Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx

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Full Length Article

Flow of a Bingham fluid in a porous bed under the action of a magneticfield: Application to magneto-hemorheology

http://dx.doi.org/10.1016/j.jestch.2016.11.0082215-0986/� 2016 Karabuk University. Publishing services by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.E-mail address: misrajc@gmail.com (J.C. Misra).

Please cite this article in press as: J.C. Misra, S.D. Adhikary, Flow of a Bingham fluid in a porous bed under the action of a magnetic field: Applicamagneto-hemorheology, Eng. Sci. Tech., Int. J. (2016), http://dx.doi.org/10.1016/j.jestch.2016.11.008

J.C. Misra a,⇑, S.D. Adhikary b

aCentre for Healthcare Science and Technology, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, IndiabCentre for Theoretical Studies, Indian Institute of Technology, Kharagpur, 721302, India

a r t i c l e i n f o

Article history:Received 30 August 2016Revised 5 November 2016Accepted 8 November 2016Available online xxxx

Keywords:Bingham plastic fluidPorous mediumPulsatile flowYield stressMagnetic fieldWall shear stress

a b s t r a c t

The study deals with an investigation of the flow of a Bingham plastic fluid in a porous bed under theaction of an external magnetic field. Porosity of the bed has been described by considering Brinkmanmodel. Both steady and pulsatile motion of this non-Newtonian fluid have been analysed. The governingequations are solved numerically by developing a suitable finite difference scheme. As an application ofthe theory in the field of magneto-hemorheology, the said physical variables have been computed by con-sidering the values of the involved parameters for blood flow in a pathological state, when the system isunder the action of an external magnetic field. The pathological state corresponds to a situation, wherethe lumen of an arterial segment has turned into a porous structure due to formation of blood clots.Numerical estimates are obtained for the velocity profile and volumetric flow rate of blood, as well asfor the shear stress, in the case of blood flow in a diseased artery, both the velocity and volumetric flowrate diminish, as the strength of the external magnetic field is enhanced. The study further shows thatblood velocity is maximum in the plug (core) region. It decreases monotonically as the particles of bloodtravel towards the wall. The study also bears the potential of providing numerical estimates for manyindustrial fluids that follow Bingham plastic model, when the values of different parameters are chosenappropriately.� 2016 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Hemorheology (also called as blood rheology) deals with vari-ous flow characteristics of blood and its constituents, e.g. plasma,erythrocytes (red cells), white cells, platelets, etc. Tissue perfusioncan take place properly, as long as the rheological properties ofblood are well within certain levels. Significant departures of theproperties of blood from those at the normal physiological statemay lead to different arterial diseases. Many health problems,including hypertension, diabetes mellitus and insulin resistance,metabolic syndrome and obesity are directly linked with the vis-cosity of whole blood. The area of studies related to the rheologicalproperties of blood under the action of magnetic fields (as in thecase of MRI) may be called as magneto-hemorheology.

Human exposure to external magnetic fields is of commonoccurrence in various clinical procedures. Patients are exposed tostrong magnetic fields during MRI (magnetic resonance imaging).An excellent review of various issues related to the exposure of

humans to static magnetic fields of high intensity during MRIwas presented by Schenck [1]. He made an important observationthat there is hardly any evidence of health hazard associated withexposure of the human body to magnetic fields, not even when thebody is exposed to a strong magnetic field in a cumulative manner,provided no ferromagnetic material is present. The reason behindthis observation is twofold: (i) Human tissues lack ferromagneticmaterials, and (ii) magnetic susceptibility of these tissues is small.

As mentioned by Schenck [1], studies on human subjected to amagnetic fields of strength up to 8 T and on sub-human under theaction of magnetic fields up to 16 T indicate that a considerablemargin of safety exists. This observation shows that the range3–4 T commonly used in clinical procedures is well within thesafety zone.

