Finite Element Simulation of Two-Dimensional Pulsatile ...

International Journal for Computational Methods in Engineering Science and Mechanics, 15:390–400, 2014Copyright c© Taylor & Francis Group, LLCISSN: 1550-2287 print / 1550-2295 onlineDOI: 10.1080/15502287.2014.915253

Finite Element Simulation of Two-Dimensional PulsatileBlood Flow Through a Stenosed Artery in the Presenceof External Magnetic Field

Haleh Alimohamadi1 and Mohsen Imani2

1Department of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran2Department of Electrical and Computer Engineering, College of Engineering, University of Tehran,Tehran, Iran

This paper introduces the impact of external magnetic fieldon blood flow patterns in a stenosis artery. Considering the fattydeposited lump, arterial walls as porous media, and pulsatile in-flow base on human-heart-beating rate closes our model to theactual stenosis blood artery. In this study, by solving transientfluid dynamic equations in coupled porous and free media, the ve-locity, temperature, and shear stress distribution along the lumpare investigated. The results show that applying 105 magneticfield intensity (MnF) creates two vortexes on the lumps’ edgesand 15X (16.6X) higher shear stress (temperature) in the stenosisregion.

Keywords Blood flow, Biomagnetic fluid, Stenosis artery, Heattransfer, Porous media, Magnetization and Lorentzforces

1. INTRODUCTIONCurrently, cardiovascular diseases, especially stenosis, are

very prevalent and take the lives of many people. This commondisease occurs by backlogging macromolecules on the arterialwalls and atherosclerotic plaques’ formation by the passage ofthe time. These plaques block the normal path of blood flow,make it narrower and, in critical conditions, close it completely.Biological research shows that these macromolecules are mostlymade from fatty material and are vulnerable to high temperatureand shear stress, so dissolving them by increasing heat transferrate or exerting higher viscous forces through applying external

Address correspondence to Mohsen Imani, Room 401, School ofElectrical and Computer Engineering, University College of Engineer-ing, University of Tehran, P.O. Box 14395-515, North Kargar St.,Tehran, Iran. E-mail: [email protected]

Color versions of one or more of the figures in the article can befound online at www.tandfonline.com/ucme.

magnetic field is a safe and new bioengineering-suggested rem-edy. Natural blood has an innate, slight magnetization featureand in order to increase this characteristic, additional injectedferro-nanoparticles have recently been used. These particles in-crease blood magnetization and are mostly organic solventswhich can act as solutes. The solution of these nanoparticles innatural and industrial fluids have wide application in bioengi-neering [1, 2], micro-electrical and mechanical structure [3],cardiovascular and articulate treatments [4, 5], and two-phaseflow [6]. During the last few decades, numerous investigationshave been done in this new field of biomechanics. Stangeby andEthier [7] worked out a coupling model of Navier-Stokes andBrinkman equations in free and porous media. The dynamicexpression for heat and mass transfer in stenosis artery is intro-duced in [8]. In [9], the velocity and temperature distribution ofbiofluid flow under the influence of an external magnetic fieldare discussed. The latter article modeled the artery as a uni-form tub with solid and impenetrable walls. In [10], differentmodels of macro blood flow were coupled with various areasand types of stenosis. The time-dependent heat transfer of two-dimensional blood flow is discussed in [11]; however, this articleneglected the porous assumption for arterial walls. The effectsof permeability, magnetic field, and body acceleration on bloodflow passing through a porous media are brought up in [12], andthe analytical solution of stationary non-Newtonian blood flowin porous media at the present of magnetic field is investigated in[13]. The innovation of this article lies in solving transient fluiddynamics equations of blood flow through stenosis geometrytaking into account the non-Newtonian viscosity of blood andboth magnetization and Lorentz forces. In this study, accordingto a real heart-beating rate, the time-dependent inlet velocityalters and the impact of the magnetic field on different heart cy-cles is described. In Part 2, the assumed geometry for a stenosisvessel in addition to the governing equations of fluid flow underthe action of magnetic field is discussed. Part 3 is developed to

390

PULSATILE BLOOD FLOW THROUGH A STENOSED ARTERY 391

TABLE 1Constant value

Parameters Numerical value Parameters Numerical value

ρ 1050 kg.m−3 Ec 8.7 × 10−6

η 3.2 × 10−3 kg.m−1.s−1 α 1.22 × 10−7 m2.s−1

Re 0.04 k 1.832 × 10−3 Joule.0K−1m−1s−1

T1 300 0K MnF [105:5×105]

δT 30 0K MnM 150Pr 24.95 Da 40-400-4000Cp 14.65J.(Kg.K)−1 χ0 0.06μ0 4π × 10−7 β 5.6×10−3 K−1

η0 35 × 10−3 kg.m−1.s−1 n 0.6

present the dimensionless equations and boundary conditionsand, finally, the simulation results and conclusion are outlinedin sections 4 and 5, respectively.

