Meccanica (2007) 42:247–262DOI 10.1007/s11012-007-9052-z

Pulsatile magneto-biofluid flow and mass transferin a non-Darcian porous medium channel

R. Bhargava · S. Rawat · H. S. Takhar ·O. Anwar Bég

Received: 25 May 2006 / Accepted: 4 November 2006 / Published online: 14 February 2007©Springer Science+Business Media B.V. 2007

Abstract A numerical study of pulsatile flow andmass transfer of an electrically conductingNewtonian biofluid via a channel containing porousmedium is considered. The conservation equationsare transformed and solved under boundary condi-tions prescribed at both walls of the channel, usinga finite element method with two-noded line ele-ments. The influence of magnetic field on the flowis studied using the dimensionless hydromagneticnumber, Nm, which defines the ratio of magnetic(Lorentz) retarding force to the viscous hydrody-namic force. A Darcian linear impedance for lowReynolds numbers is incorporated in the trans-formed momentum equation and a second orderdrag force term for inertial (Forchheimer) effects.Velocity and concentration profiles across thechannel width are plotted for various values ofthe Reynolds number (Re), Darcy parameter (λ),

R. Bhargava · S. RawatMathematics Department, Indian Instituteof Technology, Roorkee, Uttaranchal 247667, Indiae-mail: rbharfma@iitr.ernet.in

H. S. TakharEngineering Department, Manchester MetropolitanUniversity, Oxford Rd., Manchester M5 1GD, UKe-mail: h.s.takhar@mmu.ac.uk

O. Anwar Beg (B)Fire Safety Engineering Program, Leeds Collegeof Building/Leeds Metropolitan University, NorthStreet, Leeds LS2 7QT, UKe-mail: docoabeg@hotmail.com; OBeg@lcb.ac.uk

Forchheimer parameter (Nf ), hydro-magneticnumber (Nm), Schmidt number (Sc) and also withdimensionless time (T). Profiles of velocity vary-ing in space and time are also provided. The con-duit considered is rigid with a pulsatile pressureapplied via an appropriate pressure gradient term.Increasing the hydromagnetic number (Nm) from1 to 15 considerably depresses biofluid velocity (U)indicating that a magnetic field can be used as aflow control mechanism in, for example, medicalapplications. A rise in Nf from 1 to 20 stronglyretards the flow development and decreases thevelocity, U, across the width of the channel. Theeffects of other parameters on the flowfield arealso discussed at length. The flow model also hasapplications in the analysis of electrically conduct-ing haemotological fluids flowing through filtrationmedia, diffusion of drug species in pharmaceuticalhydromechanics, and also in general fluid dynamicsof pulsatile systems.

Keywords Pulsatile · Hydromagnetic · Porous ·Fluid mechanics

NomenclatureDimensionalx, y Coordinates parallel and transverse

to channel wallst Time variableVi Velocity vectorJi Current density vector

248 Meccanica (2007) 42:247–262

Bk Magnetic field vectorP Hydrodynamic pressureτij Shear stress tensorρ Density of bio-fluidεijk Permutation symbolν Kinematic viscosity of bio-fluidb Forchheimer geometric parameterC Concentration of species (e.g. Oxy-

gen)D Coefficient of mass diffusivity of spe-

ciesddt

Differential with respect to time fol-lowing the material particle

H Width of channelB0 y-direction component of magnetic

field vectorX Transformed coordinate parallel to

channel wallsY Transformed coordinate transverse

to channel wallsU Transformed velocity component in

X directionV0 Transpiration velocityσ Electrical conductivity of haemo-

fluidP∗ Transformed hydrodynamic pres-

sure (* dropped for convenience inanalysis)

K Permeability of the porous mediumT Dimensionless time� Dimensionless mass transfer (spe-

cies concentration) functionRe Reynolds numberNm Hydromagnetic numberλ Darcian porous parameterNf Forchheimer porous inertial param-

eterSc Schmidt number( )s Steady component( )o Oscillating component

1 Introduction

The analysis of pulsatile flows has been a ferventtopic of engineering hydromechanics and biofluidmechanics for more than five decades. An earlyclassical study was conducted by Uchida [1]. Excel-lent works in the context of arterial blood flows