It is known that blood is an electrically conducting fluid and sowhen blood flows under the action of an external magnetic field ofsufficient strength, a transverse EMF is developed, which is directlyproportional to the velocity of blood, as well as to the intensity ofthe applied magnetic field. Owing to this, it becomes difficult toobtain good ECGs during magnetic resonance scanning. Kinouchiet al. [2] pointed out that the said effect contributes to humantolerance of highly intensified magnetic fields.

tion to

Nomenclature

r�;/�; z� cylindrical co-ordinatesR� radius of the tuber�p radius of plug regionR0 characteristic radiusu� velocity in radial directionuc characteristic velocityq;l density and viscosity, respectivelys� shear stresss�c characteristic shear stresss�y yield stress

PðtÞ non-dimensional pressure gradientkðrÞ permeability factora Womersley parameterA amplitude of the floww angular frequencyB0;r magnetic induction and conductivity of the mediumM Hartmann number

2 J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx

Bingham plastic fluids are those in which the excess deviatoricstress over the yield stress varies linearly with shear rate. Thesefluids exhibit non-Newtonian behavior. They bear the potentialto transmit shear stress even in the absence of a velocity gradient.There exists a linear relationship between shear stress and strainfor Bingham plastic fluids; a finite yield stress is needed for themto flow. At a low stress level, their motion is very similar to thatof a rigid body, but when the stress level is high, they flow like aviscous fluid. It has been found that Bingham liquids possessthicker coating in comparison to the case of Newtonian fluids. Clay,drilling mud, printing ink, molten liquid crystalline polymers,foams, paint, toothpaste and food articles like margarine, mayon-naise, molten chocolate, yoghurts and ketchup are some typicalexamples of Bingham plastics. Since Bingham fluids are trans-formed to solids when the applied shear stress is less than the yieldstress, it is apparent that Bingham fluids behave like a solid med-ium in the core layer. That means, a solid plug moves within theflow.

A study on the settling characteristics of spherical particles inBingham plastic fluids was conducted by Ansley and Smith [3].Mitsoulis [4] presented a review of research activities of viscoplas-tic models that include Bingham plastic model, Herschel–Bulkleymodel and Casson model. He discussed, in particular, the entryand exit flows from dies, flows around spheres and cylinders, aswell as squeeze flows. A theoretical study of the free convectionin a Bingham plastic fluid on a vertical flat plate with constant walltemperature was performed by Kleppe and Marner [5]. Theyreported that Bingham plastic friction coefficients are considerablyhigher than those in the case of Newtonian fluids. This increasewas attributed to the yield stress of Bingham plastic fluids. Atwo-dimensional study on creeping flow of a Bingham plastic fluidpast a cylinder was conducted numerically by Nirmalkar et al. [6],who reported that in the limit of plastic flow, drag approaches aconstant value. Bahaduri et al. [7] developed a simple predictivetool that can be used to predict easily the boundaries of the plugof Bingham plastic fluids for laminar flow through annulus.Sayad-Ahmed et al. [8] used a numerical method to examine ther-mally laminar heat transfer based on the fully developed velocityfor Bingham fluids in the entrance region of a circular duct. Withthe help of Bingham plastic model, Liu and Mei [9] made anattempt to examine the effects of wave-induced friction on amuddy sea bed. Yu et al. [10] performed a study of oscillatory flowof magnetorheological fluid dampers subject to sinusoidal dis-placement excitation, based on Bingham plastic and Herschel–Bulkley models.

The available scientific literature reveals that the behavior ofbio-mass and some physiological fluids e.g. blood under certainpathological situations can be well described by a Bingham model.The yield stress of blood is nowwell established. A review of differ-ent studies on the yield stress of blood has been given in the articleof Paicart et al. [11]. The Bingham plastic characteristics of blood

Please cite this article in press as: J.C. Misra, S.D. Adhikary, Flow of a Binghammagneto-hemorheology, Eng. Sci. Tech., Int. J. (2016), http://dx.doi.org/10.1016

flow through a stenosed artery was recently discussed by Yadavand Kumar [12]. Singh and Singh [13] considered the fully devel-oped one dimensional Bingham plastic flow of blood through asmall artery having multiple stenoses and post-stenotic dilation.Using regular perturbation method, the oscillatory flow of a Bing-ham plastic fluid was studied theoretically by De-Chant [14],who made an attempt to develop a relationship between the veloc-ity field and dimensionless flow rate. The author mentioned thatthis study was motivated towards examining blood flow in arteriesin a pathological state and claimed that the solution reported byhim provides useful analytical models that bear the potential tosupport experimental and computational studies on arterial bloodflow.

