Flow of a Bingham Fluid in a Porous Bed Under The

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 magnetic ﬁeld: Application to magneto-hemorheology ⇑ J.C. Misra a, , S.D. Adhikary b a Centre for Healthcare Science and Technology, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India b Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur, 721302, India article info abstract

Article history: The study deals with an investigation of the flow of a Bingham plastic fluid in a porous bed under the Received 30 August 2016 action of an external magnetic field. Porosity of the bed has been described by considering Brinkman Revised 5 November 2016 model. Both steady and pulsatile motion of this non-Newtonian fluid have been analysed. The governing Accepted 8 November 2016 equations are solved numerically by developing a suitable finite difference scheme. As an application of Available online xxxx the theory in the field of magneto-hemorheology, the said physical variables have been computed by considering the values of the involved parameters for blood flow in a pathological state, when the system is Keywords: under the action of an external magnetic field. The pathological state corresponds to a situation, where Bingham plastic fluid the lumen of an arterial segment has turned into a porous structure due to formation of blood clots. Porous medium Pulsatile flow Numerical estimates are obtained for the velocity profile and volumetric flow rate of blood, as well as Yield stress for the shear stress, in the case of blood flow in a diseased artery, both the velocity and volumetric flow Magnetic field rate diminish, as the strength of the external magnetic field is enhanced. The study further shows that Wall shear stress blood velocity is maximum in the plug (core) region. It decreases monotonically as the particles of blood travel towards the wall. The study also bears the potential of providing numerical estimates for many industrial fluids that follow Bingham plastic model, when the values of different parameters are chosen appropriately. Ó 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 humans to static magnetic fields of high intensity during MRI was presented by Schenck [1]. He made an important observation Hemorheology (also called as blood rheology) deals with vari- that there is hardly any evidence of health hazard associated with ous flow characteristics of blood and its constituents, e.g. plasma, exposure of the human body to magnetic fields, not even when the erythrocytes (red cells), white cells, platelets, etc. Tissue perfusion body is exposed to a strong magnetic field in a cumulative manner, can take place properly, as long as the rheological properties of provided no ferromagnetic material is present. The reason behind blood are well within certain levels. Significant departures of the this observation is twofold: (i) Human tissues lack ferromagnetic properties of blood from those at the normal physiological state materials, and (ii) magnetic susceptibility of these tissues is small. may lead to different arterial diseases. Many health problems, As mentioned by Schenck [1], studies on human subjected to a including hypertension, diabetes mellitus and insulin resistance, magnetic fields of strength up to 8 T and on sub-human under the metabolic syndrome and obesity are directly linked with the vis- action of magnetic fields up to 16 T indicate that a considerable cosity of whole blood. The area of studies related to the rheological margin of safety exists. This observation shows that the range properties of blood under the action of magnetic fields (as in the 3–4 T commonly used in clinical procedures is well within the case of MRI) may be called as magneto-hemorheology. safety zone. Human exposure to external magnetic fields is of common It is known that blood is an electrically conducting fluid and so occurrence in various clinical procedures. Patients are exposed to when blood flows under the action of an external magnetic field of strong magnetic fields during MRI (magnetic resonance imaging). sufficient strength, a transverse EMF is developed, which is directly An excellent review of various issues related to the exposure of proportional to the velocity of blood, as well as to the intensity of the applied magnetic field. Owing to this, it becomes difficult to obtain good ECGs during magnetic resonance scanning. Kinouchi ⇑ Corresponding author. et al. [2] pointed out that the said effect contributes to human E-mail address: [email protected] (J.C. Misra). tolerance of highly intensified magnetic fields. http://dx.doi.org/10.1016/j.jestch.2016.11.008 2215-0986/Ó 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/).

Nomenclature

r; /; z cylindrical co-ordinates PðtÞ non-dimensional pressure gradient R radius of the tube kðrÞ permeability factor a rp radius of plug region Womersley parameter R0 characteristic radius A amplitude of the ﬂow u velocity in radial direction w angular frequency uc characteristic velocity B0; r magnetic induction and conductivity of the medium q; l density and viscosity, respectively M Hartmann number s shear stress s c characteristic shear stress s y yield stress