were reported by Wormersley [2,3] in the 1950s.Skalak and co-workers presented rigorous studiesof wave propagation in pulsatile blood flows [4–6].It was shown in these studies that for largeblood vessels, the flow of blood is principally New-tonian and therefore approximations of theNavier–Stokes equations are justified. Significantcomputational studies of pulsatile blood flows ina variety of different cardiovascular geometrieswere communicated in the 1970s by Hung [7–10].Later Berger and co-authors [9–13] demonstratedmajor improvements in numerical simulation ofcurved tube pulsatile flows using finite element-based methods. Other excellent studies of pulsa-tile flows include the more recent investigations byZamir [14–16] and also Majdalani and co-workers[17–20]. In mid-1980s engineers became interestedin the influence of magnetic fields on blood flowsprimarily with a view to utilizing MHD (magneto-hydrodynamics) in controlling blood flow veloc-ities in surgical procedures and also establishingthe effects of magnetic fields on blood flows inastronauts, citizens living in the vicinity of EM tow-ers etc. When applied transverse to the directionof flow, magnetic fields have been shown to im-pede blood flow, since blood as a plasma suspen-sion is electrically-conducting. Keltner et al. [21]reported an analysis of the pressure changes in ves-sels of the human vasculature under the action ofstrong magnetic fields. Their study indicated that15% Sodium Chloride solutions are retarded bytransverse magnetic fields of 2.3 and 4.7 Tesla forfluxes below 0.5 l/min. A seminal paper on hydro-magnetic blood flow via a rigid tube network wascommunicated by Sud and Sekhon [22]. Using fin-ite elements they showed that both the intensityand orientation of a magnetic field has substan-tial effect on flow profiles. Other studies of hydro-magnetic blood flow include those by Rao andDesikachar [23] who investigated using a vortic-ity formulation the MHD oscillatory flows in var-iable cross-section channels, reporting a distinctreduction in flow strength with applied magneticfield. [24] Bég et al. studied the Newtonian bio-magnetic blood flow in a two-dimensional porous(tissue) medium using a ferrohydrodynamic for-mulation. All the above studies considered onlymomentum transfer. The diffusion of a species inpulsatile flows is also however of major interest

Meccanica (2007) 42:247–262 249

to engineers. Very recently, therefore, Bég et al.[25] simulated the pulsatile hydromagnetic flowand species diffusion of a biviscosity haemotolog-ical fluid in a non-Darcian porous channel. Theycomputed finite element solutions for the spatialdistribution at fixed times rather than the velocityand concentration distribution variation with time,indicating a distinct decrease in velocity with incre-asing magnetic field, even for a shear-thinning typerheological model. In the present study, we shallconsider the transient flow regime for Newtonianhydromagnetic blood flow in a porous channel.Temporal and spatial velocity distributions will beprovided and also several special cases computedfor comparison, e.g., pulsating hydromagneticnon-porous flow. The present study is importantin further elucidating the basic physics of magneto-pulsatile blood flows in large arterial models whichmay contain blockage material, residue or suspen-sions which can be idealized as a drag-inducingporous medium.

2 Mathematical model

The biofluid (blood) considered in this paper iselectrically conducting. The conservation equationsfor Newtonian magneto-fluid mechanics aretherefore the starting point for analyzing the hemo-dynamics of such a fluid. These equations com-prise the Maxwell electrodynamic, fluid continuity,and momentum conservation (modified Navier–Stokes) equations, which are not reproduced here,for brevity. The porous medium comprising thebio-channel is modeled using a drag force modelfollowing Bég et al. [26]. We note that the vastmajority of pulsatile blood flow studies consideronly velocity fields and ignore the diffusion of spe-cies in the blood stream. Motivated by an interestin pharmaceutical or tracer transport in hemody-namics in the cardiovascular system, we study inthis paper the coupled flow and species transferunder a pulsating pressure gradient, through a non-Darcian porous medium. The species is taken asbeing inert, i.e., non-reacting and possessing hom-ogenous dispersion characteristics. While manystudies have been conducted focusing on purelydiffusive pharmaco-dynamics, the case of pulsa-tile flows of pharmaceutical fluids has not thusfar

received much attention. Pharmaceutical transportin porous media is of great interest in transder-mal drug delivery [27]. The present study consti-tutes, to the authors’ knowledge, the first finiteelement investigation of one-dimensional pulsatilemagneto-hemodynamic flow and species diffusionin a non-Darcian porous channel and is, as such,an important addition to the literature on electri-cally conducting biofluid dynamics. In the analysis,the pulsatory character of the flow is caused by asource at infinity, following the so-called “pistonsat infinity” simplification [28] which enabled thesolution of the pulsatory channel flow problemwith arbitrary pressure distribution. Additionallythe issue of hydrodynamic stability is excludedfrom the study. We study the transport of a pharma-ceutical species in laminar, incompressible,magneto-hydrodynamic, pulsating flow through atwo-dimensional channel with porous walls, con-taining a non-Darcian porous material. Periodicpressure is exerted on the flow in the longitudi-nal direction, parallel to the channel, as depictedbelow in Fig. 1. The channel walls are located adistance H apart with reference to an (x, y, t)coordinate system, where x defines the longitudi-nal coordinate, parallel to the walls and y is thetransverse coordinate, perpendicular to x, and t istime. A uniform magnetic field, Bo, is applied trans-verse to the walls. Transpiration takes place via thechannel walls with uniform velocity, Vo. The gov-erning equations for the mass, momentum, andspecies conservation for the pulsating poroushydromagnetic flow regime may be derived, fol-lowing substantial simplification of the generalizedflow equations, to the following:

Conservation of mass:

∂ρ

∂t+ ρVi,j = 0, (1)

Momentum conservation

∂Vi

∂t= − 1

ρ

∂P∂xi

+ 1ρ

∂τij

∂xi− 1ρεijkJiBk

− νρ

Vi − b√

ViViVi, (2)

Species conservation

250 Meccanica (2007) 42:247–262

Upperwall

Lower wall

x

y

H

y = 0

y = H

Newtonian fluid-saturated,isotropic, homogenous porousmedium channel

Transversemagnetic field,Bo

Pulsepropagation

Pharmaceutical agent diffusing in biofluid

transpiration

Fig. 1 Physical model and co-ordinate system

∂C∂t

= D[∂2C∂x2 + ∂2C

∂y2

]. (3)

In (2) the third-term on the right-hand-sidedesignates the hydromagnetic body force, the pen-ultimate term on the right-hand-side defines theDarcian bulk drag force (which dominates in thecreeping flow regime) and the final term representsthe Forchheimer quadratic inertial resistance. Con-vective terms in the momentum and also speciesconservation equations, i.e., u ∂u

∂x , u ∂v∂x , u ∂C

∂x , u ∂C∂y are

neglected since in the present pulsatile flow study,the viscous diffusion terms are dominant. Weassume that magnetic Reynolds number is small sothat induced magnetic field effects can be ignored.Heat transfer in the flowing blood is also ignoredso that Joule thermo-electrical heating effects donot require consideration. The magnetic field isnot strong and therefore Hall current effects canalso be neglected [29]. The flow variables (velocity,concentration) are purely functions of y and t sincethe walls of the channel are assumed to be infinitelylong. We therefore also ignore x-direction velocityand concentration gradients. The y-direction veloc-ity is ignored in the momentum equation and app-ears only via a transpiration velocity through thewalls, i.e., V = Vo everywhere along the lower wall(injection) and V = −Vo on the upper wall (suc-tion). The flow is assumed to be fully developed.The partial differential equations (1)–(3) are there-fore further reduced under these assumptions tothe following form:

∂u∂t

+ Vo∂u∂y

+ 1ρ

∂P∂x

= ν∂2u∂y2 − σB2

o

ρu − ν

Ku − bu2, (4)

∂C∂t

+ Vo∂C∂y

= D∂2C∂y2 , (5)

where all parameters are defined in the nomencla-ture. The corresponding boundary conditions areprescribed as follows at the lower and upper chan-nel walls:

u = 0, C = C1 at y = 0, (6)

u = 0, C = C2 at y = H. (7)

3 Transformation of mathematical model

To facilitate a numerical solution to the non-linearmomentum equation and the linear concentrationequation, we introduce a set of dimensionless vari-ables. The power of dimensional analysis in bloodflow modeling is well-documented in the literatureand allows global behavior for general flow scenar-ios to be studied with good accuracy. Introducing:

U = uVo

, X = xH

, Y = yH

, T = Vo

Ht,

P∗ = PρV2

o, � = C − C2

C1 − C2, Re = HV0

νB,

Nm = σB20H

ρV0, λ = KV0

νH, NF = Hb,

Meccanica (2007) 42:247–262 251

Sc = HV0

D. (8)

The governing equations now reduce to the follow-ing pair of non-linear partial differential equationsin terms of dimensionless longitudinal velocity, Uand dimensionless species concentration parame-ter, �, viz:

∂U∂T

+ ∂U∂Y

+ ∂P∂x

= 1Re

∂2U∂Y2 − NmU − 1

λU − NFU2, (9)

∂�

∂T+ ∂�

∂Y= 1

Sc∂2�

∂Y2 . (10)

The corresponding transformed boundary condi-tions are now given by:

U = 0, � = 1 at Y = 0 (lower channel wall),

(11)

U = 0, � = 1 at Y = 1 (upper channel wall).

(12)

Equations (9) and (10) have been solved using thefinite element method. To simplify a numericalsolution, the pressure gradient in (9) is decom-posed into a steady component and an imposed(oscillatory) component as follows:

− ∂P∂X

=[∂P∂X

]

s+

[∂P∂X

]

Oeiωt. (13)

Many studies have appeared using this approach.For example, Mazumder and Das examined thepulsatile mass transfer of a contaminant in a con-duit [30] using this approach. Their study howeverdoes not consider magnetic field or porous effects.In the present study, we shall examine the inter-action of pulsation pressure and species diffusionand also the influence of porous media drag forces,magnetic body force, on biofluid flow velocity pro-file in a channel. We have also studied the variationof maximum velocity in the channel with time.