Several benchmark contributions have been made by Misraet al. (cf. [15–40]) to explore a variety of information in hemorhe-ology. Since a single non-Newtonian model is inadequate todescribe the complexity of blood and its flow in normal/diseasedarteries, they have used different non-Newtonian models, e.g. Cas-son model, micropolar fluid model, power law fluid model, couplestress fluid model, viscoelastic fluid model and Herschel–Bulkleyfluid model. In each case, the choice of a particular model has beenmade by keeping an eye on the objective of the study. Some of theirstudies pertain to arterial blood flow in the normal physiologicalstate, while some others are concerned with blood flow in arteriesunder pathological conditions. Some of the hemorheological stud-ies conducted by Misra et al. that have been mentioned above alsoinvolve application of transport theory in porous media.

An investigation on peristaltic motion of blood in the micro-circulatory system was recently conducted by Misra and Maiti[33], in which the non-Newtonian behaviour of blood has beenmodelled as a Herschel–Bulkley fluid and the vessel has been con-sidered to be of varying cross-section. In this communication, theauthors mentioned that the Herschel–Bulkley fluid model is moregeneral than most other non-Newtonian models and that theresults for a fluid represented by the Bingham plastic fluid modelcan be derived from those of any study conducted with considera-tion of blood as a Herschel–Bulkley fluid.

An interesting study on the effect of thermal radiation on themagnetohydrodynamic flow of blood and heat transfer in a perme-able capillary in stretching motion has been reported by Misra andSinha [34]. In this study, the lumen of the capillary has been con-sidered to have turned into a porous structure due to some arterialdisease. The authors have developed a suitable numerical model tostudy the problem. The results obtained on the basis of the studyhave an important bearing on the therapeutic procedure of electro-magnetic hyperthermia, particularly in understanding/regulatingblood flow and heat transfer in capillaries.

A similar model for blood flow and vessel geometry was formu-lated and analyzed by Maiti and Misra [35] in their recent investi-gation of the non-Newtonian characteristics of peristaltic flow ofblood through micro-vessels, e.g. arterioles and venules. On the

fluid in a porous bed under the action of a magnetic field: Application to/j.jestch.2016.11.008

J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx 3

basis of this study, the authors discussed some novel features ofSSD wave propagation that affect significantly the flow behaviourof blood in arterioles. Another study was carried out by Maitiand Misra [36] to investigate the peristaltic flow of an incompress-ible viscous fluid in a porous channel, by considering slip boundaryconditions of Saffman type. This study was motivated towardsexploring some important information concerning the flow of bilewithin ducts in a pathological state, when numerous stones areformed in the bile.

A separate study of a dually stenosed artery was conducted bySinha and Misra [37], in which they took into account the variationof blood viscosity, variable hematocrit and velocity-slip at the arte-rial wall. Two different research investigations on electro-osmoticflow of blood having the promise of important applications tomicro-biofluidics were recently carried out by Misra et al.[38,39]. Another study depicting the effect of chemical reactionon heat and mass transfer of blood was reported by Misra andAdhikary [40].

A problem related to the dynamics of blood flow was recentlyinvestigated by Chandra and Misra [41] for the situation whenthe influence of Hall current and the rotation of microparticles ofblood, such as erythrocytes and thrombocytes co-exist. Theauthors made an important observation that the micro rotationof the erythrocytes of blood is enhanced due to the effect of Hallcurrent, when blood flows in the arterial system. Effect of heattransfer during MHD flow of blood was reported by Sinha et al.[42] for the case when a non-uniform heat source is present. Stag-nation point flow and heat transfer of blood treated as an electri-cally conducting fluid was investigated by Misra et al. [43] on thebasis of which the authors reported that blood flow is reduced,as the strength of the induced magnetic field increases. Theyobserved that reduction in blood velocity is followed by anenhancement of the temperature field. By considering thetemperature-dependent properties of blood, heat and mass trans-fer of blood flowing peristaltically in asymmetric channels was alsostudied by Misra et al. [44]. This study reveals that the volume ofthe trapped bolus reduces when the Hartmann number/Reynoldsnumber/fluid viscosity/Scmidt number increases, and that thebolus size is enhanced, when the viscosity of the blood diminishes.Another magnetohydrodynamic systematic study of blood flowwas presented by Korchevskii and Marochnik [45] in a very elabo-rate manner.

