Convective Flow of Blood in Square and Circular Cavities
P. Senel and M. Tezer-Sezgin
Abstract
In this study, the fully developed, steady, laminar flow of blood is studied in a long pipe with square and circular cross-sections subjected to a magnetic field generated by an electric wire. Temperature difference between the walls causes heat transfer within the fluid by the displace- ment of the magnetizable fluid particles in the cavity. The governing equations are the coupled Navier-Stokes and energy equations including magnetization terms. The axial velocity is also computed with the ob- tained plane velocity. The Dual Reciprocity Boundary Element Method (DRBEM) is used by taking all the terms other than Laplacian as in- homogeneity which transforms the partial differential equations into the boundary integral equations. Numerical results are given for increasing values of Magnetic (M n) and Rayleigh (Ra) numbers. The numeri- cal results reveal that an increase inM naccelerates the plane velocity in the cavity but decelerates the axial velocity around the magnetic source. Pressure increases through the channel starting from the mag- netic source. Isotherms show the cooling of the channel with highM n andRaonly leaving a thin hot layer near the top heated wall. AsRa increases viscous effect is reduced leaving its place to convection in the channel. The use of DRBEM has considerably small computational ex- pense compared to domain type methods.
Key Words: Biomagnetic fluid, heat transfer, magnetic field, DRBEM Received: November, 2016.
Revised: March, 2017.
Accepted: June, 2017.
209
1 Introduction
Biomagnetic fluid dynamics (BFD) is the area, investigating the biological fluid in the presence of magnetic fields. The most characteristic biomagnetic fluid is blood. Blood is a magnetizable fluid due to the cell membrane and the hemoglobin molecule, iron-based protein, carried by the red blood cells [1]. An externally applied magnetic field changes the flow and the heat transfer char- acteristics of blood significantly, resulting in some applications in biomedicine as cell separation, drug targeting, magnet therapies and controlling bleeding during surgeries. Tzirtzilakis et al. [2] investigated the fully developed blood flow in a square cavity. They solved the governing equations with a pressure- linked pseudotransient method. Biomagnetic fluid flow model is extended by taking into account both magnetization and Lorentz forces by Tzirtzilakis [3].
Kenjeres [4] carried the blood flow analysis to realistic arteries. He reported that an imposed non-uniform magnetic field could have created significant changes in the secondary flow patterns which made it possible to optimize the targeted drug delivery. Khashan et al. [5] presented a numerical simulation for magnetically mediated separation of labeled biospecies from a native fluid flowing through a two-dimensional channel using finite volume method. The influence of a dipole like field and a magnetic field generated by a thin electric wire on the flow of biomagnetic fluid flow in a circular duct with stenosis is studied by Tzirakis et al. [6]. They have used a method based on pressure correction scheme combining discontinuous and continuous Galerkin approxi- mations.
The heat transfer through the blood flow in the presence of magnetic field can be used in hypothermic sessions, thermal simulation and thermal ther- apy applications [7]. The stretching disk flow of a heated biomagnetic fluid is investigated as a special case by Tzirtzilakis and Kafoussias [8]. They trans- formed the nonlinear system of equations to ordinary differential equations by introducing appropriate non-dimensional variables. The blood flow between two parallel plates is numerically simulated by Loukopoulos and Tzirtzilakis [9] using a finite difference scheme. They reported that the temperature and the rate of heat transfer are increasing the area where the magnetic source is placed. A finite element study of flow and temperature disturbance between two parallel plates subjected to multiple point magnetic sources is presented by Morega and Four [10]. Alimohamadi et al. [11] investigated the influence of numerous numbers of magnetic dipoles on the heat transfer in a rectangular duct. They compared MHD (Magnetohydrodynamics) and FHD (Ferrohydro- dynamics) analysis of the flow which are obtained by a commercial software.
The gravitational acceleration effect on the flow and heat transfer of blood flow between parallel plates is studied by Idris et al. [12] using a finite difference
method.
The Dual Reciprocity Boundary Element Method (DRBEM) is essentially a generalized way of constructing particular solutions to the inhomogeneity of the differential equation that can be used to solve non-linear and time depen- dent problems [13]. Senel and Tezer-Sezgin [14] presented DRBEM solutions of Stokes and Navier-Stokes equations subjected to spatially varying point source magnetic field in circular and rectangular lid-driven cavities. Biomagnetic fluid flow between parallel plates imposed to a magnetic source was investigated by Tezer-Sezgin et al. [15]. They used both finite element and dual reciprocity boundary element method for solving momentum and energy equations in terms of stream function and vorticity. Recently, Senel and Tezer-Sezgin [16]
studied the forced convection biomagnetic fluid flow in a square cavity.
In this paper, the fully developed, steady, flow of blood in the cross-section of a long pipe (cavity) under the influence of magnetization and buoyancy forces are investigated. The flow configuration, pressure and the temperature dis- turbance are visualized for various values of magnetic and Rayleigh numbers in square and circular cavities. The axial velocity profile is also presented.