4 Numerical solution

The governing equations amount to a pair of non-linear and linear partial differential equations with

corresponding boundary conditions. Finite elementsolutions have been obtained for the present two-point boundary value problem. An excellentdescription of the method is available in Ref. [31].Herein we shall briefly describe the numerical tech-nique. In the computations, pressure gradient isre-defined as:

− ∂P∗

∂X= Ps + Po(cosω∗T). (14)

Using (14), the momentum equation (9) takes theform:∂U∂T

+ ∂U∂Y

− Ps − Po(cosω∗T)

= 1Re∂2U∂Y2 − NmU − 1

λU − Nf U2. (15)

The corresponding boundary conditions are:

U = 0, � = 1 at Y = 0 (lower channel wall),

(16)

U = 0, � = 0 at Y = 1 (upper channel wall).

(17)

The initial conditions in time, are:

U = 0, � = 1 at T = 0. (18)

For numerical robustness Y = ∞ has been fixedas 1. The whole domain is divided into a set of 100line elements of equal width, each element beingtwo-noded.

4.1 Variational formulation

The variational form associated with (15) and (10)over a typical two-noded linear element

(Ye, Ye+1

)

is given by

Ye+1∫

Ye

w1

{−∂U∂T

− ∂U∂Y

+ Ps + Po(cos ω∗T)

+ 1Re∂2U∂Y2 − 1

λU − NmU − Nf U2

}dY

= 0, (19a)

Ye+1∫

Ye

w2

{1

Sc∂2�

∂Y2 − ∂�

∂Y− ∂�

∂T

}dY = 0, (19b)

252 Meccanica (2007) 42:247–262

where w1 and w2 are arbitrary test functions andmay be viewed as the variation in U and�, respec-tively.

4.2 Finite element formulation

The finite element model may be obtained from(19a) to (19b) by substituting finite element approx-imations of the form:

U =2∑

j=1

Ujψj, � =2∑

j=1

�jψj, (20)

where w1 = w2 = ψi(i = 1, 2) and ψi are the shapefunctions for a typical element

(Ye, Ye+1

)and are

taken as:

ψ(e)1 = Ye+1 − Y

Ye+1 − Ye,

ψ(e)2 = Y − Ye

Ye+1 − Ye, Ye ≤ Y ≤ Ye+1. (21)

The finite element model of the equations thusformed is given by:[KUij

][{Ui}] + [

MUij] [{

U′i}] = [{

FUi

}](22)

and[K�ij

][{�i}] + [

M�ij] [{

�′i}] = [{

F�i

}],

(23)

where ([KUij

],

[K�ij

]) and (

[{FUi

}],

[{F�i

}])

(i.j = 1, 2) are the matrices of order 2×2, and 2×1,respectively. Also U′

i and �′i are derivatives of Ui

and �i with respect to T. All these matrices maybe defined as follows:

KUij = − 1Re

Ye+1∫

Ye

−dψi

dY

dψj

dYdY −

Ye+1∫

Ye

ψidψj

dYdY

−Nm

Ye+1∫

Ye

ψiψjdY − 1λ

Ye+1∫

Ye

ψiψjdY

−Nf U1

Ye+1∫

Ye

ψiψ1ψjdY

−Nf U2

Ye+1∫

Ye

ψiψ2ψjdY, (24)

MUij = −Ye+1∫

Ye

ψiψjdY, (25)

K�ij = − 1Sc

Ye+1∫

Ye

dψi

dY

dψj

dYdY −

Ye+1∫

Ye

ψidψj

dYdY, (26)

M�ij = −Ye+1∫

Ye

ψiψjdY, (27)

FUi = − 1Re

(ψi

dUdY

)Ye+1

Ye

−Ye+1∫

Ye

ψi(Ps + Po(cosω ∗ T))dY, (28)

F�i = − 1Sc

(ψi

d�dY

)Ye+1

Ye

, (29)

where

U =2∑

i=1

Uiψi, � =2∑

i=1

�iψi. (30)

Each element matrix is of the order 2 × 2. Theentire flow domain is discretized into a set of 100line elements. Post-assembly of all the elementequations leads to a matrix of order 101 × 101.The system of equations generated after assemblyof the element equations is non-linear and an itera-tive scheme is employed for a robust solution. Thesystem is linearized by incorporating the functionsU and �, which are assumed to be known. Afterapplying the given initial and boundary conditionsa reduced system of 99 equations remains whichis solved using the Gauss elimination method. Inthe present paper, we have computed the veloc-ity and mass transfer distributions in the channelover space and time. The present analysis servesto provide in particular a systematic examinationof the interactive effects of the problem controlparameters, i.e., Darcian parameter (λ), Forchhe-imer parameter (Nf ), Schmidt number (Sc), Rey-nolds number (Re), hydromagnetic number (Nm),non-dimensional time (T) and pulsatile coefficients(Ps, Po) on the flow regime.

