magnetohydrodynamic viscous flow in a rotating

MAGNETOHYDRODYNAMIC VISCOUS FLOW IN A ROTATING POROUS MEDUM CYLINDRICAL ANNULUS WITH AN APPLIED RADIAL MAGNETIC FIELD O.D. Makinde 1, O.A. Bég2, and H.S. Takhar3 1

Faculty of Engineering, Cape Peninsula University of Technology, P. O. Box 1906, Bellville 7535, South Africa. 2 Engineering Magnetohydrodynamics Research, Mechanical Engineering Department, Sheaf Building ,Room 4112, Sheffield Hallam University, Sheffield, S1 1WB, England, UK. Email: [email protected] 3 Applied Mathematics and Materials Modelling Research, 75 The Avenue, Sale, Manchester, M33 4GA,England, UK. Received 31 October 2008; accepted 2 March 2009

ABSTRACT We study the steady, axisymmetric, magnetohydrodynamic (MHD) flow of a viscous, Newtonian, incompressible, electrically-conducting fluid through an isotropic, homogenous porous medium located in the annular zone between two concentric rotating cylinders in the presence of a radial magnetic field. Transformation variables are introduced to render the tangential and axial momentum equations non-dimensional with corresponding no-slip boundary conditions. Closed-form solutions are obtained using Bessel and Lommel functions. Axial velocity (UZ) is found to decrease markedly with an increase in Hartmann number (Ha) with radial coordinate, with profiles becoming increasingly curved for stronger magnetic field. An approximately linear decay in axial velocity from the inner cylinder to the outer cylinder is computed for the non-magnetic case (Ha = 0). Dimensionless tangential velocity (Uθ) is also suppressed with magnetic field increasing; however profiles increase consistently from the inner cylinder to the outer cylinder i.e. over the domain 0.5 ≤R ≤ 1. Again for the electrically non-conducting case, the tangential velocity distribution is approximately linear. An increase in relative rotation parameter, N, which embodies the relative speeds of rotation ωa of the inner to the outer cylinders (= 1 ) strongly boosts the tangential velocity (Uθ) ω2b throughout the annular regime, in particular at the interior wall of the annulus (R = 0.5). Increasing Darcy number induces a strong acceleration in both axial and tangential flow i.e. increases both components of velocity, although for low Da (= 0.01) there is a significant reversal in axial flow further from the inner wall of the annulus. An increase in axial pressure gradient induces a marked increase in the axial velocity component, (Uz). Applications of the analysis include magnetic field control of rotating annular filter porous media membrane systems, drilling flows in geothermal operations, MHD power generators, geophysical hydromagnetics and fundamental magnetofluid dynamic studies. Keywords: Annular hydromagnetic flow; porous media; radial magnetic field; Bessel functions; Lommel functions; relative rotation; energy systems. Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.