The transport theory in porous media involving various models,such as Darcy and Brinkman models for momentum transport hasbeen found to be quite useful in describing the flow of various bio-logical fluids. It has been found that models for flow through por-ous media are widely applicable in the simulation of blood flow intumors and in modeling blood flow when fatty plaques of choles-terol and artery-clogging clots are formed in the lumen of anartery.

Kim and Tarbell [46] have discussed a simple macromolecularmodel on the basis of a two-parameter strain-dependent perme-ability function developed by Klancher and Tarbell [47], by usinga pore theory in order to determine the transport properties ofmacromolecules in the arterial wall. Vankan et al. [48,49] havecompared a hierarchical mixture model of blood perfused biologi-cal tissue that utilizes an extended Darcy equation for blood flow.They also have performed a simulation for blood flow through acontracting muscle with a hierarchical structure of pores and havefound that blood pressures calculated by them are approximatelyequal to blood pressures measured in skeletal muscle.

Another important domain of research in hemorheology is theone that deals with the application of the Darcy model to bloodflow in a tumor, which consists of tissues that are highly heteroge-neous as compared to the case of normal tissues. Baish et al. [50]have used the Darcy model to represent the flow through the vas-

Please cite this article in press as: J.C. Misra, S.D. Adhikary, Flow of a Binghammagneto-hemorheology, Eng. Sci. Tech., Int. J. (2016), http://dx.doi.org/10.1016

cular/porous media in arterial tumors. Since Darcy model ignoresthe boundary effects on the flow, application of this model is notvalid when the effect of the boundaries of the porous medium isaccounted for. In order to incorporate the boundary effects, theBrinkman model is more appropriate for modeling blood flowthrough a porous matrix.

As mentioned earlier, there has been a recent trend of using theinteraction of a magnetic field with tissue fluids, in many diagnos-tic devices, especially those which are used to diagnose cardiovas-cular diseases. Magnetic fields are employed to develop devices forcell separation, targeted transport of drug using magnetic particlesas drug carriers, reduction of bleeding during surgeries and provo-cation of occlusion of feeding vessels of cancerous tumors anddevelopment of magnetic tracers. For this reason, studies of flowproperties of human blood as well as deformation of blood vesselssubject to an applied external magnetic field have been carried outby several researchers (cf. [51–55]).

Two-dimensional pulsatile blood flow through a stenosedartery in the presence of an external magnetic field was studiedby Alimohamadi and Imani [56] by means of finite element simu-lation. Another study of non-Newtonian blood flow in magnetictargeting drug delivery in stenosed carotid artery having bifurca-tion was performed by Alimohamadi et al. [57]. Akbarzadeh [58]studied pulsatile magneto-hydrodynamic blood flow through por-ous blood vessels using a third grade non-Newtonian fluid model.

In the present paper, the flow of a Bingham plastic fluid througha porous bed under the action of a magnetic field has been inves-tigated. Although such a problem bears the potential of importantapplications in hemorheology, particularly in pathological fluiddynamics and also in the study of industrial fluids, it has escapedthe attention of previous researchers. A mathematical analysis iscarried out for the flow of a Bingham plastic fluid through a porousmatrix using Brinkman model. The problem is investigated for aparticular situation, when an externally applied magnetic field ispresent. Computational results for the variation in velocity distri-bution, volumetric flow rate and wall shear stress with porosityparameter, magnetic parameter and yield stress are presentedgraphically. As indicated above, the present study bears a strongpotential of a wide variety of applications in polymer industriesas well as in pathological fluid dynamics, biomedical engineering(e.g.MRI,ECG etc.), clinical diagnosis and surgery, which involveflows through porous media in presence of magnetic fields. As anillustrative example, the theory developed has been applied toinvestigate the dynamics of blood flow in diseased arteries underthe action of an external magnetic field, when the arterial lumenturns into a porous structure and the behavior of blood changesto that of a plastic fluid.