Meccanica (2007) 42:247–262 253

5 Graphical results and discussion

The physics of the flow regime can be controlledby varying the following parameters: ω* (dimen-sionless angular frequency), Re (Reynolds num-ber), λ (Darcian linear drag parameter), P0 (pul-sating amplitude), Ps (steady component of pres-sure gradient), Nm (Hydromagnetic number), Nf(Forchheimer second-order parameter), Nm (Hyd-romagnetic number), Sc (Schmidt species transfernumber). Velocity and species functions are plot-ted against Y and/or T (dimensionless time). Forthe sake of comparison numerical results have alsobeen computed with an efficient finite differencealgorithm for parameter values of ω∗ = 8, Re = 1,T = 0.5, λ = 1, Po = 7, Ps = 10, Nm = 1,Nf = 1. Both U and � distributions were com-puted and demonstrate very good correlation forboth numerical methods, as shown in Table 1.

In all computations the molecular diffusivity ofthe drug species is assumed to be constant. Alsono reactivity or dissolution function has been ass-igned to the species. Numerical solutions have beenobtained for the general case by adopting a par-ticular time T = 0.5 and computing the velocityand mass transfer function profiles at this instant.The variation of these flow quantities in time hasalso been evaluated and is illustrated in severalfigures. Throughout the computations we employω* (angular frequency of oscillation) = 8, Re(Reynolds number) = 1, λ (Darcian linear drag par-ameter) = 1, Nf (Forchheimer second-order par-ameter) = 1, Nm (hydromagnetic number) = 1, Sc

(Schmidt species transfer number) = 0.15 which isa reasonable approximation for a pharmaceuticalspecies [32], Ps (steady component of pressure gra-dient) = 10 and P0 (pulsating amplitude) = 7, unlessotherwise stated. Following other studies of hydro-magnetic blood flow a limiting upper value for Nmof 15 is used, as magnetic fields of higher strengthwould generally induce Hall current effects in theconducting fluid [33].

Figures 2–5 shows the computations for the gen-eral flow case. Figure 2 shows the distribution ofvelocity U with both space (Y-coordinate) andtime (T). A distinct oscillatory profile can be obs-erved. Maximum velocities are observed between35 and 40 within a time-span of 0 < T < 2. Thepeak velocities clearly occur at the centerline of thechannel, i.e., at Y = 0.5. While we have includedtranspiration at the channel walls, this parameteris absorbed in the transformations. The dimension-less angular frequency, i.e., ω* is prescribed as 8 inthe computations. The undulating profile is a directfunction of the parameters Po and Ps. Velocity asc-ends to a peak at Y = 0.5 (center of channel) anddescends back to zero at Y = 1. The velocity alsocommences from 0 at T = 0 (initiation of the pulse)and the oscillatory flow nature is witnessed in time,rather than due to space, since the pulsatile gradi-ent is a time-imposed effect. This plot correspondsto a weak magnetic field, Nm = 1, which reason-ably characterizes biofluid dynamic transport inthe human circulation. A weak Darcian resistanceis also present and also weak second order iner-tial impedance, as defined by λ = 1 and Nf = 1.

Table 1 FEM (Finite Element) versus FD (Finite Difference) Computations

Y U(ω∗ = 8, Re = 1, T = 0.5, λ = 1, Po = 7, Ps = 10, Nm = 1, Nf = 1) �(T = 0.5, Sc = 5)

FE FD FE FD

0 0 0 1 10.1 8.59615 8.59609 0.997961 0.9978990.2 13.7737 13.7731 0.994174 0.9941620.3 16.6878 16.6874 0.98686 0.986790.4 18.196 18.196 0.97281 0.972780.5 18.7823 18.7823 0.946452 0.9464440.6 18.6074 18.6074 0.898397 0.8983850.7 17.5164 17.5163 0.813196 0.8131800.8 14.9488 14.9486 0.665847 0.6657910.9 9.74434 9.74431 0.41622 0.416161 0 0 0 0

254 Meccanica (2007) 42:247–262

Fig. 2 Velocity variation with dimensionless transverse coordinate and time

Fig. 3 Darcian effects (λ): U verses Y for various λ-values at T = 0.5

We note that Reynolds number of unity implies acreeping flow regime, i.e., where the inertial coreencountered in higher velocity porous flows hasnot developed, as discussed by Dybbs and Edwards[34]. The effects of the control parameters are illus-trated in Figs. 3–5.