Magnetohydrodynamic Viscous Flow in a Rotating Porous Medum Cylindrical

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1 INTRODUCTION Hydromagnetic flows in rotating systems continue to stimulate significant research in the engineering science and applied mathematics fields. Numerous investigations have been reported presenting analytical and also numerical solutions for a plethora of different geometrical scenarios. Such flows are important in rocket propulsion control, crystal growth technology, astrophysical plasma fluid dynamics, tribological regulation in moving machine parts and also magnetohydrodynamic (MHD) energy generators (Bég 2007). Relatively few studies however have considered transport in annular regimes, for example in the annulus intercalated between rotating spheres or rotating cylinders or even rectangular annular ducts. Flow in rotating annular spaces frequently arise for example in drilling operation of oil and gas wells. The accurate simulation of the flow of the drilling mud in the annular space between the well wall and the drill pipe is critical in evaluating the variation in the mud pressure within the well-bore, the frictional pressure drop and the efficiency of the transport of the rock drill cuttings. Magnetic fields may also be utilized to control such flow operations where the exiting fluid is electrically-conducting i.e. ionized. In the manufacturing of certain chemicals, excellent control is also achievable with radial magnetic field imposition on concentric rotating cylinder systems. Although modern computational fluid dynamics (CFD) codes have revolutionized the simulation of these and other hydromagnetic systems, there remain an exceedingly large group of problems which still require more detailed mathematical investigation. Such investigations also provide an excellent reference for benchmarking more complex simulations and validating CFD analyses. Interest in rotating hydromagnetic flow in annular spaces was initiated in the late 1950s with an important analysis by Globe (Globe 1959) who considered fully developed laminar MHD flow in an annular channel. Jain and Mehta (Jain and Mehta 1962) examined wall suction/injection effects on the Globe problem. Uberoi and Chow (Uberoi and Chow 1966) further investigated the hydromagnetic viscous flow in an annular pipe region. Nath (Nath 1970) obtained closedform solutions for fully developed axial flow through two concentric rotating cylinders with radial magnetic field presence, showing that both axial and tangential velocity components are significantly decreased with Hartmann number increasing. Baylis and Hunt (Baylis and Hunt 1971) presented a detailed theoretical and experimental study of hydromagnetic Newtonian flow in a rectangular annular channel, extending the earlier model of Hunt and Stewartson (Hunt and Stewartson 1965) with conducting walls aligned with the magnetic field and non-conducting walls transverse to it. Bathaiah and Venugopal (Bathaiah and Venugopal 1982) examined the MHD flow between two concentric rotating cylinders under the influence of a uniform magnetic field, with non-erodable and non-conducting porous lining on the inner wall of the outer cylinder under the influence of a uniform radial magnetic field of the form, Br = A/r where r is radial coordinate, discussing the effects of ratio of the velocities of the cylinders, the slip parameter, the porosity parameter and the Biot number on flow variables. Basu and Mandal (Basu and Mandal 1977) obtained closed form finite Hankel transform solutions for the transient hydromagnetic flow between two co-axial cylinders including induced magnetic field effects and showing that frictional couple per unit length increases with the increase in the strength of the magnetic field. Antimirov and Kolyshkin (1984) studied the unsteady magnetohydrodynamic flow in an annular channel with radial magnetic field. Takhar et al. (Takhar et al. 1989) studied numerically the hydromagnetic stability of dissipative Couette flow in a rotating narrow gap annulus showing that the effect of the magnetic field is to inhibit the onset of instability, with a more pronounced effect in the presence of conducting walls than in the presence of non-conducting walls. Kandaswamy et al. (Kandaswamy et al. 1994) analyzed magnetohydrodynamic flow in Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.

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the annulus between rotating eccentric cylinders. Panja et al. (Panja et al 1996) studied computationally the effects of wall mass flux on MHD viscoelastic flow in a cylindrical annulus. Li et al (Li et al. 2005) considered the magnetohydrodynamic heat transfer and flow in a liquid metal-gas annular regime under a transverse magnetic field, showing that the pressure drop in the MHD annular flow is of the same order of magnitude as in the singlephase MHD pipe flow under similar liquid metal flow conditions. These studies were restricted to purely fluid regimes in the annulus. However many applications in geomechanics, geophysics and industrial systems exist in which the rotating annulus contains a porous medium. For example in chemical engineering processes, the rotating annular membrane filter exploits porous media to achieve better control of membrane fouling; the porous medium coupled with a porous wall enables transmembrane pressure to be decoupled from axial flow in such systems and fouling components can be identified with the aid of permeation fluxtransmembrane pressure plots (Belfort et al. 1993). The most popular model employed in low-velocity porous media hydrodynamics is the classical Darcy law, which is well-suited to implementation in viscous fluid dynamics (Bear 1988). It is generally valid for pore Reynolds numbers up to 10; thereafter inertial porous drag effects may become significant. Analyses of annular porous flows are of interest in petroleum engineering, contamination technologies and also solar energy collector systems. Chang and Papadopoulos (Chang and Papadopoulos 2000) analyzed the electro-osmotic hydromechanics in a Darcian porous medium concentric annular geometry as a simulation of the hydraulic wash column liquid separator. Prasad and Venkatasiva Murthy (Prasad and Venkatasiva Murthy 1991) used a Laplace transform technique to study the unsteady flow in a rotating porous annulus due to suction at the walls wherein the flow is disturbed by imposing suction/injection at the outer/inner cylindrical boundaries respectively. Deiber and Bortolozzi (Deiber and Bortolozzi 1998) studied free convection heat transfer in a porous medium annulus with boundary vorticity effects. Mota et al (Mota et al. 2000) considered the free convection heat transfer in an elliptic annular porous medium regime. The Darcian model has also been employed successfully by Bég et al. (Bég et al. 2007) in investigating the micro-structural double-diffusive convective heat and mass transfer in a porous regime with buoyancy effects using the finite element method. In the past two decades considerable attention has also been devoted to modeling hydromagnetic flows in porous media saturated with electrically-conducting fluids. Applications are diverse for such flows including magnetic-metallurgy, control of propulsion combustion processes, tissue biomagnetics, pollution regulatory systems exploiting contaminant conductivity properties, intelligent magnetic materials design etc. An excellent investigation of hydromagnetic fluid dynamics in Darcian porous materials was presented by Geindreau and Auriault (Geindreau and Auriault 2002) who have provided a rigorous derivation of the permeability tensor. Bég et al. (Bég et al. 2005) obtained perturbation solutions for the transient hydromagnetic convection boundary layer flow in a Darcian porous medium. Khan et al (Khan et al 2007) studied the unsteady hydromagnetic flow in a rectangular cross-section duct containing a Darcian isotropic porous medium. More recently Bég et al. (Bég et al. 2009) obtained comprehensive network computational solutions for the two-dimensional transient hydromagnetic flow in a parallel-plate channel containing a Darcian porous material with dissipation and Hall/ionslip currents. Bég et al . (Bég et al . 2009) have also studied numerically the hydromagnetic free convection from a spherical body to a porous medium including inertial porous drag and heat source/sink effects. Rotating porous media flows are also of significant interest in materials processing, chemical engineering and geophysics and have been thoroughly reviewed by Vadasz (Vadasz 1998) for both isotropic and anisotropic regimes. For example in the context of hydromagnetics, Ghosh et al. (Ghosh et al. 2009) recently analyzed the radiative-convection magnetohydrodynamic gas flow in a rotating Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.