2. The problem and the Governing equations

Let us consider the fully developed flow of a Bingham plasticfluid in a porous matrix formed inside a circular tube (cf. Fig. 1),subject to an external magnetic field B0. Using cylindrical coordi-nates (r�;/�; z�), the momentum equation can be written as

q@u�

@t�¼ � @p�

@z�� 1r�

@ðr�s�Þ@r�

� lk�ðr�Þ þ B2

0r� �

u� ð1Þ

and the constitutive equation for a Bingham plastic fluid as

s� ¼ s�y þ �l @u�@r�

� �; s� P s�y

� @u�@r� ¼ 0; s� 6 s�y

ð2Þ

Definitions of all the symbols appearing in the Eqs. (1) and (2)are included in the Nomenclature given earlier.

The Eqs. (1) and (2) are required to be solved subject to the fol-lowing conditions:

fluid in a porous bed under the action of a magnetic field: Application to/j.jestch.2016.11.008

R

z

r

pr

B0

u (r, t)

Fig. 1. Schematic diagram of flow in a porous matrix.

4 J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx

u� ¼ 0atr� ¼ R� ðno� slipconditionÞs� is finiteat r� ¼ 0 ðregularityconditionÞ ð3Þ

Let us first introduce the non-dimensional variables

r ¼ r�R0; R ¼ R�

R0; u ¼ u�

uc; s ¼ s�

sc ;@p�@z� ¼ �p0PðtÞ; t ¼ xt�

sc ¼ lucR0

; M2 ¼ R20B20rl ; sy ¼ s�y

sc ; kðrÞ ¼ k�ðr�ÞR20

; a2 ¼ qxR20l ;

ð4Þ

where uc ¼ � p0R20

2l represents a dimensional characteristic velocity,

p0 the dimensional pressure-gradient.In terms of the non-dimensional variables defined in (4), Eqs.

(1) and (2) may be re-written in the form

a2 @u@t

¼ 2PðtÞ � 1r@ðrsÞ@r

� 1kðrÞ þM2

� �u ð5Þ

s ¼ sy þ � @u@r

� �; s P sy

� @u@r ¼ 0; s 6 sy

ð6Þ

while the boundary conditions (3) are transformed to

u ¼ 0 at r ¼ R

s is finiteatr ¼ 0ð7Þ

In the section that follows, we have analyzed the problem in thecase of steady and unsteady flows.

3. Analysis

3.1. Steady flow

In the steady state, the non-dimensional form of the momen-tum Eq. (5) reduces to

0 ¼ 2Ps � 1r@ðrsÞ@r

� 1kðrÞ þM2

� �u; ð8Þ

where Ps stands for the non-dimensional pressure gradient in thesteady-state.

Case I: When M = 0, and kðrÞ ¼ 1, Eq. (8) further simplifies to

0 ¼ 2Ps � 1r@ðrsÞ@r

: ð9Þ

Integrating (9) with respect to r, we obtain

s ¼ Psr þ Cr;

where C is an arbitrary constant of integration.In view of condition (7), since s has to be finite at r = 0, we have

to take C = 0, so that

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s ¼ Psr: ð10ÞFrom (6), (7) and (10), we have

u ¼ Ps

2ðR2 � r2Þ � syðR� rÞ rp 6 r 6 R ð11Þ

and

u ¼ Ps

2ðR2 � r2pÞ � syðR� rpÞ; 0 6 r 6 rp ð12Þ

where rpis the radius of the plug flow region.

If sy=0, Eqs. (11) and (12) become

u ¼ Ps2 ðR2 � r2Þ; rp 6 r 6 R

whileu ¼ Ps2 ðR2 � r2pÞ; 0 6 r 6 rp:

ð13Þ

Case II: When M – 0, from Eqs. (6) and (8) we have for finitevalues of kðrÞ,.@2u@r2

þ 1r@u@r

� f 1kðrÞ þM2gu ¼ sy

r� 2Ps; rp 6 r 6 R ð14Þ

and

u ¼ up; 0 6 r 6 rp ð15Þwhere up is the value of uðrÞ at r ¼ rp and can be obtained from Eq.(14), by using (6).

If kðrÞ ¼ 1, the Eq. (14) reduces to the form

@2u@r2

þ 1r@u@r

�M2u ¼ syr� 2Ps

By employing finite difference method, we now proceed tosolve numerically the Eq. (14) subject to the boundary conditions

u ¼ 0 at r ¼ R

and @u@r ¼ 0 at r ¼ rp

ð16Þ

The plug radius rpis undetermined at this stage.