The influence of the porous bulk matrix resis-tance embodied in the Darcian parameter, λ,(directly proportional to permeability) on the spa-tial distribution of velocity U versus Y is depictedin Fig. 3. Such an effect may occur as a result of apossible blockage material in arterial flows or

non-perfused tissue, which has been discussed byKhaled and Vafai [35]. Increasing λ correspondsto a rise in permeability and therefore physicallyimplies that the porous channel material, progres-sively diminishes in propensity, i.e., less and lessimpeding material is present in the channel flow.Figure 3 implies that as λ rises, the Darcian bulkresistance falls markedly. The permeating fluid rec-eives less resistance and consequently is effectivelyaccelerated. Therefore, longitudinal velocities areboosted as λ rises from 0.1 (low permeability, i.e.,densely-packed porous bio-material) through 0.3,

Meccanica (2007) 42:247–262 255

0.5 to 1 and 2. All profiles are symmetrically par-abolic owing to the prescription of no-slip condi-tions at either walls. Maximum longitudinal velo-cites arise in the centre-line of the channel at Y =0.5, as with the velocity profile in Fig. 2. The orderof magnitude of the Darcian resistance is similar tothe hydromagnetic body force i.e. both terms arelinear. The latter however as we shall see from Fig. 4has a greater impeding effect on the flow.

Hydromagnetic number influence on velocity(U) profiles is depicted in Fig. 4. As Nm is increasedfrom 1 (weak transverse field) to 3, 5, 10, and 15(maximum field strength), flow velocity U clearlyplummets considerably. Increasing the magneticbody force clearly impedes the flow which there-fore decelerates as U falls from a maximum valueof about 18 at the channel centre for Nm = 1,to about 14 for Nm = 15. Once again Re = 1 so

Fig. 4 Hydromagnetic effects (Nm): U verses Y for various Nm-values at T = 0.5

Fig. 5 Forchheimer effects (Nf ): U verses Y for various Nf -values at T = 0.5

256 Meccanica (2007) 42:247–262

that the flow regime corresponds to the Darcian-dominated or creeping flow regime where velocitydistributions are determined mainly by the localporous media geometry. Above this value of Rey-nolds number more complex porous effects takeplace.

The influence of Forchheimer inertial parame-ter, Nf , on the velocity distribution across the chan-nel is shown in Fig. 5. A rise in Nf from 1 (weakquadratic drag) to 5, 10, 15, and finally 20, stronglyretards the flow development and decreases thevelocity, U, across the width of the channel.Increasing Forchheimer inertial drag effects there-fore systematically swamps out the influence ofDarcian viscous-dominated drag. Velocity U is seento fall from a peak value at the channel centre of18, (Y = 0.5) for Nf = 1, to approximately 4 forNf = 20, i.e., strong inertial drag. Our results there-fore confirm the studies of many previous research-ers including Dybbs and Edwards [34] and Vafaiand Tien [36]. The porous filtration material in thechannel has a significant deceleration effect on flowmomentum at high Nf values. Although we havefixed the Reynolds number at 1 for the presentcomputation, at values above this boundary layersbegin to form in the porous medium, a phase whichis followed by a chaotic laminar stage as Reynoldsnumbers increase further.

To gain a perspective of the influence of mag-netic field on the flow field, we have next con-

sidered the special case of non-conducting flow,i.e., Nm = 0. We observe from Fig. 6 that peakvelocities are slightly higher than for the magne-tohydrodynamic case (Fig. 2) indicating that theabsence of magnetic field does indeed acceleratethe flow. For a higher magnetic field, say Nm > 5,peaks are even lower for the magnetic case (Fig.2) although we have not plotted these. Again theoscillatory profile is evident from Fig. 6.

Figures have also been provided to study theinfluence of Darcian (λ) parameter and Forchhei-mer parameter (Nf ) on the longitudinal flow veloc-ity.

Figure 7 shows the distribution of U versus Yfor various λ values. Comparing with the magne-tohydrodynamic case (Fig. 3) we observe againthat velocities are slightly higher for all Darcianparameter values. The general trends are clearlyalso similar, i.e., an increase in Darcian parameterconsiderably elevates flow velocity as this corre-sponds to a decrease in Darcian drag, as the flowregime becomes more and more porous (i.e., solidmatrix particle presence progressively diminishes).Figure 8 shows the influence of the Nf parameter(quadratic inertial effect) which as with the magne-tohydrodynamic case (Fig. 5) continuously reducesflow velocity as Nf increases from 1 to 20.

Another special case has also been investigatedin Figs. 9 and 10, namely pulsating non-magneticflow and mass transfer in a Darcian porous medium,

Fig. 6 Non-dimensional velocity profile (U) versus dimensionless transverse coordinate and time (pulsatile, non-magneticflow in a non-Darcian channel)