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system using the Laplace transform method. In the context of porous media studies, Prasad et al. (Prasad et al. 2008) studied numerically the influence of Hall currents and oblique applied magnetic field on rotating plasma magneto-hydrodynamic flow in a two-dimensional channel at generalized Ekman numbers. In the present study we shall extend the seminal work of Nath (Nath 1970) to consider the effect of Darcian drag forces on the axial fully developed MHD flow in the annular regime between two concentric rotating cylinders containing an isotropic porous medium, under the action of a radial magnetic field. Such a study has to the authors’ knowledge not appeared in the scientific communications thusfar and is an interesting addition to the existing body of knowledge.

2 MATHEMATICAL MODEL Let us consider the Newtonian, steady, incompressible, electrically-conducting viscous axial MHD flow between two concentric cylinders composed of insulating material and containing an isotropic, homogenous porous material saturated with the fluid, in an (r, θ, z) coordinate system. Magnetic Reynolds number is assumed to be small so that magnetic induction effects can be ignored, as described by Shercliff (Shercliff 1965). Also the applied radial magnetic field is weak enough to neglect Hall current and ionslip current effects. No separation of charges occurs and the body surface is not charged so that electrical field everywhere vanishes i.e. E = 0. An axial pressure gradient is applied and the cylinders are rotated at velocities which are lower than the threshold required to initiate Taylor vortices. The physical regime is illustrated in figure 1 below. z Rotating concentric cylinders of radii, a (inner) and b (outer)

ω2 ω1 r

Radial magnetic field, Br

Isotropic saturated annular porous medium

Figure 1: Physical Model and Coordinate System As shown by Nath (Nath 1970) the tangential velocity profile generated in a viscous liquid by the rotating cylinders (in the absence of Taylor vortex motions) becomes a function of the axial coordinate when an axial velocity is superimposed, irrespective of whether fully developed flow has been attained or not. For an annulus of sufficient extent, the tangential profile will approach the fully developed state when it becomes independent of axial position. The radial magnetic field, Br is generated by passing a steady electric current parallel to the axis of the coaxial cylinders, where the cylinders terminate at perfect electrodes which are connected via a load. As indicated by Nath (Nath 1970), an alternative technique to achieve the desired magnetic field is through the implementation of a permeable core within the annulus and a permeable cylinder shell outside the annulus. As such the magnetic flux lines would close via these permeable paths at significant distances from the regime of interest. Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.