2. ARTERY MODEL AND MATHEMATICALFORMULATION

2.1 Stenosis Artery ModelIn this article, we consider a viscous, laminar, incompress-

ible, transient and two-dimensional blood flow between twoporous plates. Uniform flow enters the domain and, in the mid-dle of the vessel (Lf < x < Lf + Ls), one porous lump on thelower plate blocks the normal blood flow path. Both up anddown vessel walls are assumed at the constant temperature andthe length and height of the artery are shown by L and h, re-spectively. The flow is subjected to an external magnetic forceproducing an infinite current plate along the z axis. The non-symmetric stenosis blood artery is shown in Fig. 1 and presumedas below function [14]:

y

h=

⎧⎨⎩

0.7 − 3d

2hL4 [11(x − Lf )L3s − 47(x − Lf )2L2

s

+72(x − Lf )3Ls − 36(x − Lf )4]1 otherwise

Lf ≤ x ≤ Ls + Lf .

The porous assumption for arterial walls and the fatty de-posited plaque is conforming to actual behavior of our bloodartery. As depicted in Fig. 1, 15% total artery thickness is as-signed to each porous arterial wall and “d” is the maximumheight of stenosis, which depends on the disease progression,and its increment can create a critical condition for patients.

2.2 Heat Transfer and Fluid Flow EquationsUnder the action of an external magnetic field, two kinds

of forces affect the blood flow. One of them is magnetizationforce, which appears due to magnetic field gradient, and thesecond is the Lorentz force, which should be taken into accountbecause of the high electrical conductivity of blood flow andferronanoparticles’ solution. The governing equations on the

non-Newtonian flow under the action of the external magneticfield are described as below:

Continuity equation:

∇. �V ∗ = 0 (1)

Momentum equation [15]:

ρD �V ∗

Dt∗= −∇p∗ + η∇2 �V ∗ + J ∗ × B∗ + μ0M

∗∇H ∗ (2)

Energy equation [15]:

ρCp

DT ∗

Dt∗+ μ0T

∗ ∂M∗

∂T ∗DH ∗

Dt∗− J ∗.J ∗

σ= k∇2T ∗ + ηφ∗ (3)

where �V ∗ = (u∗, v∗)is the two-dimensional velocity field,D

Dt∗ = ∂∂t∗ + V ∗∇ is the material derivative, ∇ = ( ∂

∂x∗ ,∂

∂y∗ )

is the gradient operator, ∇2 = ∇.∇ = ( ∂2

∂x∗2 ,∂2

∂y∗2 ) is the Lapla-cian operator, ρ is the fluid density, P is the pressure, η is thedynamic viscosity, μ0 is the magnetic permeability of vacuum,M∗ is the magnetization, H∗ is the magnetic field intensity, B∗ isthe magnetic induction where B∗ = μ0(M∗ + H∗), σ is the elec-trical conductivity of the fluid, J∗ is the density of the electricalcurrent, T∗ is the temperature, k is the thermal conductivity, Cp

is the specific heat at constant pressure, and ϕ∗ is the dissipationfactor. For the two-dimensional problem, it is written as:

φ∗ = 2

(∂u∗

∂x∗

)2

+ 2

(∂v∗

∂y∗

)2

+(

∂v∗

∂x∗ + ∂u∗

∂x∗

)2

(4)

The μ0M∗∇H ∗ and J∗×B∗ terms in the momentum equation

demonstrate the magnetization and Lorentz forces, respectively,which appear because of Magneto Hydro Dynamic (MHD) andFerro Hydro Dynamic (FHD) effects. In this article, the powerlaw model is used for modeling non-Newtonian blood flowviscosity [16]:

η = η0 |γ |n (5)

η0 and n are constant parameters and their values are cited inTable 1.