Meccanica (2007) 42:247–262 257

Fig. 7 Darcian effects (λ): U verses Y for various λ-values at T = 0.5

Fig. 8 Forchheimer effects (Nf ): U verses Y for various Nf -values at T = 0.5 (pulsatile, non-magnetic flow in a non-Darcianchannel)

achieved by setting the Forchheimer parameter Nfto zero and also Nm = 0 (non-conducting regime).For this case, Fig. 9 clearly indicates that velocitiesare massively increased with the absence of theForchheimer drag as the latter is a second ordereffect. Peak values are generally greater than 100,compared with the non-magnetic case, shown inFig. 6. Also while the peaks in Fig. 6 were approx-imately the same we observe a sharp rise from thefirst peak velocity in Fig. 9 to the second peak—an increase of over 40 in the value of U to nearly170. The third peak however indicates a slight dec-rease in maximum velocity (in all cases this occ-

urs at the centre of the channel) from 170 at thesecond peak to 160 approximately for the third.Clearly therefore the Forchheimer parameter is adominant control mechanism in pulsatile porousflow and has a stronger influence than the mag-netic field, since although both parameters haveunity values, the Forchheimer parameter howeverexerts a much more dramatic influence. This factis again confirmed by studying the plot of U ver-sus Y for various Darcian parameters, i.e., Fig. 10.Comparing this non-magnetic, Darcian case withthat illustrated in Fig. 7 (non-magnetic, non-Dar-cian, i.e., where Nf = 1), we see that velocities

258 Meccanica (2007) 42:247–262

Fig. 9 Non-dimensional velocity profile (U) versus dimensionless transverse coordinate and time (Pulsatile, non-magneticflow in a Darcian channel)

Fig. 10 Darcian effects (λ): U verses Y for various λ valuesat T = 0.5

for the Darcian case are greatly increased with theabsence of the Forchheimer inertial term. The min-imum velocity U in the Darcian case (Fig. 10), i.e.,25 is infact greater than the maximum velocity U inthe non-Darcian case, i.e,. 18 (Fig. 7). Clearly there-fore the presence of Forchheimer drag has a majorinfluence on the flow regime and tremendouslydecelerates the flow in the porous medium. As withall other graphs depicting the Darcian influence(λ), increasing this parameter always boosts thevelocity value. In the non-magnetic, Darcian case(Fig. 10) we see that rise in λ from 0.1 to 2 increasesU from 25 to about 50 (i.e., doubles the velocity ofthe flow).

The special case where no porous effects arepresent and the flow is non-conducting has alsobeen studied. The flow regime is therefore purelyfluid without any porous material fibers and themagnetic body force vanishes. The velocity distri-bution with Y and T is shown in Fig. 11. Compar-ing with the Darcian, non-magnetic case (Fig. 9)we see that velocity peaks are slightly higher forthe purely fluid case, since the Darcian drag forceis absent. The oscillatory flow profiles are similarto Fig. 9—again parabolic in profile with all max-ima at the channel centre (Y = 0.5). The influenceof Reynolds number on the flow velocity versus Ycoordinate across the channel is shown in Fig. 12for this special case. A rise in Re from 0.5 (veryslow, viscous flow) to 1, 2, and 5 causes a clearrise in velocity from about 20 to 250. Increasing Reimplies that the inertial force begins to dominatethe viscous force, but due to an absence in porouseffects, Forchheimer drag is not invoked for thiscase. Hence the fluid is actually accelerated withRe increasing for the purely fluid case. We notethat the profiles are actually skewed parabolas tothe right of the channel centre line.

For the simplest case considered, i.e., steady,non-magnetic, non-porous flow(Po = 0, Nm = 0,Nf = 0, λ = ∞), we have computed the influ-ence of Reynolds number and the steady pres-sure gradient coefficient (Ps) on velocity profiles inFigs. 13 and 14. Over the same increase in Reynolds

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Fig. 11 Non-dimensional velocity profile (U) versus dimensionless transverse coordinate and time (Pulsatile non-magneticflow in a purely fluid (non-porous) channel)

Fig. 12 Reynolds Number Effects (Re): U verses Y forvarious Re-values at T = 0.5

number (0.5 to 5) comparing with the pulsatile,non-magnetic, non-porous flow case (Fig. 12), wesee that much lower velocities are generated inthe channel. Clearly, therefore, the presence of apulsating pressure gradient component (Po=7) inFig. 12 has a great influence on flow acceleration,compared with the steady case in Fig. 13, wherethe maximum velocity is less than 5 for Re = 5.We have illustrated also the influence of the steadypressure gradient component, Ps, on U versus Ydistribution in Fig. 14 for the steady, non-magnetic,non-porous flow case (Po= 0, Nm = 0, Nf = 0,λ = ∞). We observe a distinct rise in U values asPs increases from 10, through 12, 14, 16, and 20; the

00 0.5 1

2.5

5

Y

U

Ps = 10

Re = 0.5

Re = 1

Re = 2

Re = 3

Re = 5

Fig. 13 Reynolds Number Effects (Re): U verses Y for var-ious Re-values (steady, non-magnetic flow in a purely fluidchannel) (λ = ∞, Po = 0, Ps = 10, Nm = 0, Nf = 0)

peak value of U is elevated from approximately 1to 2.4 over this range.

