figure 11, strongly increased. The secondary momentum Equation (6) is coupled to the primary Equation (5), only via the rotational body force term, 2K2u1. The inclination term in (5), i.e., M2u1cos2h, therefore indirectly affects the secondary momentum field. For h ¼ 0 this force will be maximized, which will have a maximum deceleration effect on both the primary and secondary flows. For h ¼ p= 2, the reverse effect will be apparent with minimized magnetic drag and maximized primary and secondary velocity magnitudes. An increase in Darcy number (Figure 12), as with the primary flow, causes a substantial increase in secondary velocity, v1. In the secondary momentum Equation 1 (6), Darcian porous impedance, Da v1 , is inversely related to Darcy number. With increasing Da values the regime will become increasingly permeable and progressively less resistance will be offered to the viscous flow. Secondary flow velocity is therefore substantially stifled with low permeability, i.e., low Darcy number. Finally in Figure 13, an increase in inverse Ekman number, K2, is found to consistently boost the secondary flow velocity. Unlike with the primary flow response, there is no critical rotation parameter value for which flow behavior is reversed. Increasing rotational velocity, X, and therefore K2, serves to continuously accelerate the secondary flow. However, we do observe that a much greater acceleration is achieved by increasing K2 from 1 to 3, as opposed to the increase from 3 to 5, showing that while greater rotational intensity will aid the secondary flow, the benefits of very large rotational parameters will be much less pronounced.

Conclusions Laplace transform solutions have been derived for the viscous, rotating, magnetohydrodynamic (MHD), oscillating Couette flow in a porous medium parallel plate channel under the action of an inclined magnetic field. Asymptotic behavior of the solutions has also been addressed. Our numerical results have indicated that: (i) Back flow is always sustained in the primary flow (u1) for the transient case (xs ¼ p=2) but never arises for the steady-state case (xs ¼ 0.0).



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(ii) Primary velocity magnitudes are much greater in the steady-state case (xs ¼ 0.0) than the transient case (xs ¼ p=2). (iii) Back flow never arises in the secondary flow regime for the transient case (xs ¼ p=2). (iv) Increasing magnetic field (Hartmann number squared, M2) generally acts to impede both the primary (u1) and secondary flow (v1). (v) Increasing magnetic field inclination (h) serves to accelerate both the primary flow and the secondary flow. (vi) Increasing Darcy number (Da) accelerates both the primary flow and the secondary flow. (vii) Increasing inverse Ekman number (K2), in the transient case, is associated with increasing reversal in the primary flow up to a critical value, after which it is reduced. However, with increasing K2 the secondary flow is continuously accelerated and flow reversal never arises. The present study finds possible applications in, for example, hybrid designs for an MHD rotating power generator exploiting porous materials and is currently being extended to consider more complex geometries, CFD (computational fluid dynamics) simulations, and also compressibility effects for gaseous working media in hybrid chemical engineering applications (Cramer and Pai, 1973). The results of these studies will be communicated in the near future.

Acknowledgments The authors are grateful to the reviewer for his positive and constructive comments, which have served to improve the present article.

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