From (8), we have

s ¼ sþ ¼ Psr � 1r

Z r

0

1kðrÞ þM2

� �urdr; rp 6 r 6 R ð17Þ

and

s ¼ s� ¼ Psr � up

r

Z r

0

1kðrÞ þM2

� �r dr; 0 6 r 6 rp: ð18Þ

Now by using Newton–Raphson method, it is possible to deter-mine rp by considering continuity of s at r ¼ rp, i.e.

sþðr¼rpÞ ¼ sy ¼ s�ðr¼rpÞ ð19Þ

fluid in a porous bed under the action of a magnetic field: Application to/j.jestch.2016.11.008

J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx 5

3.2. Pulsatile flow

For studying the pulsatile flow of the fluid, substituting (6) into(5), we have

a2 @u@t

¼ @2u@r2

þ1r@u@r

�f 1kðrÞþM2guþfsy

r�2PðtÞg; rp 6 r6 R ð20Þ

andu ¼ upðtÞ; 0 6 r 6 rp ð21Þwhere upðtÞ is the value of uðr; tÞ at r ¼ rp, that is to be calculated bysolving Eq. (20), subject to the boundary condition (6) .

We have developed in the next section a finite differencescheme for solving numerically the Eqs. (20) and (21) subject tothe boundary conditionsu ¼ 0 at r ¼ R

and @u@r ¼ 0 at r ¼ rp;

ð22Þ

by taking the value of rp as that in the steady case.The volumetric flow rate can then be obtained by using the

formula

QðtÞ ¼ 2pZ R

0ru dr; ð23Þ

while the wall shear stress may be determined from the equation

swðtÞ ¼ PðtÞR� 1R

Z R

0

1kðrÞ þM2

� �ur dr: ð24Þ

Fig. 2. Velocity distribution for different values of magnetic parameter in the caseof steady flow of the fluid, where sy ¼ 0:05; k0 ¼ 0:5. It may be noted that thevelocity of blood diminishes, when the magnetic field strength rises. Also, for anyvalue of the magnetic parameter, the velocity is maximum in the plug (core) regionand its magnitude monotonically decreases as the fluid particles travel towards thewall.

Fig. 3. Distribution of velocity for different values of magnetic parameter inpulsatile flow, when sy ¼ 0:05; k0 ¼ 0:5; t ¼ 1:0. Here too, the observations aresimilar to that in Fig. 2, but one may note that at a particular location of the fluidmass, the velocity diminishes as the magnetic field strength gradually increases.

4. Application to magneto-hemorheology: estimates of bloodflow in a diseased artery under the action of a magnetic field

The object of the present study has been to investigate the flowcharacteristics of a Bingham plastic fluid through a porous matrixin the presence of an external magnetic field. We now develop anumerical scheme, with an aim to examine the variations of differ-ent flow variables. Since the study is motivated towards investigat-ing the characteristics of blood flow in a pathological state, andsince as discussed earlier in this paper, under certain pathologicalconditions, the nature of blood resembles that of a Bingham fluid,the numerical estimates have been found by using the analyticalexpressions presented in the preceding section, considering bloodas the working fluid. For the computational work, the followingdata presented in [59–61] have been used.

PðtÞ ¼ 1þ Acost ðAbeing aconstantÞ; 0:1 6 k0 6 15;0:8 6 M 6 4; A ¼ 0:2; Ps ¼ 1:0;a ¼ 0:04; 0:0 6 sy 6 0:1;M ¼ 1;2;3;4

and kðrÞ is a function of r, given by

kðrÞ ¼ k0 1�rr , where k0 is a constant.

The values/ranges of values match fairly well with blood flow ina coronary artery in a pathological state. It may be noted thatkðrÞ ¼ k0 when r ¼ 0:5; kðrÞ < k0 when r > 0:5 and kðrÞ > k0 whenr < 0:5. This implies that the permeability increases for r < 0:5and decreases for r > 0:5 for the present study.