Bingham plastic fluids are those in which the excess deviatoric flow through a stenosed artery was recently discussed by Yadav stress over the yield stress varies linearly with shear rate. These and Kumar [12]. Singh and Singh [13] considered the fully devel- fluids exhibit non-Newtonian behavior. They bear the potential oped one dimensional Bingham plastic flow of blood through a to transmit shear stress even in the absence of a velocity gradient. small artery having multiple stenoses and post-stenotic dilation. There exists a linear relationship between shear stress and strain Using regular perturbation method, the oscillatory flow of a Bing- for Bingham plastic fluids; a finite yield stress is needed for them ham plastic fluid was studied theoretically by De-Chant [14], to flow. At a low stress level, their motion is very similar to that who made an attempt to develop a relationship between the veloc- of a rigid body, but when the stress level is high, they flow like a ity field and dimensionless flow rate. The author mentioned that viscous fluid. It has been found that Bingham liquids possess this study was motivated towards examining blood flow in arteries thicker coating in comparison to the case of Newtonian fluids. Clay, in a pathological state and claimed that the solution reported by drilling mud, printing ink, molten liquid crystalline polymers, him provides useful analytical models that bear the potential to foams, paint, toothpaste and food articles like margarine, mayon- support experimental and computational studies on arterial blood naise, molten chocolate, yoghurts and ketchup are some typical flow. examples of Bingham plastics. Since Bingham fluids are trans- Several benchmark contributions have been made by Misra formed to solids when the applied shear stress is less than the yield et al. (cf. [15–40]) to explore a variety of information in hemorhe- stress, it is apparent that Bingham fluids behave like a solid med- ology. Since a single non-Newtonian model is inadequate to ium in the core layer. That means, a solid plug moves within the describe the complexity of blood and its flow in normal/diseased flow. arteries, they have used different non-Newtonian models, e.g. Cas- A study on the settling characteristics of spherical particles in son model, micropolar fluid model, power law fluid model, couple Bingham plastic fluids was conducted by Ansley and Smith [3]. stress fluid model, viscoelastic fluid model and Herschel–Bulkley Mitsoulis [4] presented a review of research activities of viscoplas- fluid model. In each case, the choice of a particular model has been tic models that include Bingham plastic model, Herschel–Bulkley made by keeping an eye on the objective of the study. Some of their model and Casson model. He discussed, in particular, the entry studies pertain to arterial blood flow in the normal physiological and exit flows from dies, flows around spheres and cylinders, as state, while some others are concerned with blood flow in arteries well as squeeze flows. A theoretical study of the free convection under pathological conditions. Some of the hemorheological stud- in a Bingham plastic fluid on a vertical flat plate with constant wall ies conducted by Misra et al. that have been mentioned above also temperature was performed by Kleppe and Marner [5]. They involve application of transport theory in porous media. reported that Bingham plastic friction coefficients are considerably An investigation on peristaltic motion of blood in the micro- higher than those in the case of Newtonian fluids. This increase circulatory system was recently conducted by Misra and Maiti was attributed to the yield stress of Bingham plastic fluids. A [33], in which the non-Newtonian behaviour of blood has been two-dimensional study on creeping flow of a Bingham plastic fluid modelled as a Herschel–Bulkley fluid and the vessel has been con- past a cylinder was conducted numerically by Nirmalkar et al. [6], sidered to be of varying cross-section. In this communication, the who reported that in the limit of plastic flow, drag approaches a authors mentioned that the Herschel–Bulkley fluid model is more constant value. Bahaduri et al. [7] developed a simple predictive general than most other non-Newtonian models and that the tool that can be used to predict easily the boundaries of the plug results for a fluid represented by the Bingham plastic fluid model of Bingham plastic fluids for laminar flow through annulus. can be derived from those of any study conducted with considera- Sayad-Ahmed et al. [8] used a numerical method to examine ther- tion of blood as a Herschel–Bulkley fluid. mally laminar heat transfer based on the fully developed velocity An interesting study on the effect of thermal radiation on the for Bingham fluids in the entrance region of a circular duct. With magnetohydrodynamic flow of blood and heat transfer in a perme- the help of Bingham plastic model, Liu and Mei [9] made an able capillary in stretching motion has been reported by Misra and attempt to examine the effects of wave-induced friction on a Sinha [34]. In this study, the lumen of the capillary has been con- muddy sea bed. Yu et al. [10] performed a study of oscillatory flow sidered to have turned into a porous structure due to some arterial of magnetorheological fluid dampers subject to sinusoidal dis- disease. The authors have developed a suitable numerical model to placement excitation, based on Bingham plastic and Herschel– study the problem. The results obtained on the basis of the study Bulkley models. have an important bearing on the therapeutic procedure of electro- The available scientific literature reveals that the behavior of magnetic hyperthermia, particularly in understanding/regulating bio-mass and some physiological fluids e.g. blood under certain blood flow and heat transfer in capillaries. pathological situations can be well described by a Bingham model. A similar model for blood flow and vessel geometry was formu- The yield stress of blood is now well established. A review of differ- lated and analyzed by Maiti and Misra [35] in their recent investi- ent studies on the yield stress of blood has been given in the article gation of the non-Newtonian characteristics of peristaltic flow of of Paicart et al. [11]. The Bingham plastic characteristics of blood blood through micro-vessels, e.g. arterioles and venules. On the