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The “source” of the magnetic flux would be discs of permanently magnetized material between the paths and the annulus channel. Following Nath (Nath 1970) and implementing the Darcy drag force model, following Bég et al (Bég et al. 2009), the Navier-Stokes equations in cylindrical coordinates for the magnetohydrodynamic saturated porous regime can be shown to reduce to the following form:

ρuθ 2 r

=

∂p ∂r

(1)

∂uθ νu  ∂ 2 uθ 1 ∂uθ uθ ∂ 2 uθ  σuθ Br =ν  + − + − − θ  2 2 2 ρ ∂z r ∂r K r ∂z   ∂r 2

uθ

2

(2)

 ∂ 2 u z 1 ∂u z  σu z Br νu 1 ∂p =ν  + − − z  2 ρ r ∂r  K ρ ∂z  ∂r

(3)

∂ (rBr ) = 0 ∂r

(4)

2

in which uθ, uz are the tangential and axial velocity components, respectively, p is hydrostatic pressure, ρ is fluid density, σ is fluid electrical conductivity, ν is fluid kinematic viscosity, K is the porous medium permeability (isotropic hydraulic conductivity i.e. the same in r- and zdirections). Following Nath (Nath 1970), equation (4) implies that the radial magnetic field 1 component, Br is proportional to . Since at r = a (inner cylinder radius), Br = A/a, therefore r Br = A/r. To generate solutions independent of the dimensions of the regime, we introduce the following non-dimensional variables:

 p P =  2 2  0.5 ρω 2 b

 u u  ,U θ = θ , U z = z , R = r ,  ω2b ω2b b  2

σB b 2 B ω b2 z 1 Z = , β = r = , Ha 2 = 0 , Re = 2 , b B0 R µ ν Re

∂P a K = −α (α > 0) , λ = , Da = 2 ∂Z b b

(5)

in which P is dimensionless pressure, Uθ is dimensionless tangential velocity, Uz is dimensionless axial velocity, R is the dimensionless radial coordinate, Z is dimensionless axial coordinate, β is the magnetic field ratio, B0 is the characteristic magnetic induction, Ha ∂P is Hartmann number, Re is rotational Reynolds number, is dimensionless axial pressure ∂Z gradient (to be prescribed), λ is the cylinder radius ratio (outer: inner), Da is the Darcy number. Implementing (5) in equations (1) to (4) we obtain the following dimensionless equations: Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.


∂P 1 U θ = ∂R 2 R

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2

(6)

∂U θ ∂ 2U θ ∂ 2 U θ 1 ∂U θ U U ReU z − = + − (1 + Ha 2 ) θ2 − θ 2 2 R ∂R Da ∂Z ∂Z ∂R R

(7)

2 2 U ∂ U z 1 ∂U z Ha + − 2 U z − z = −α 2 R ∂R Da ∂R R

(8)

The appropriate boundary conditions are specified as follows: Uθ (R,0) = 0; Uθ (1,Z) = 1; Uθ (λ, Z) =

ω1a = N ; Uθ (R,∞) = 1 ω2b

UZ (1) = 0; UZ (λ) = 1

(9a) (9b) (9c)

3 EXACT SOLUTIONS USING BESSEL AND LOMMEL FUNCTIONS Equation (8) together with the boundary conditions can be easily solved analytically and we obtain the axial velocity component as:

(10) However, equation (7) is a non-linear second order partial differential equation and is solved numerically using finite difference method implemented in MAPLE. In both sets of computations, the following default values are prescribed: Ha2 = 5, N = 0.4, Re =10 (limit if Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.

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Darcy law validity), Da = 1, λ = 0.5, α =1. Such values correspond to a highly permeable regime. We are primarily concerned with the radial spatial distributions of velocity profiles in the regime. λ is defined as radius of the inner to outer cylinder, maintained at 0.5 throughout the computations. α =1 corresponds to a constant axial pressure gradient of unity. Here we dwell on the influence of Ha2, N, Da, and α on UZ and Uθ. Of course shear stress profiles may also be derived from these velocities although for brevity they are omitted in the present study. Details of the Lommel and Bessel function are given in Appendix 1.