Viscous dissipation effect on the flow and the heat transfer is studied. To the best of authors’ knowledge this is the first application of DRBEM to the math- ematical model for the blood flow in square and circular cavities in primitive variables. DRBEM has the advantage of discretizing only the boundary of the cavity and results in considerably low computational expense.
2 Mathematical Formulation
Consider a fully developed, steady flow of biofluid in a long impermeable pipe with square or circular cross-section. The fluid is flowing through the axis of the pipe due to an imposed constant pressure gradient Pz in the same direction. The flow and the heat transfer in the pipe are also affected by mag- netization and buoyancy forces. Magnetization force is generated by a long wire carrying electric current placed ¯c units below and parallel to the axis of the pipe. Buoyancy force occurs due to the temperature differences between the walls of the pipe and the gravitational acceleration. Being a hydrody- namically and thermally fully developed flow the problem is considered in the two-dimensional cross-section of the pipe (cavity) and in this case the electric wire serves as a point magnetic source. There is a longitudinal heat transport, but isotherms have the same profile on each cross-section of the pipe [17].
Magnetization, buoyancy forces and the constant pressure gradient in the ax- ial direction develops a forced convection flow in the cavity. The axial velocity can be separately obtained with the computed plane velocity components.
The governing equations in dimensional form are given by continuity, Navier-
Stokes and the energy equations in terms of pressure ¯P, velocity (¯u,v,¯ w) and¯ the temperature ¯T of the fluid which are two-dimensional
∂¯u
∂¯x+∂¯v
∂¯y = 0 (1)
¯ u∂¯u
∂¯x+ ¯v∂¯u
∂y¯ =−1 ρ
∂P¯
∂x¯ +ν ∂2u¯
∂¯x2 +∂2u¯
∂y¯2
+µ0M¯ ρ
∂H¯
∂x¯ (2)
¯ u∂¯v
∂¯x+ ¯v∂v¯
∂y¯ =−1 ρ
∂P¯
∂y¯ +ν ∂2v¯
∂¯x2 +∂2v¯
∂y¯2
+µ0M¯ ρ
∂H¯
∂y¯ +gβ( ¯T−Tcold) (3)
¯ u∂w¯
∂¯x + ¯v∂w¯
∂¯y =−1 ρ
∂P¯
∂z¯ +ν ∂2w¯
∂¯x2 +∂2w¯
∂y¯2
(4)
ρcp
¯ u∂T¯
∂¯x + ¯v∂T¯
∂y¯
= −µ0T¯∂M¯
∂T¯
¯ u∂H¯
∂¯x + ¯v∂H¯
∂y¯
+k ∂2T¯
∂x¯2 +∂2T¯
∂¯y2
+µ
"
2 ∂u¯
∂x¯ 2
+ 2 ∂¯v
∂y¯ 2
+ ∂¯v
∂x¯ +∂u¯
∂¯y 2#
. (5) Here, µ,ν,ρ, cp, k andβ are the dynamic viscosity, the kinematic viscosity, the density, the specific heat, the thermal conductivity and the thermal ex- pansion coefficient of the fluid,µ0is the magnetic permeability of vacuum and gis the magnitude of the gravitational acceleration.
H¯ is the magnetic field strength and ¯M =χH¯(Tc−T¯) is the magnetization whereχ is the magnetic susceptibility of the blood and Tc is the Curie tem- perature of iron.
The terms µ0M¯ ρ
∂H¯
∂x¯ and µ0M¯ ρ
∂H¯
∂¯y are the components of the magnetization force. gβ( ¯T −Tcold) is the buoyancy force, where Tcold is the temperature of the cold wall. The second term in the energy equation is the heating due to magnetization and the last term is the viscous dissipation which is a heat source caused by the friction between the fluid particles.
In fully developed flows the pressure is split as in [18]
P¯(¯x,y,¯ z) = ¯¯ p(¯x,y) + ¯¯ P1(¯z) (6)
∂P¯
∂z¯ =∂P¯1
∂z¯ = ¯Pz=constant . (7) The components of the magnetic fieldH~¯ = ( ¯Hx,H¯y) generated by an infinitely long electric wire is defined as in [19]
H¯x= −I 2π
¯ y−¯b
(¯x−¯a)2+ (¯y−¯b)2, H¯y = I 2π
¯ x−¯a
(¯x−¯a)2+ (¯y−¯b)2 (8)
whereI is the electric current, (¯a,¯b) is the place of the point magnetic source.