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: Application to magneto-hemorheology, Eng. Sci. Tech., Int. J. (2016), http://dx.doi.org/10.1016/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 of cular/porous media in arterial tumors. Since Darcy model ignores SSD wave propagation that affect significantly the flow behaviour the boundary effects on the flow, application of this model is not of blood in arterioles. Another study was carried out by Maiti valid when the effect of the boundaries of the porous medium is and Misra [36] to investigate the peristaltic flow of an incompress- accounted for. In order to incorporate the boundary effects, the ible viscous fluid in a porous channel, by considering slip boundary Brinkman model is more appropriate for modeling blood flow conditions of Saffman type. This study was motivated towards through a porous matrix. exploring some important information concerning the flow of bile As mentioned earlier, there has been a recent trend of using the within ducts in a pathological state, when numerous stones are interaction of a magnetic field with tissue fluids, in many diagnos- formed in the bile. tic devices, especially those which are used to diagnose cardiovas- A separate study of a dually stenosed artery was conducted by cular diseases. Magnetic fields are employed to develop devices for Sinha and Misra [37], in which they took into account the variation cell separation, targeted transport of drug using magnetic particles of blood viscosity, variable hematocrit and velocity-slip at the arte- as drug carriers, reduction of bleeding during surgeries and provo- rial wall. Two different research investigations on electro-osmotic cation of occlusion of feeding vessels of cancerous tumors and flow of blood having the promise of important applications to development of magnetic tracers. For this reason, studies of flow micro-biofluidics were recently carried out by Misra et al. properties of human blood as well as deformation of blood vessels [38,39]. Another study depicting the effect of chemical reaction subject to an applied external magnetic field have been carried out on heat and mass transfer of blood was reported by Misra and by several researchers (cf. [51–55]). Adhikary [40]. Two-dimensional pulsatile blood flow through a stenosed A problem related to the dynamics of blood flow was recently artery in the presence of an external magnetic field was studied investigated by Chandra and Misra [41] for the situation when by Alimohamadi and Imani [56] by means of finite element simu- the influence of Hall current and the rotation of microparticles of lation. Another study of non-Newtonian blood flow in magnetic blood, such as erythrocytes and thrombocytes co-exist. The targeting drug delivery in stenosed carotid artery having bifurca- authors made an important observation that the micro rotation tion was performed by Alimohamadi et al. [57]. Akbarzadeh [58] of the erythrocytes of blood is enhanced due to the effect of Hall studied pulsatile magneto-hydrodynamic blood flow through por- current, when blood flows in the arterial system. Effect of heat ous blood vessels using a third grade non-Newtonian fluid model. transfer during MHD flow of blood was reported by Sinha et al. In the present paper, the flow of a Bingham plastic fluid through [42] for the case when a non-uniform heat source is present. Stag- a porous bed under the action of a magnetic field has been inves- nation point flow and heat transfer of blood treated as an electri- tigated. Although such a problem bears the potential of important cally conducting fluid was investigated by Misra et al. [43] on the applications in hemorheology, particularly in pathological fluid basis of which the authors reported that blood flow is reduced, dynamics and also in the study of industrial fluids, it has escaped as the strength of the