The governing Eq. (20) subject to the boundary conditions (16)is solved numerically using finite difference implicit Crank–Nichol-son scheme given by

@2u@r2

¼ unþ1iþ1 � 2unþ1

i þ unþ1i�1

Dr2þ OðDr2Þ;

@u@r

¼ unþ1iþ1 � unþ1

i�1

2Drþ OðDr2Þ;

and@u@t

¼ unþ1i � un

i

Dtþ OðDtÞ;

Please cite this article in press as: J.C. Misra, S.D. Adhikary, Flow of a Binghammagneto-hemorheology, Eng. Sci. Tech., Int. J. (2016), http://dx.doi.org/10.1016

where n; i and ðnþ 1Þ; ðiþ 1Þ represent the conditions at ðt; rÞ andfðt þ DtÞ; ðr þ DrÞg respectively. The computational work has beencarried out by taking Dr ¼ 0:005;Dt ¼ 0:05. It has checked that fur-ther reduction in the values of Dr and Dt does not bring about anysignificant change in the results.

The convergence criteria of the numerical solution are given by

maxi

juni ðjþ 1Þ � un

i ðjÞj < e

maxn

juni ðkþ 1Þ � un

i ðkÞj < e

where j; k represent iteration steps for r and t, while e stands for theerror of tolerance.

Figs. 2 and 3 present the variation of the velocity distribution inthe case of steady and pulsatile flows for different values of themagnetic parameter M. It may be noted that in both steady andpulsatile cases, velocity decreases as the strength of the magneticfield increases. This implies that it is possible to reduce the flowof blood by suitably increasing the strength of the externally

fluid in a porous bed under the action of a magnetic field: Application to/j.jestch.2016.11.008

Fig. 4. Distribution of velocity for different values of porosity parameter in pulsatileflow, when sy ¼ 0:05;M ¼ 1:0; t ¼ 1:0.

Fig. 5. Comparison of velocity distribution in Cases I and II in the steady state,where sy ¼ 0:05. Case I refers to the situation when the magnetic field is absent andporosity of the medium is disregarded. The results presented for this situation arebased upon the closed form solution. The results for Case II correspond to thesituation where the system is under the action of an external magnetic field(M ¼ 1:2) and the porosity (k0 ¼ 1:0) of the bed is accounted for. The numericalresults for this case have been obtained by using the numerical solution presentedearlier in this section.

Fig. 6. Variation of volumetric flow rate with yield stress for different values of themagnetic field parameter in case of steady flow, when k0 ¼ 1:0.

Fig. 7. Variation of volumetric flow rate with yield stress for different values ofpermeability parameter in the case of steady flow, when M ¼ 1:2.

6 J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx

applied magnetic field. This observation has an important bearingon the act of minimizing blood flow during surgery and also on thetreatment of various health hazards. However, reduction in volu-metric flow of blood must be made within safety limits, so that itdoes not lead to cardiac/cerebral stroke for the individual. Volu-metric flow measurement with Doppler ultrasound was performedby Hoyt et al. [62] with an aim to assess blood flow in the range of100 to 1000 ml/min.

Fig. 4 gives the distribution of velocity for different values of theporosity parameter during pulsatile flow. One may note that veloc-ity increases with a rise in the value of the porosity parameter.Fig.5 gives a comparison between the velocity profiles in Case Iand Case II. The Case I is based upon the analytical solution fornon-porous case in the absence of the magnetic field, while forCase II the results are based on our computational solution thattakes into account the effects of porosity and the action of theexternal magnetic field.

Figs. 6 and 7 respectively give the distribution of volumetricflow rate in the case of steady flow with the yield stress, for differ-

Please cite this article in press as: J.C. Misra, S.D. Adhikary, Flow of a Binghammagneto-hemorheology, Eng. Sci. Tech., Int. J. (2016), http://dx.doi.org/10.1016

ent values of magnetic and permeability parameters. It is impor-tant to observe that the volumetric flow rate decreases as thevalue of the magnetic parameter increases, but increases as perme-ability rises. Time-variations of volumetric flow rate for pulsatileflow are shown in Figs. 8 and 9 for different values of the perme-ability parameter and the yield stress. One may observe fromFig.9 that the flow rate decreases, when yield stress increases.

The wall shear stress plays a very important role in the determi-nation of the location of platelet aggregation in the case of bloodflow. Keeping this in mind we have computed the values of thewall shear stress for pulsatile flow and presented in Fig. 10.

These results correspond to the situation when the permeabilitychanges with radius according to the law

kðrÞ ¼ k01� rr

It is to be noted that at any instant of time, the wall shear stressgets enhanced with increase in permeability.