Finally Fig. 15 indicates the distribution of masstransfer function, φ, with Y across the channel forvarious Sc values, for the case of pulsating mag-netic (Nm = 1), Darcy-Forchheimer flow. Unlikewith the velocity plots, the profile is not parabolicbut rather follows a monotonic pattern, decay-ing from the lower channel wall (Y = 0) to theupper channel wall (Y = 1). A rise in Schmidtnumber substantially increases the mass transfer

260 Meccanica (2007) 42:247–262

00 0.5 1

1.25

2.5

Y

U

Re = 1

Ps = 10

Ps = 12 Ps = 14

Ps = 16 Ps = 20

Fig. 14 Steady Pressure Gradient effects (Ps): U verses Yfor various Ps-values (steady, non-magnetic flow in a purelyfluid, i.e., non-porous channel) (λ = ∞, Po= 0, Re = 1,Nm = 0, Nf = 0)

00

0.5

0.5

1

1

Y

Sc = 1

Sc = 2

Sc = 3

Sc = 5

Sc = 10

Sc = 0.5

Φ

Fig. 15 � verses Y for various Schmidt number values (Sc)(general flow case) (ω∗ = 8, T = 0.5, λ = 1, Po= 7, Ps= 10,Re = 1, Nm = 1, Nf = 1)

function. The Schmidt number defines the relativeeffectiveness of momentum and mass transport bydiffusion in the velocity and concentration fields.Lower Sc values physically represent for examplein industrial applications, hydrogen gas as the spe-cies diffusing (Sc = 0.5); larger values to Methanoldiffusing in air (Sc = 1.0) and Ethylbenzene in air(Sc = 2.0) [37]. As Sc rises, the chemical molecu-lar diffusivity of the species is reduced markedlyso that it becomes easier for the species to movein the flow field. φ values are therefore increasedas is evident from Fig. 15. With regard to phar-maceutical diffusion in a pulsating blood flow, itwould appear that better mass transfer and disper-sion performance can be achieved by using higherSc agents introduced into the blood stream.

Fig. 16 Maximum velocity versus Time (pulsatile, non-magnetic, Darcian case)

Fig. 17 Maximum velocity versus Time (pulsatile, non-magnetic, non-porous case)

Figures 16 and 17 show the variations of max-imum velocity (Umax) in the channel with time(T) for the non-magnetic, Darcian case (Fig. 16,Nm = 0, Nf = 0, λ = 1) and the non-magnetic,purely fluid case (Fig. 17, Nm = 0, Nf = 0, λ→ ∞),respectively. All other parameter values are iden-tical for these two graphs. The oscillatory profilesare clearly visible in both cases with a rise in Umax

between the first and second peaks; however val-ues of Umax are notably lower in the Darcian case(Fig. 16) than in the purely fluid case (Fig. 17) asin the former the Darcian drag force is presentwhich decelerates the flow. The second and thirdpeaks in Fig. 17 have values of approximately 190whereas in Fig. 16 they are clearly less than 180. Fig-ure 18 shows the results for the general flow case(Nm �= 0, Nf �= 0). Comparing with Figs. 16 and 17,

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Fig. 18 Maximum velocity versus Time (general case, i.e.,pulsating, magnetic, non-Darcian flow)

the presence of magnetic and inertial porous drag(Fig. 18) clearly massively reduces the peak valuescomputed for maximum velocity which drop fromover 100 for Figs. 16 and 17, to less than 40 forFig. 18. These results confirm the inhibiting effectsof magnetic field on pulsatile flow and also thedominating influence of non-linear inertial drag onthe flow.

The present study is currently being extendedfor reacting species and also for flows where thechannel geometry is perturbed or constricted, i.e.,stenotic flows. Results of these investigations willbe communicated in the near future.

6 Conclusions

A finite element numerical solution for the hydro-magnetic Newtonian bio-fluid flow in a porouschannel with simultaneous mass transfer (repre-senting for example the dispersion of a drug) hasbeen presented. A drag force porous media modelhas been employed to simulate a rigid-walled ideal-ized arterial geometry containing blockage mate-rials, or the flow via perfused tissue systems. Aone-dimensional spatial and transient model hasbeen derived. The model has been solved usingboth finite element and finite difference numericalmethods. The computations have been shown toagree well for the influence of magnetic field andalso Darcian and Forchheimer effects on the flow,

with previous studies. The numerical simulationshave shown that:

(a) Increasing magnetic field serves to reduce flowvelocities across the width of the channel.

(b) Increasing Reynolds number increases the flowvelocity across the channel width.

(c) Increasing the Darcian drag has an imped-ing effect on the flow but less so than theForchheimer second order drag which substan-tially decelerates flow in the channel.

(d) Increasing unsteady component of pressure(Po) increases velocity substantially.

(e) Increasing the steady component of pressure(Ps) also has a positive influence on flow.

(f) Increasing Schmidt number for the generalmagnetohydrodynamic, pulsating, non-Darcian flow case, increases the mass transferfunction considerably.

Acknowledgements The authors are grateful to thereviewers for their excellent comments which helped toimprove the paper.

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