392 H. ALIMOHAMADI AND M. IMANI

TABLE 2Time variation of Nusselt number in different magnetic fields

with MnM = 102

Time Step

MnF 2 4 6 8

0 0.756 0.089 0.011 −0.001105 0.692 0.059 −0.012 −0.0232×105 0.609 −0.006 −0.070 −0.0803×105 0.504 −0.090 −0.147 −0.1554×105 0.389 −0.179 −0.232 −0.2375×105 0.282 −0.269 −0.317 −0.322

2.3 The Brinkman Equations for the Porous Media FlowAs was mentioned before, the artery walls and the lump are

considered as porous media in this study. For simulating thecombination of porous and free flow, the Brinkman equationshave great accuracy. These equations for the porous regions areexpressed as:

Continuity equation:

∇. �V ∗p = 0 (6)

Momentum equation:

ρ

εp

δ �V ∗p

δt∗= −∇p∗ + η

εp

∇2 �V ∗p − η

kbr

�V ∗p +J ∗ ×B∗ +μ0M

∗∇H ∗

(7)where

−→V ∗

p = (u∗p, v∗

p),εp and kbr are velocity field, porosity,and permeability of the porous regions. The magnetization andLorentz force are exerted to the porous regions as well. After ob-taining the velocity field in the porous regions, the temperaturedistribution in these domains is calculated by Eq. (3).

2.4 Magnetic FormulaFor controlling blood flow in the artery, the magnetic field as

an external force is applied to the stenosis region. This magne-tization is created by a current plate along the z axis and belowthe stenosis region.

TABLE 3Effect of porosity value on maximum temperature with

different percent of artery stenosis and MnF = 105, MnM = 102

at time = 1

d/h

Darcy Number 0.1 0.15 0.2 0.25

40 10.43 10.40 10.38 10.36400 10.25 10.24 10.21 10.194000 10.23 10.21 10.19 10.17

FIG. 1. Modeled stenosis artery.

According to Maxwell’s law, the external magnetic field isexplained based on the equations below:

∇ × H ∗ = J ∗ = σ ( �V ∗ × B∗) (8)

∇.B∗ = ∇.(H ∗ + M∗) = 0 (9)

The magnetic field intensity components (H ∗x ,H ∗

y ) of a cur-rent plate are given by the expressions below:

H ∗y = −H0

2

[Ln

((x∗ − x∗

2 )2 + (y∗ − y∗0 )2

(x∗ − x∗1 )2 + (y∗ − y∗

0 )2

)](10)

H ∗x = H0

[tan−1

((x∗ − x∗

2 )

y∗ − y∗0

)− tan−1

((x∗ − x∗

1 )

y∗ − y∗0

)](11)

H ∗ =√

H ∗y

2 + Hx∗2 (12)

H0 is the magnetic field strength which depends on the ap-plied magnetics induction (B∗ = μ0 (H∗+M∗)) and x∗

1, x∗2, y∗

0

are the positions of the horizontal plate, as shown in Fig. 2. Themagnetic field and magnetization force components in the en-tire 2D space of the artery are illustrated in Fig. 3. As describedearlier, magnetization force consists of multiplying magneticfield and its gradient in different points of space. Thus, mag-netic force is severely high at the edges of the plate where themagnetic field starts and stops. This force affects blood flowand diverts it from direct motion. The x-component of this forceattempts to stop the fluid motion and return it backward, whilethe y-component drags ferroparticles upwards. These two forcescan produce circulating flow near the stenosis region, which isthe focus of this article.

FIG. 2. The plate with current passes through the z direction.


FIG. 3. (a) Normalized total magnetic field intensity; (b) normalized x and ymagnetization force components produced by a fixed current plate.

2.5 Magnetization EquationMagnetization property (M) is a feature of biofluid, which

determines the impression of magnetic field on the flow. Variousequations for magnetization property have been introduced; inthis article, the linear formula that relates the magnetization tomagnetic field strength and temperature is used [17]:

M∗ = χ∗mH ∗ (13)

χ∗m is the magnetic susceptibility and varies with temperature:

χ∗m = χ0

1 + β(T ∗ − T0)(14)

χ0, βandT0 are constant parameters that are obtained by exper-imental data [17].

3. TRANSFORMATION OF EQUATIONSIn order to solve the coupled systems of equations, the fol-

lowing non-dimensional variables are introduced:

t = α

h2t∗ x = x∗

hy = y∗

hu = u∗

ur

v = v∗

ur

p = p∗

ρu2r

H = H ∗

H0T = T ∗

δT

(15)where α = K

ρcpis the thermal diffusivity of fluid and ur = α

his

the characteristic velocity.By substituting these non-dimensional variables into Eqs.