4 RESULTS AND DISCUSSION Graphical illustrations of the axial and tangential velocity profiles are shown in figures 2 to 7. Figure 2 indicates that axial velocity (UZ) is found to decrease markedly with an increase in Hartmann number (Ha) with radial coordinate, with profiles becoming increasingly curved for stronger magnetic field. Ha2 is directly proportional to the characteristic magnetic induction intensity, B0. Therefore the change in magnetic field is amplified considerably by altering Ha2. Inspection of the profile for the non-conducting case (also presented by Nath (Nath 1970) and in exact agreement with the present solutions) i.e. for Ha = 0, shows that the axial velocity decays in an almost linear fashion from the inner cylinder to the outer cylinder; the effect of increasing radial magnetic field is to reduce both the axial velocity and the rate of decay as we progress from the inner to the outer cylinder. Peak axial velocity is therefore always at the wall with the minimal response at the outer cylinder. Of course for this case, we have constrained the inner cylinder to rotate at 40 % of the outer cylinder rotation (N = 0.4), stifling the possibility of Taylor vortex effects. Of course the porous medium is highly permeable with Da = 1 and tortuosity effects are neglected. Dimensionless tangential velocity (Uθ) as depicted in figure 3, is also considerably inhibited with an increase in radial magnetic field i.e. Hartmann number squared. Contrary to the axial velocity distribution however profiles ascend from the inner cylinder to the outer cylinder i.e. over the domain 0.5 ≤R ≤ 1, for all values of Ha2. As in figure 2, the maximum response is observed for the non-magnetic i.e. purely viscous hydrodynamic case (Ha2 =0), for which the tangential velocity distribution is linear. The profile for Ha2 = 10 is however more markedly affected than for the axial velocity; tangential velocity almost vanishes at R ~ 0.6 and remains much less than for Ha2 =5 or 1 for the entire domain. For both components of velocity, the regulatory influence of magnetic field is apparent and indeed would appear to achieve excellent control in practical applications. In figure 4, the tangential velocity response to an increase in relative rotation parameter, N, is presented. N simulates the relative speeds of rotation of the inner to the outer cylinders ωa (= 1 ) and allows the investigation of operational speeds of the cylinders in actual industrial ω2b systems on the flow fields. For relatively low rotation rates, we prescribe values of N < 1, following Nath (Nath 1970). N > 0 implies both cylinders are rotating in the same direction. The case for N < 0 i.e. counter-rotation, is not considered here. Increasing N is seen to strongly accelerate the tangential flow. Tangential velocity (U θ ) is greatly enhanced throughout the annular regime, although the most dramatic escalation is concentrated at the interior wall of the annulus (R = 0.5). A rise in N is seen to elevate the tangential velocity component at the inner cylinder wall from 0.1 for N = 0.1, to 0.2 for N = 0.2, to 0.4 for N = 0.4 and to the maximum computed value of 0.6 for N =0.6. The peak velocities, as in figure 3, are however consistently at the outer cylinder, where all profiles converge to unity. In Figures 5 and 6, the axial and tangential velocity distributions for various permeability parameters i.e. Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.


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Darcy number, are provided.

Figure 2: UZ (axial velocity) versus R with N = 0.4, Re =10 (limit if Darcy law validity), Da = 1, λ = 0.5, α =1, at Z = 1 for various squares of the Hartmann number (Ha2).

Figure 3: Uθ (tangential velocity) versus R with N = 0.4, Re =10, Da = 1, λ = 0.5, α =1, at Z = 1 for various squares of the Hartmann number (Ha2).

Figure 4: Uθ (tangential velocity) versus R with Ha2 = 5, Re =10, Da = 1, λ = 0.5, α =1, at Z = 1 for various relative rotation parameter values (N).

Figure 5: UZ (axial velocity) versus R with N= 0.4, Ha2 = 5, Re =10, λ = 0.5, α =1, at Z = 1 for various Darcy numbers (Da).

Figure 6: Uθ (tangential velocity) versus R with N= 0.4, Ha2 = 5, Re =10, λ = 0.5, α =1, at Z = 1 for various Darcy numbers (Da).

Figure 7: UZ (axial velocity) versus R with N= 0.4, Ha2 = 5, Re =10, Da = 1, λ = 0.5, , at Z = 1 for various pressure gradient parameter values (α).