induced magnetic field increases. They the attention of previous researchers. A mathematical analysis is observed that reduction in blood velocity is followed by an carried out for the flow of a Bingham plastic fluid through a porous enhancement of the temperature field. By considering the matrix using Brinkman model. The problem is investigated for a temperature-dependent properties of blood, heat and mass trans- particular situation, when an externally applied magnetic field is fer of blood flowing peristaltically in asymmetric channels was also present. Computational results for the variation in velocity distri- studied by Misra et al. [44]. This study reveals that the volume of bution, volumetric flow rate and wall shear stress with porosity the trapped bolus reduces when the Hartmann number/Reynolds parameter, magnetic parameter and yield stress are presented number/fluid viscosity/Scmidt number increases, and that the graphically. As indicated above, the present study bears a strong bolus size is enhanced, when the viscosity of the blood diminishes. potential of a wide variety of applications in polymer industries Another magnetohydrodynamic systematic study of blood flow as well as in pathological fluid dynamics, biomedical engineering was presented by Korchevskii and Marochnik [45] in a very elabo- (e.g.MRI,ECG etc.), clinical diagnosis and surgery, which involve rate manner. flows through porous media in presence of magnetic fields. As an The transport theory in porous media involving various models, illustrative example, the theory developed has been applied to such as Darcy and Brinkman models for momentum transport has investigate the dynamics of blood flow in diseased arteries under been found to be quite useful in describing the flow of various bio- the action of an external magnetic field, when the arterial lumen logical fluids. It has been found that models for flow through por- turns into a porous structure and the behavior of blood changes ous media are widely applicable in the simulation of blood flow in to that of a plastic fluid. tumors and in modeling blood flow when fatty plaques of cholesterol and artery-clogging clots are formed in the lumen of an 2. The problem and the Governing equations artery. Kim and Tarbell [46] have discussed a simple macromolecular Let us consider the fully developed flow of a Bingham plastic model on the basis of a two-parameter strain-dependent perme- fluid in a porous matrix formed inside a circular tube (cf. Fig. 1), ability function developed by Klancher and Tarbell [47], by using subject to an external magnetic field B0. Using cylindrical coordi- a pore theory in order to determine the transport properties of nates (r; /; z), the momentum equation can be written as macromolecules in the arterial wall. Vankan et al. [48,49] have @u @p 1 @ rs l compared a hierarchical mixture model of blood perfused biologi- q ð Þ 2r ¼ þ B0 u ð1Þ cal tissue that utilizes an extended Darcy equation for blood flow. @t @z r @r k ðrÞ They also have performed a simulation for blood flow through a and the constitutive equation for a Bingham plastic fluid as contracting muscle with a hierarchical structure of pores and have s s l @u ; s P s found that blood pressures calculated by them are approximately ¼ y þ @ y r ð2Þ equal to blood pressures measured in skeletal muscle. @u ; s 6 s @ ¼ 0 Another important domain of research in hemorheology is the r y one that deals with the application of the Darcy model to blood Definitions of all the symbols appearing in the Eqs. (1) and (2) flow in a tumor, which consists of tissues that are highly heteroge- are included in the Nomenclature given earlier. neous as compared to the case of normal tissues. Baish et al. [50] The Eqs. (1) and (2) are required to be solved subject to the fol- have used the Darcy model to represent the flow through the vas- lowing conditions:

rp z

u (r, t)

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

u ¼ 0atr ¼ R ðno slipconditionÞ s ¼ Psr: ð10Þ ð3Þ s isfiniteat r ¼ 0 ðregularityconditionÞ From (6), (7) and (10), we have Let us ﬁrst introduce the non-dimensional variables Ps 2 2 u ¼ ðR r ÞsyðR rÞ rp 6 r 6 R ð11Þ r ; R ; u ; s s ; @p ; x 2 r ¼ R ¼ u ¼ ¼ s @ ¼p PðtÞ t ¼ t R0 R0 uc c z 0 l R2B2r s qxR2 and s uc ; 2 0 0 ; s y ; k ðr Þ ; a2 0 ; c ¼ M ¼ l y ¼ s kðrÞ¼ 2 ¼ l R0 c R 0 P u ¼ s ðR2 r2Þs ðR r Þ; 0 6 r 6 r ð12Þ ð4Þ 2 p y p p

p R2 0 0 where rpis the radius of the plug flow region. where uc ¼ 2l represents a dimensional characteristic velocity, s p0 the dimensional pressure-gradient. If y=0, Eqs. (11) and (12) become In terms of the non-dimensional variables defined in (4), Eqs. Ps 2 2 u ¼ ðR r Þ; rp 6 r 6 R (1) and (2) may be re-written in the form 2 ð13Þ whileu ¼ Ps ðR2 r2Þ; 0 6 r 6 r : @u 1 @ðrsÞ 1 2 p p a2 ¼ 2PðtÞ þ M2 u ð5Þ @t r @r kðrÞ Case II: When M – 0, from Eqs. (6) and (8) we have for finite values of kðrÞ,. @u s ¼ sy þ ; s P sy @r ð6Þ @2 @ s @u ; s 6 s u 1 u 1 2 y @ ¼ 0 y þ f þ M gu ¼ 2P ; r 6 r 6 R ð14Þ r @r2 r @r kðrÞ r s p while the boundary conditions (3) are transformed to and u ¼ 0 at r ¼ R ð7Þ u u ; 0 6 r 6 r 15 s isfiniteatr ¼ 0 ¼ p p ð Þ In the section that follows, we have analyzed the problem in the where up is the value of uðrÞ at r ¼ rp and can be obtained from Eq. case of steady and unsteady flows. (14), by using (6). If kðrÞ¼1, the Eq. (14) reduces to the form 3. Analysis @2u 1 @u s þ M2u ¼ y 2P @r2 r @r r s 3.1. Steady flow By employing finite difference method, we now proceed to In the steady state, the non-dimensional form of the momen- solve numerically the Eq. (14) subject to the boundary conditions tum Eq. (5) reduces to u ¼ 0 at r ¼ R @ s 16 1 ðr Þ 1 2 @u ð Þ 0 ¼ 2P þ M u; ð8Þ and @ ¼ 0 at r ¼ rp s r @r kðrÞ r The plug radius rpis undetermined at this stage. where Ps stands for the non-dimensional pressure gradient in the steady-state. From (8), we have Z 1 r 1 Case I: When M = 0, and kðrÞ¼1, Eq. (8) further simplifies to s ¼ s ¼ P r þ M2 ur dr; r 6 r 6 R ð17Þ þ s r kðrÞ p 1 @ðrsÞ 0 0 ¼ 2P : ð9Þ s r @r and Z r Integrating (9) with respect to r, we obtain up 1 2 s ¼ s ¼ Psr þ M rdr; 0 6 r 6 rp: ð18Þ C r 0 kðrÞ s ¼ Psr þ ; r Now by using Newton–Raphson method, it is possible to deter- s where C is an arbitrary constant of integration. mine rp by considering continuity of at r ¼ rp, i.e. In view of condition (7), since s has to be finite at r = 0, we have s ¼ s ¼ s ð19Þ to take C = 0, so that þðr¼rpÞ y ðr¼rpÞ

3.2. Pulsatile flow where n; i and ðn þ 1Þ; ði þ 1Þ represent the conditions at ðt; rÞ and fðt þ DtÞ; ðr þ DrÞg respectively. The computational work has been For studying the pulsatile flow of the fluid, substituting (6) into carried out by taking Dr ¼ 0:005; Dt ¼ 0:05. It has checked that fur- (5), we have ther reduction in the values of Dr and Dt does not bring about any @u @2u 1 @u 1 s significant change in the results. a2 2 y ; 6 6 ¼ þ f þ M gu þf 2PðtÞg rp r R ð20Þ The convergence criteria of the numerical solution are given by @t @r2 r @r kðrÞ r n n < e and maxjui ðj þ 1Þui ðjÞj i u ¼ upðtÞ; 0 6 r 6 rp ð21Þ ; maxjunðk þ 1ÞunðkÞj < e where upðtÞ is the value of uðr tÞ at r ¼ rp, that is to be calculated by n i i solving Eq. (20), subject to the boundary condition (6) . We have developed in the next section a finite difference where j; k represent iteration steps for r and t, while e stands for the scheme for solving numerically the Eqs. (20) and (21) subject to error of tolerance. the boundary conditions Figs. 2 and 3 present the variation of the velocity distribution in u ¼ 0 at r ¼ R the case of steady and pulsatile flows for different values of the ð22Þ @u ; magnetic parameter M. It may be noted that in both steady and and @ ¼ 0 at r ¼ rp r pulsatile cases, velocity decreases as the strength of the magnetic by taking the value of rp as that in the steady case. field increases. This implies that it is possible to reduce the flow The volumetric flow rate can then be obtained by using the of blood by suitably increasing the strength of the externally formula Z R QðtÞ¼2p ru dr; ð23Þ 0 while the wall shear stress may be determined from the equation Z R 1 1 2 swðtÞ¼PðtÞR þ M ur dr: ð24Þ R 0 kðrÞ