(1)–(7) and (14), we have:In the free flow:Continuity:

∂u

∂x+ ∂v

∂y= 0 (16)

x-momentum:

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y= −∂p

∂x+ Pr × |γ |

(∂2u

∂x2+ ∂2u

∂y2

)

+MnF χmH∂H

∂x+ MnM

Re(χm + 1)2(vHxHy − uH 2

y ) (17)

y-momentum:

∂v

∂t+ u

∂v

∂x+ v

∂v

∂y= −∂p

∂y+ Pr × |γ |

(∂2v

∂x2+ ∂2v

∂y2

)

+ MnF χmH∂H

∂y+ MnM

Re(uHxHy − vH 2

y ) (18)

Also, the energy equation is transformed into:

∂T

∂t+ u

∂T

∂x+ v

∂T

∂y=

(∂2T

∂x2+ ∂2T

∂y2

)+ Ec × Pr × |γ |(

2

(∂u

∂x

)2

+ 2

(∂v

∂y

)2

+(

∂v

∂x+ ∂u

∂y

)2)


FIG. 4. Stream function and temperature contours with velocity and temperature profiles at time = 1 in different locations.

−MnF × Ec × ∂χ

∂TT × H

(u

∂H

∂x+ v

∂H

∂y

)+ MnM

Re

×Ec × (χm + 1)2 × (uHy − vHx)2 (19)

For the porous region, the continuity and energy equa-tions are similar to the free flow, but the momentum equationis:

x momentum:∂up

∂t= −∂p

∂x+ Pr × |γ |(

∂2up

∂x2+ ∂2up

∂y2

)+ Pr × |γ | × Da × up + MnF

×χm × H∂H

∂x+ MnM

Re(χm + 1)2(vpHxHy − upH 2

y )

(20)


FIG. 5. Shear stress in the stenosis region in different d/h at time = 1.

y momentum:

∂vp

∂t+ up

∂vp

∂x+ vp

∂vp

∂y= −∂p

∂y+ Pr × |γ |

(∂2vp

∂x2+ ∂2vp

∂y2

)+ Pr × |γ | × Da × vp + MnF

χmH∂H

∂y+ MnM

Re(uHxHy − vH 2

y ) (21)

where

χm = χ0

1 + (βδT )(T − T0δT

)(22)

The non-dimensional parameters which appear in the aboveequations are:

Re = hρur

η= ρα

η(Reynolds number) (23)

Ec = u2r

CpδT= α2

CpδT h2(Eckert number) (24)

Pr = η

ρα(Prandtl number) (25)

Da = h2

kbr

(Darcy number) (26)

MnF = μ0H20

ρu2r

=μ0H20 h2

ρα2(Magnetic number of FHD)(27)

MnM = μ20H

20 h2σ

η(Magnetic number of MHD) (28)

The two magnetic numbers appear because of magnetizationand Lorentz forces, and if they are set to zero, the problem re-duces to an ordinary flow between two plates with heat transfer.

3.1 Boundary ConditionsIn order to solve Eqs. (16)–(21) simultaneously, the non-

dimensional boundary conditions below are applied:

• For the upper wall (y = 1 and 0 < x < 10), the tem-perature is constant (T1) and the velocity is zero (noslip condition);

• For the lower plate (y = 0 and 0 < x < 10), thetemperature is constant (T1) and the velocity is zero(no slip condition);

• For outlet (x = 10 and 0 < y < 1 ), the temperaturegradient ( ∂T

∂x) and the pressure are zero;

FIG. 6. Streamlines and temperature contours with MnF = 105 and MnM = 102 in different time steps.


FIG. 7. The time variation of “u” velocity profile for different locations with MnF = 105 and MnM = 102.

• For inlet (x = 0 and 0 < y < 1 ), the temperature isconstant (T1) and the uniform inflow velocity changeswith time according to [18];

• For the free-porous interface, it is assumed that velocityis equal in these two regions (

−→Vp = �V .). This condi-

tion means that the velocity field remains continuousbetween free and porous media.

4. SIMULATION RESULTIn this article, the current plate is fixed at x1 = 3.3, x2 = 6.7,

and y0 = −0.1 in order to encompass the entire critical regionand produce the maximum magnetic forces at the starting andending edges of the lump. The fluid is assumed to have non-Newtonian behavior and its electrical conducting is noticeable.The time-dependent flow in the described geometry is solvedwith COMSOL 4.3 code using a direct UMFPACK linear sys-tem solver. The constant values that are used in this paper arearchived in Table 1 [9, 11].