An increase in Da corresponds to a direct rise in the permeability of the porous medium i.e. to a reduction in the Darcian drag force components in the tangential and axial momenta

Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.

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Uθ U and − z . Increasing Darcy number from 0.01 through 0.1 to Da Da 1 causes a substantial rise in UZ. For Da > 0.1, the axial velocity remains positive i.e. there is no back flow. However for low Da (= 0.01) there is a dramatic reversal in axial flow further from the inner wall of the annulus. The lowest velocity is computed therefore for Da = 0.01 and arises at R ~ 0.85 with a value of -0.67 approximately. Increasing Da reduces the magnitude of the Darcian drag force i.e. as the medium becomes more and more permeable, the matrix fibers have a progressively lower resistance to the flow causing an acceleration in axial flow. In Figure 6, the tangential velocity is also seen to be suppressed considerably with low Darcy number (Da = 0.01) and accelerated with an increase in Da to 0.1 through 0.5 to 1.0. Unlike the axial velocity evolution, flow reversal is not caused in the tangential flow field although at R = 0.7, for Da = 0.01, Uθ almost vanishes. For the case of infinite permeability, Da →∞, the present model reduces to exactly the equations solved by Nath (Nath 1970), with the boundary conditions unchanged, viz:

equations (7) and (8) i.e. −

∂P 1 U θ = ∂R 2 R

2

(11)

∂U θ ∂ 2U θ ∂ 2 U θ 1 ∂U θ U ReU z − = + − (1 + Ha 2 ) θ2 2 2 ∂Z R ∂R ∂Z ∂R R

(12)

2 2 ∂ U z 1 ∂U z Ha + − U z = −α R ∂R ∂R 2 R2

(13)

Although excellent agreement has been computed between our solutions and the Nath case, this was ommitted for conservation of space. The effect of the dimensionless axial pressure gradient (α) on axial velocity profiles is depicted in figure7. This parameter arises only in the axial momentum equation (8). The case of vanishing pressure gradient obviously corresponds therefore to α = 0. An increase in axial pressure gradient from 0 through 0.5, 1 and to 2, generates, as expected, a noticeable rise in the axial velocity component, (Uz). With increasing α the profiles become less parabolic. Effectively therefore the fluid flow may be successfully controlled with increasing the radial magnetic field, decreasing permeability and also reducing the pressure gradient.

5 CONCLUSIONS Closed-form solutions have been obtained for the dimensionless flow equations governing the magnetohydrodynamic viscous flow through a rotating porous medium annulus in the presence of a radial magnetic field and axial pressure gradient. The solutions have been presented in terms of Bessel and Lommel functions and numerical calculations have revealed that: (i) Increasing Hartmann number parameter (Ha2) elevates radial magnetic field strength which suppresses both axial (Uz) and tangential velocity (Uθ). Magnetic field therefore is an effective regulatory mechanism for the flow.



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(ii) Increasing Darcy number (Da) enhances the permeability of the annular porous medium regime and simultaneously decreases the Darcian matrix retarding force, causing a strong acceleration in both velocity components axial (Uz, Uθ). A porous medium therefore can be used to control the flow dynamics. (iii) Increasing the relative rotation parameter (N), which implies co-rotating cylinders for N >0, strongly boosts the tangential velocity (Uθ) throughout the annular regime, with the maximum acceleration in tangential flow identified as being located at the inner cylinder wall. (iv) Increasing axial pressure gradient parameter (α) causes a rise in axial velocity (Uz) The current study has neglected inertial porous media effects and also transient effects, both of which are under investigation. Results will be communicated in the near future (Bég et al 2009).