Fig. 11 gives the time-variation of the wall shear stress in thecase of pulsatile flow, for different values of the yield stress. Theplots given in this figure depict very clearly that for the presentstudy, the wall shear stress increases with a rise in yield stress at

fluid in a porous bed under the action of a magnetic field: Application to/j.jestch.2016.11.008

Fig. 9. Variation of volumetric flow rate with time in the case of pulsatile flow ofblood for different values of yield stress, where k0 ¼ 1:0;M ¼ 1:4.

Fig. 10. Time-variation of wall shear stress in the case of pulsatile flow of blood fordifferent values of variable permeability parameter, when sy ¼ 0:05 and M ¼ 1:0.

Fig. 11. Time-variation of the wall shear stress in the case of pulsatile flow of bloodfor different values of yield stress, when k0 ¼ 1:0;M ¼ 1:0.

Fig. 8. Variation of volumetric flow rate with time in the case of pulsatile flow ofblood for different values of permeability parameter, when sy ¼ 0:05;M ¼ 1:4.

Table 1Comparison of velocity distribution with a previous study.

r 0 0.2 0.4 0.6 0.8 1.0

Our results 0.151 0.145 0.132 0.104 0.075 0.0Dash et al. [59] 0.118 0.105 0.084 0.076 0.055 0.0

J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx 7

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all points of time, and further that it is minimum for a Newtonianfluid (for which sy ¼ 0:0).

5. Validation of the model

With a view to validating the results of the present study, wehave compared the computed results for the fluid velocity withthose of a previous study of similar problem for the flow of Cassonfluid [59]. The results presented in Table 1 have been computed bybringing both the problems as close as possible with k0 ¼ 0:1 andsy ¼ 0:1 for both the studies. It may be noted that the results ofpresent study match fairly well with those reported in [59]. Thisobservation may be considered as the validation of our model.The small differences may be attributed to be due to the fact thatthe fluid model of the present study is different from that usedin [59].

6. Summary and conclusion

The paper is devoted to an investigation the steady flow as wellas the pulsatile flow of a Bingham plastic fluid in the presence of anexternally applied magnetic field. The governing equations aresolved by using finite difference technique. The effects of porosity,magnetic field and yield stress on the velocity, volumetric flow rateand wall shear stress are examined. The said effects have been esti-mated numerically. The computational results for blood flow in adiseased artery have been presented graphically.

On the basis of the present study, we can conclude that both thevolumetric flow rate of blood (if considered as a Bingham plasticfluid) and the wall shear stress are enhanced as the permeabilityof the diseased arterial wall increases. But the velocity and volu-metric flow rate of blood reduce, when there is an increase in themagnetic field strength. One can further conclude that as the yieldstress of blood increases, both the volumetric flow rate and thewall shear stress are enhanced.

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8 J.C. Misra, S.D. Adhikary / Engineering Science and Technology, an International Journal xxx (2016) xxx–xxx

The study serves as a first step towards exploring blood flow ina pathological situation, when fatty plaques of cholesterol areformed and/or when artery-clogging takes place in the lumen ofan artery. The study is also useful in investigating blood flow inand around a region, where a tumor has been formed. Cliniciansdealing with treatment of various hemodynamical diseases bythe application of an external magnetic field will also find theresults interesting. Moreover, the results of the present study willbe of immense benefit in validating the results of similar experi-mental studies and those of more complex theoretical studies tobe carried out in future.

Acknowledgment

Author Prof. J.C.Misra is thankful to the Alexander von Hum-boldt Foundation, Germany for supporting his visit to the TechnicalUniversity of Hamburg during May–July 2016, where a part of thework was carried out. He is also thankful to his Host Professor Dr.Robert Seifried, Director of the Institute of Mechanics and NavalEngineering, TU Hamburg for some valuable discussions withhim. Both the authors wish to express their deep sense of gratitudeto the Science and Engineering Research Board, Department ofScience and Technology, Government of India for the financial sup-port through Grant No. SB/S4/MS864/14. The authors wish tothank all the three reviewers (anonymous) for their words of pro-found appreciation of the quality and application potential of thepaper. They also appreciate the comments of the reviewers onthe basis of which the manuscript has been revised.

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