4.1 Flow in Stenosis Artery without Magnetic ForceIn order to reach the fully developed flow before and after

the stenosis region, the ratio of stenosis length to total length

(Ls/L) is assumed to be small enough in this study. Thus, theentry length is too small and the flow soon reaches the fullydeveloped condition, but in the stenosis region, due to a narrow-ing of the cross-sectional area, the velocity magnitude increases.Fig. 4 shows the non-dimensional velocity and temperature con-tours with 20% artery stenosis. The dimensionless “u” velocitycomponent and temperature profiles in specific locations alongthe vessel are shown in this figure. As expected, the flow hasfully developed behavior in each cross-section, and the veloc-ity magnitude near the arterial walls (x = 2 graph in Fig. 4)and inside the formed plaque (x = 4, 5, 6 graph in Fig. 4) isextremely low because of the porosity feature of the walls andlump.

In fluid mechanics, the temperature directly relates to the ve-locity magnitude and the fluid temperature increase if it movesfaster. As seen in Fig. 4, the stenosis region has the high-est temperature because of stream line contraction and flowacceleration in this area. As the added temperature graphs show,the maximum increment of temperature in the stenosis region isequal to 12mK (x = 6 in Fig. 4).

The dimensionless shear stress distribution on the lump iselucidated in Fig. 5. By reaching to the first throat, the shearstress value jumps severely due to increasing velocity gradientabove the lump. The shear stress decreases until the middle of


FIG. 8. The time variation of temperature profile in different x-positions with MnF = 105 and MnM = 102.

the lump region; after that, it rises up through the second throat.As depicted in the figure, by increasing the stenosis percent, thelump must tolerate higher shear stress because, with increasingstenosis height, the available cross-sectional area for blood flowbecomes narrower and higher velocity gradients are exerted onthe lump. The shear stress trend and its increment, by increasingthe stenosis percent, coincides with calculated results in thepaper [14].

4.2 Time-Dependent Effect of Magnetic Field on BloodFlow

The stream function and temperature contours at the presentexternal magnetic field and at three different time steps (t = 0.1,0.3, 0.8) are demonstrated in Fig. 6. The primary effect of theexerting magnetic field is the formation of two main vortexesapproximately at the edges of the current plate (x = 3.5 andx = 6.6). The first vortex rotates counter-clockwise while thesecond one revolves in the opposite direction. At t = 0.1, theinlet velocity is negative, so the number of streamlines between

the two vortices are much fewer than at other times. With thepassage of time, the inlet velocity becomes positive and peaksat t = 0.3 and t = 08; the power of viscosity and inertia forcesbecomes remarkable and competes with magnetic forces. Asthe figure at t = 0.8 shows, these two forces partly overcomethe magnetic force, increasing the number of streamlines in thechannel and weakening the vortices’ size and effect.

On the other hand, the temperature contours depict that, withthe passage of the time, not only does the maximum temper-ature go up, but it also focuses on the first and end edges ofthe lump, where the vortices are created and the flow circu-lates. Atherosclerotic plaques are usually formed from fattydeposits and can be dissolved in high ambient temperature. Infact, the current plate acts as an external heat term that producestwo concentrated heat sources on the lump’s edges and can bebeneficial for eliminating these fatty plaques. Furthermore, theformed vortices can be useful for targeting drug delivery sincethe flow circulation creates stagnant flow above the specific areaand drug carriers could have more accessible time for activationand absorption.


FIG. 9. Temperature and v-velocity profile in the stenosis region with differentmagnetic field intensity and MnM = 102 at time = 1.

The time variation of dimensionless axial velocity at the ex-ternal magnetic field and for specific locations is depicted in Fig.7. The location x = 2 is far from the magnetic plate and the effectof the formed vortices in this region is less than in other areas. Asseen, with the passage of time the effect of the magnetic forcesdeclines because the inlet velocity increases and the viscous andinertia forces become prominent such that the velocity profileat t = 0.8 is similar to its parabolic profile without impact ofmagnetization forces (see Fig. 4). The trend of velocity profilesat x = 4 and x = 6 is reversed since the vortices in these regionsrotate in opposite directions. At x = 5, two opposite vortices neu-tralize each other’s impact and the “u” velocity profile in thislocation resembles the figure with its non-magnetization effect(see Fig. 4).