REFERENCES Antimirov MY and Kolyshkin AA (1984). Nonstationary MHD flow in an annular channel in a radial magnetic field, Magnetohydrodynamics, 20, 3, pp. 285-288. Basu B. and Mandal KK. (1977). Unsteady hydromagnetic flow between two co-axial cyinders, Revue Roumaine des Sciences Techniques: Serie de Mecanique Appliquee, 22, pp. 211-225. Bathaiah D and Venugopal R. (1982) Effect of porous lining on the MHD flow between two concentric rotating cylinders under the influence of a uniform magnetic field, Acta Mechanica, 44, 3-4, pp. 141-158. Baylis JA and Hunt JCR (1971). MHD flow in an annular channel; theory and experiment. Journal of Fluid Mechanics, 48, pp. 423-428 Bear J (1988). Dynamics of Fluids in Porous Media, Dover, New York. Bég, OA, Takhar HS, and Singh SK (2005). Multiparameter perturbation analysis of unsteady oscillatory magnetoconvection in porous media with heat source effects. International Journal of Fluid Mechanics Research, 32, 6, pp 635-661. Bég OA (2007). Applications of MHD in Supersonic Aerodynamics, Technical Report, MFD08-2, BAE Systems, 144 pages, Aerosciences, Academic Wing, KFAA, Riyadh, KSA, July. Bég OA, Bhargava R, Rawat, R, Takhar HS, and Bég TA (2007). A study of steady buoyancy-driven dissipative micropolar free convection heat and mass transfer in a Darcian porous regime with chemical reaction, Nonlinear Analysis: Modelling and Control, 12(2), pp. 157–180. Bég, OA., Zueco J, and Takhar HS (2009). Unsteady magnetohydrodynamic Hartmann– Couette flow and heat transfer in a Darcian channel with Hall current, ionslip, viscous and Joule heating effects: Network numerical solutions. Communications in Nonlinear Science and Numerical Simulation, 14, pp. 1082-1097. Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.

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Bég OA, Bhargava R, Zueco J and Takhar HS (2009). Magnetohydrodynamic natural convection flow from sphere to non-Darcian porous medium with heat generation or absorption: network simulation. International Journal of Thermal Sciences, 48, pp. 913-921. Belfort G, Pimbley JM, Greiner A and Chung KY (1993). Diagnosis of membrane fouling using a rotating annular filter. I: Cell culture media, Journal of Membrane Science, 77, 1, pp. 1-22. Chang W R and Papadopoulos KD (2000). Electro-osmotic flow through porous media: cylindrical and annular models, Colloids and Surfaces A: Physicochemical and Engineering Aspects, 161 (3), pp. 469-476. Deiber JA and Bortolozzi RAA (1998). Two-field model for natural convection in a porous annulus at high Rayleigh numbers - a numerical study for Brinkman-extended Darcy formulation. Chemical Engineering Science, 53, 8, pp. 1505-1516. Geindreau, C and Auriault, J-L (2002). Magnetohydrodynamic flows in porous media, Journal of Fluid Mechanics, 466, pp. 343-363. Globe S (1959). Laminar steady-state magnethydrodynamic flow in an annular channel, Physics of Fluids, 2 (4), 404-407 (1959). Ghosh SK, Bég OA and Zueco J (2009). Rotating hydromagnetic optically-thin gray gas flow with thermal radiation effects. Journal of Theoretical and Applied Mechanics, 39, 1, pp. 101120. Hunt JCR and Stewartson K (1965). Magnetohydrodynamic flow in a rectangular duct. II, Journal of Fluid Mechanics, 23, pp. 563-581. Jain RK and Mehta KN (1962). Laminar hydromagnetic flow in an annulus with porous walls, Physics of Fluids, 5(10), pp. 1207-1211. Kandaswamy P, Murugesan K, and Debnath L (1994). On the hydromagnetic flow between two rotating eccentric cylinders. J. Applied Mathematics and Mechanics (Zeitschrift für Angewandte Mathematik und Mechanik), 74(1), pp. 57 - 61. Khan M, Fetecau C, and Hayat T (2007). MHD transient flows in a channel of rectangular cross-section with porous medium, Physics Letters: Section A, 369 (1) pp. 44-54. Li FC, Kunugi T, and Serizawa A (2005). MHD effect on flow structures and heat transfer characteristics of liquid metal-gas annular flow in a vertical pipe. International Journal of Heat and Mass Transfer, 48, pp. 2571-2581. Mota JPB, Esteves IAAC, Portugal CAM, Esperanca JMSS, and Saatdjian E (2000). Natural convection heat transfer in horizontal eccentric elliptic annuli containing saturated porous media. International Journal of Heat and Mass Transfer, 43(24), pp. 4367-4379. Nath G (1970). Tangential velocity profile for axial flow through two concentric rotating cylinders with radial magnetic field, Source, Defence Science Journal, 20(4), pp. 207-212. Int. J. of Appl. Math and Mech. 5 (6): 68-81, 2009.