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

The object of the present study has been to investigate the flow characteristics of a Bingham plastic fluid through a porous matrix in the presence of an external magnetic field. We now develop a numerical scheme, with an aim to examine the variations of different flow variables. Since the study is motivated towards investigating the characteristics of blood flow in a pathological state, and since as discussed earlier in this paper, under certain pathological conditions, the nature of blood resembles that of a Bingham fluid, Fig. 2. Velocity distribution for different values of magnetic parameter in the case of steady flow of the fluid, where sy ¼ 0:05; k0 ¼ 0:5. It may be noted that the the numerical estimates have been found by using the analytical velocity of blood diminishes, when the magnetic field strength rises. Also, for any expressions presented in the preceding section, considering blood value of the magnetic parameter, the velocity is maximum in the plug (core) region as the working fluid. For the computational work, the following and its magnitude monotonically decreases as the fluid particles travel towards the data presented in [59–61] have been used. wall.

PðtÞ¼1 þ 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

1r kðrÞ¼k0 r , where k0 is a constant. The values/ranges of values match fairly well with blood flow in a coronary artery in a pathological state. It may be noted that kðrÞ¼k0 when r ¼ 0:5; kðrÞ < k0 when r > 0:5 and kðrÞ > k0 when r < 0:5. This implies that the permeability increases for r < 0:5 and 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 unþ1 2unþ1 þ unþ1 ¼ iþ1 i i1 þ OðDr2Þ; @r2 Dr2 @u unþ1 unþ1 ¼ iþ1 i1 þ OðDr2Þ; @r 2Dr Fig. 3. Distribution of velocity for different values of magnetic parameter in pulsatile flow, when s ¼ 0:05; k ¼ 0:5; t ¼ 1:0. Here too, the observations are @u unþ1 un y 0 and ¼ i i þ OðDtÞ; similar to that in Fig. 2, but one may note that at a particular location of the fluid @t Dt mass, the velocity diminishes as the magnetic field strength gradually increases.

Fig. 4. Distribution of velocity for different values of porosity parameter in pulsatile Fig. 6. Variation of volumetric flow rate with yield stress for different values of the s : ; : ; : flow, when y ¼ 0 05 M ¼ 1 0 t ¼ 1 0. magnetic field parameter in case of steady flow, when k0 ¼ 1:0.

Fig. 5. Comparison of velocity distribution in Cases I and II in the steady state, s : where y ¼ 0 05. Case I refers to the situation when the magnetic field is absent and Fig. 7. Variation of volumetric flow rate with yield stress for different values of porosity of the medium is disregarded. The results presented for this situation are permeability parameter in the case of steady flow, when M ¼ 1:2. based upon the closed form solution. The results for Case II correspond to the situation 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 numerical results for this case have been obtained by using the numerical solution presented ent values of magnetic and permeability parameters. It is impor- earlier in this section. tant to observe that the volumetric flow rate decreases as the value of the magnetic parameter increases, but increases as perme- applied magnetic field. This observation has an important bearing ability rises. Time-variations of volumetric flow rate for pulsatile on the act of minimizing blood flow during surgery and also on the flow are shown in Figs. 8 and 9 for different values of the perme- treatment of various health hazards. However, reduction in volu- ability parameter and the yield stress. One may observe from metric flow of blood must be made within safety limits, so that it Fig.9 that the flow rate decreases, when yield stress increases. does not lead to cardiac/cerebral stroke for the individual. Volu- The wall shear stress plays a very important role in the determi- metric flow measurement with Doppler ultrasound was performed nation of the location of platelet aggregation in the case of blood by Hoyt et al. [62] with an aim to assess blood flow in the range of flow. Keeping this in mind we have computed the values of the 100 to 1000 ml/min. wall shear stress for pulsatile flow and presented in Fig. 10. Fig. 4 gives the distribution of velocity for different values of the These results correspond to the situation when the permeability porosity parameter during pulsatile flow. One may note that veloc- changes with radius according to the law ity increases with a rise in the value of the porosity parameter. 1 r Fig.5 gives a comparison between the velocity profiles in Case I kðrÞ¼k0 and Case II. The Case I is based upon the analytical solution for r non-porous case in the absence of the magnetic field, while for It is to be noted that at any instant of time, the wall shear stress Case II the results are based on our computational solution that gets enhanced with increase in permeability. takes into account the effects of porosity and the action of the Fig. 11 gives the time-variation of the wall shear stress in the external magnetic field. case of pulsatile flow, for different values of the yield stress. The Figs. 6 and 7 respectively give the distribution of volumetric plots given in this figure depict very clearly that for the present flow rate in the case of steady flow with the yield stress, for differ- study, the wall shear stress increases with a rise in yield stress at

Fig. 8. Variation of volumetric flow rate with time in the case of pulsatile flow of Fig. 11. Time-variation of the wall shear stress in the case of pulsatile flow of blood blood for different values of permeability parameter, when sy ¼ 0:05; M ¼ 1:4. for different values of yield stress, when k0 ¼ 1:0; M ¼ 1:0.