Increasing the temperature in respect to time and in differentlocations is shown in Fig. 8. As the trend of this figure elucidates,with the passage of time the fluid temperature increases suchthat at x = 6, where the second vortex is formed, this increment

FIG. 10. Shear stress distribution in the stenosis region with different valuesof magnetic field intensity and MnM = 102 at time = 1.

is maximized and the fluid temperature is 0.2 K higher than thewalls.

4.3 Effect of Magnetic Field Intensity on Fluid FlowThe magnetic field intensity has a direct impact on the pene-

tration velocity, temperature, vortex size, and shear stress. Theeffect of this parameter on the temperature and vertical velocityalong the stenosis region is demonstrated in Fig. 9. As shown inthis figure, by doubling and tripling the magnetic field intensityfrom 105, the maximum velocity (temperature) increases about2.25X and 3.4X (1.02X and 10.1X), respectively.

By strengthening the magnetic field, the bigger vortices influ-ence a wider area and higher velocity gradient applied strongershear stress to the lump. By applying the magnetic field withtwo (three) times stronger from 105, the maximum shear stresswill become 2X (2.8X) higher in the stenosis region.

Our simulation shows, at first time steps, that the blood tem-perature is lower than the arterial walls and heat transfers fromthe walls to the fluid, but with time, the blood becomes warmerand the heat transfer direction is reversed.

This occurrence is elucidated in Table 2, such that the Nusseltnumber varies from a positive value to a negative value throughone heart cycle. The heat transfer rate speeds up by strength-ening magnetic field intensity and bigger vortices formation,and as the table shows, at t = 8 with a five-times increase inmagnetic field intensity from 105 to 5×105, the value of theNusselt number goes up about 13.5X. Generally, the results ofthis section indicate that by applying a more powerful externalmagnetic field, the magnitude of temperature, penetration ve-locity, and shear stress on the stenosis region go up remarkablyand the high value of these parameters can be useful for cutting,corrosion, and dissolving the fatty lump from the vessel’s wall.


FIG. 11. Velocity arrows in the artery with different porosity values of walls and stenosis region with MnF = 105 and MnM = 102 at time = 1.

4.4 Effect of Porosity on Magneto-Therapy PerformanceThe porous permeability effect on the vortex power and pen-

etration velocity in the stenosis area is illustrated in Fig. 11and Fig. 12, respectively. Based on Fig. 11, a decreasing Darcynumber value causes bigger and stronger vortices to form in thestenosis region, reducing the Darcy number about two orders ofmagnitude and creating a six-times-higher vertical velocity andpenetration flow into the fatty lump (see Fig. 12). These resultsmake it clear that stenosis disease has better treatment undera magneto-therapy process in the preliminary level because thefatty lumps are more penetrable (lower Darcy number), and evenone weak external magnetic field can affect them.

Table 3 demonstrates the maximum blood temperature fordifferent porosity factors and stenosis heights. Increasing bothporosity and stenosis height parameters makes the magneto-therapy less efficient because, by increasing the value of theDarcy number, the porous regions become more rigid, hardly re-sisting against the flow passage, and the available cross-sectionalarea for blood flow declines. Thus, based on continuity law, thevelocity magnitude and, consequently, the inertia and viscosity

FIG. 12. V-velocity on the stenosis region with different Darcy numbers andMnF = 105 and MnM = 102 at time = 1.

forces become powerful enough to attenuate the external mag-netic field effect such that, by increasing the value of the Darcynumber about two orders of magnitude, the maximum bloodtemperature declines at least 2%. In addition, stenosis progres-sion has the same negative effect on the magneto-therapy effec-tiveness, as a lower temperature is produced on the fatty lump,and obtaining higher temperature on the stenosis region canhelp to dissolve the lump tissue faster and more easily. Hence,according to this table, using magnetic field and magneto ther-apy can be more effective for patients who have a low percentof stenosis and whose blood vessels or formed lumps on theirarterial walls are not yet rigid.

5. CONCLUSIONIn this article, a transient study of magnetic field effect on a

stenosed artery, considering the arterial walls and atheroscleroticplaque as a porous media, is investigated. The blood viscositytreats as a non-Newtonian fluid and an infinite current plateis fixed under the stenosis region to produce two vortices onthe lump’s edges. These vortices act as additional heat sources,making the lump warmer and creating a suitable condition fordissolving it. The applied external magnetic field increases shearstress value on the stenosis region, which can rub out and washaway the fatty deposited plaques. In addition, the simulationresults show that, for a magneto-therapy process, a lower porouslump is more appropriate, since bigger vortices can create highertemperature and penetration velocity along the target site.

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