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Panja S, Sengupta PR, and Debnath L (1996). Hydromagnetic flow of Reiner-Rivlin fluid between two coaxial circular cylinders with porous walls. Computers and Mathematics with Applications, 32(2), pp. 1-4. Prasad TBA and Venkatasiva Murthy KN (1991). Transient flow in a rotating porous annulus due to suction at the walls. Zeitschrift für Angewandte Mathematik und Physik: (Journal of Applied Mathematics and Physics, ZAMP), 42(4), pp. 605-613. Prasad VR, Takhar HS, Zueco J, Ghosh SK, and Bég OA (2008). Numerical study of hydromagnetic viscous plasma flow with Hall current effects in rotating porous media. 53rd Congress Indian Society Theoretical and Applied Mechanics, University College of Engineering, Osmania University, Hyderabad, India, December, pp. 147-157. Shercliff JA (1965). A Textbook of Magnetohydrodynamics, Pergamon, Oxford. Takhar HS, Ali MA, and Soundalgekar VM (1989). Stability of MHD Couette flow in a narrow gap annulus. Applied Scientific Research, 46, pp. 1-24. Uberoi MS and Chow CY (1966). Magnetohydrodynamics of flow between two coaxial tubes. Physics of Fluids, 9(5), pp. 927-32. Vadasz P (1998). Free convection in rotating porous media. Transport in Porous Media, Pergamon, Oxford, UK. Zueco J, Bég OA, and Makinde OD (2009). Axial fully-developed hydromagnetic flow in rotating annular porous media with radial magnetic field and Forchheimer drag effects: Network numerical analysis, submitted.


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APPENDIX 1: BESSEL AND LOMMEL FUNCTIONS "Lommel functions" were first introduced by Lommel (Lommel 1884-1886). One type of Lommel function appears in the solution to the Lommel differential equation and is given by:

 1 1 1 F2  1; ( µ − nu + 3), ( µ + v + 3);− z 2 2 4  2 S µ(1,)v ( z ) = 2 2 ( µ + 1) − v 1

 1 1 1 F2  1; ( µ − v + 3), ( µ + v + 3); z 2 2 4  2 S µ( 2,v) ( z ) = 2 2 ( µ + 1) − v 1

   × z µ +1

   × z µ +1

 1   1  2 µ + v −1 Γ(v)Γ ( µ + v + 1)  F1  : 1 − v : − z 2  4  −v  2 0  ×z + 1  Γ (− µ + v + 1)  2   1  1   2 µ −v −1 Γ ( µ − v + 1) Γ(−v) 0 F1  : 1 + v : − z 2  2 4  v    ×z + 1  Γ (− µ + v + 1)  2   1 1 1 F2  1 : ( µ − v + 3), ( µ + v + 3) : z 2 2 2 4  = 2 2 ( µ + 1) − v 1

(

+

)

   × z v +1

2 µ −1 π 2 cos ec(πv) 1 1   ×  J − v ( z ) sec( π ( µ + v)) − J v ( z ) sec( π ( µ − v )) 2 2 1  1   Γ (− µ − v + 1) Γ (− µ + v + 1)   2  2  (A1)

where 1F2 and 0F1 are generalized and confluence hypergeometric functions, respectively Sµ,v (z) is defined by the function: z z  1  S µ ,v ( z ) = π  Yv ( z ) ∫ z µ J v ( z )dz − J v ( z ) ∫ z µ Yv ( z )dz  2  0 0 

(A2)

where Jv(z) and Yv(z) denote Bessel functions of the first and second kinds . Where a minus sign precedes the term in the general form of Lommel differential equation, then the solution is given by: c3 z   S µ( −,v) ( z ) =  I v ( z ) ∫ z µ K v ( z )dz − J v ( z ) ∫ z µ I v ( z )dz    z c2


(A3)


81

where Kv(z) and Iv(z) are modified Bessel functions of the first and second kinds. Lommel functions of two variables are related to the Bessel function of the first kind and arise in, for example, diffraction theory.