Table 1 Comparison 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 0.0 Dash et al. [59] 0.118 0.105 0.084 0.076 0.055 0.0

all points of time, and further that it is minimum for a Newtonian

ﬂuid (for which sy ¼ 0:0).

5. Validation of the model

With a view to validating the results of the present study, we have compared the computed results for the fluid velocity with those of a previous study of similar problem for the flow of Casson fluid [59]. The results presented in Table 1 have been computed by

bringing both the problems as close as possible with k0 ¼ 0:1 and

sy ¼ 0:1 for both the studies. It may be noted that the results of Fig. 9. Variation of volumetric flow rate with time in the case of pulsatile flow of present study match fairly well with those reported in [59]. This blood for different values of yield stress, where k0 ¼ 1:0; M ¼ 1:4. observation may be considered as the validation of our model. The small differences may be attributed to be due to the fact that the fluid model of the present study is different from that used in [59].

6. Summary and conclusion

The paper is devoted to an investigation the steady flow as well as the pulsatile flow of a Bingham plastic fluid in the presence of an externally applied magnetic field. The governing equations are solved by using finite difference technique. The effects of porosity, magnetic field and yield stress on the velocity, volumetric flow rate and wall shear stress are examined. The said effects have been esti- mated numerically. The computational results for blood flow in a diseased artery have been presented graphically. On the basis of the present study, we can conclude that both the volumetric flow rate of blood (if considered as a Bingham plastic fluid) and the wall shear stress are enhanced as the permeability of the diseased arterial wall increases. But the velocity and volumetric flow rate of blood reduce, when there is an increase in the magnetic field strength. One can further conclude that as the yield Fig. 10. Time-variation of wall shear stress in the case of pulsatile flow of blood for stress of blood increases, both the volumetric flow rate and the different values of variable permeability parameter, when sy ¼ 0:05 and M ¼ 1:0. wall shear stress are enhanced.

The study serves as a first step towards exploring blood flow in [18] J.C. Misra, B. Pal, A Mathematical model for the study of pulsatile flow of blood a pathological situation, when fatty plaques of cholesterol are under an externally imposed body acceleration, Mathl. Comp. Model. 29 (1) (1999) 89–106. formed and/or when artery-clogging takes place in the lumen of [19] J.C. Misra, G.C. Shit, Role of slip-velocity in blood flow through stenosed an artery. The study is also useful in investigating blood flow in arteries: a non-Newtonian model, J. Mech. Med. Biology (JMMB) 7 (3) (2007) and around a region, where a tumor has been formed. Clinicians 337–353. [20] J.C. Misra, S. Chandra, G.C. Shit, P.K. Kundu, Electro-osmotic flow of a dealing with treatment of various hemodynamical diseases by viscoelastic fluid in a channel, Appl. Math. Comput. 217 (2011) 7932–7939. the application of an external magnetic field will also find the [21] J.C. Misra, M.K. Patra, S.C. 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Shit, Multiphase flow of blood through arteries with a branch capillary: a theoretical study, J. Mech. Med. Biol. 7 (4) (2007) Author Prof. J.C.Misra is thankful to the Alexander von Hum- 395–414. boldt Foundation, Germany for supporting his visit to the Technical [26] A. Sinha, J.C. Misra, Numerical study of flow and heat transfer during oscillatory blood flow in diseased arteries in presence of magnetic fields, University of Hamburg during May–July 2016, where a part of the Appl. Math. Mech. Engl. Ed. 33 (2012) 649–662. work was carried out. He is also thankful to his Host Professor Dr. [27] J.C. Misra, B.K. Kar, Momentum integral method for studying flow Robert Seifried, Director of the Institute of Mechanics and Naval characteristics of blood through a stenosed vessel, Biorheology 26 (1989) 23–35. Engineering, TU Hamburg for some valuable discussions with [28] J.C. Misra, S.K. Pandey, Peristaltic transport of a non-Newtonian fluid with a him. 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