For square cavity (¯a,¯b) = (h/2,−¯c) and for circular cavity (¯a,¯b) = (0,−h−
¯
c). Here, ¯c denotes the distance between the source and the cavity. Then, magnetic field strength is
H¯ =
qH¯x2+ ¯Hy2= I 2π
1
p(¯x−¯a)2+ (¯y−¯b)2 (9) For simplicity the non-dimensional variables are given as
x=x¯
h, y= y¯
h, z= ¯z
h, u=uh¯
ν , v=vh¯
ν , w= wh¯
ν , P = P h¯ 2
ρν2, H =
H¯ H0
, T =
T¯−Tcold
Thot−Tcold
(10) wherehis the width of the square cavity and the radius of the circular cavity, H0is the magnetic field strength at (h/2,0) for square and (0,−h) for circular cavity. Thot is the temperature of the hot wall. The reference values for the physical properties areρ = 1050 kgm−3, µ = 3.2×10−3 kgm−1s−1, Thot = 316.15◦K,Tcold= 276.65◦K as in [9] andh= 1.1×10−3 m.
The magnetic field strengthH in non-dimensional form for square cavity H(x, y) = |b|
p(x−a)2+ (y−b)2 (11) and for circular cavity
H(x, y) = |c|
p(x−a)2+ (y−b)2 (12) where (a, b) = (¯a/h,¯b/h) andc= ¯c/h.
Thus, the non-dimensional equations are
∂u
∂x+∂v
∂y = 0 (13)
∂2u
∂x2 +∂2u
∂y2 = ∂p
∂x+u∂u
∂x +v∂u
∂y −M n(Tc−T)H∂H
∂x (14)
∂2v
∂x2 +∂2v
∂y2 = ∂p
∂y +u∂v
∂x+v∂v
∂y −M n(Tc−T)H∂H
∂y −Ra
P rT (15)
∂2w
∂x2 +∂2w
∂y2 =Pz+u∂w
∂x +v∂w
∂y (16)
∂2T
∂x2 +∂2T
∂y2 = P r(u∂T
∂x +v∂T
∂y)−M nEcP r(+T)H(u∂H
∂x +v∂H
∂y)
−EcP r(2 ∂u
∂x 2
+ 2 ∂v
∂y 2
+ ∂v
∂x+∂u
∂y 2
). (17) In order to simulate the flow in terms of velocity and the pressure of the fluid pressure equation is derived by differentiating thex−and
y−components of Navier-Stokes equations and adding them with the use of continuity equation
∂2p
∂x2 +∂2p
∂y2 = Ra P r
∂T
∂y − ∂u
∂x 2
− ∂v
∂y 2
−2∂v
∂x
∂u
∂y
−M n(∂T
∂xH∂H
∂x +∂T
∂yH∂H
∂y) (18)
+M n(Tc−T)(
∂H
∂x 2
+ ∂H
∂y 2
+H∇2H). The non-dimensional parameters entering the problem are
M n= µ0χH02(Thot−Tcold)h2
ν2ρ (Magnetic number) (19)
P r=ρcpν
k (Prandtl number) (20)
Ra=gρcpβ(Thot−Tcold)h3
νk (Rayleigh number) (21)
= Tcold
Thot−Tcold (Temperature number) (22) Ec= ν2
h2cp(Thot−Tcold) (Eckert number). (23) Magnetic number expresses the ratio of the magnetic forces and the inertia forces. Prandtl number is the ratio of the momentum and thermal diffusivi- ties. Rayleigh number is the product of the Grashof number and the Prandtl number where Grashof number denotes the ratio of the buoyancy forces to viscous forces. Eckert number defines the kinetic energy of the flow relative to the boundary layer enthalpy difference.
2D stream function is introduced satisfying the continuity equation as u =
∂Ψ
∂y, v = −∂Ψ∂x. To visualize the flow pattern in the cavity, the stream function equation is obtained
∂2Ψ
∂x2 +∂2Ψ
∂y2 =∂u
∂y −∂v
∂x . (24)
The walls of the cavities are motionless and the velocity has no-slip boundary conditions. The square cavity is heated from the top and cooled from the bottom walls, horizontal walls are adiabatic. The circular cavity has no adia- batic walls. The pressure boundary conditions are approximated throughx−
andy−components of the momentum equations using a forward difference for the pressure gradients and the DRBEM coordinate matrixFfor all the other terms.
The problem geometry and the boundary conditions are presented in Figure 1 for both square and circular cavities.
y
0 1 x
1
u= 0, v= 0 w= 0, Ψ = 0, T = 1
u= 0, v= 0 w= 0, Ψ = 0
∂T
∂x = 0 ∂T
∂x = 0 w= 0,Ψ = 0
u= 0, v= 0
u= 0, v= 0 w= 0, Ψ = 0, T = 0 point magnetic source
~g
-1 1 x
0
point magnetic source
uθ= 0, ur= 0 w= 0, Ψ = 0, T= 1
uθ= 0, ur= 0 w= 0, Ψ = 0, T= 0
~ g
Figure 1: Problem geometry and boundary conditions
Eqs. (14)-(18) and (24) together with the mentioned boundary conditions define the two-dimensional flow and heat transfer of biomagnetic fluid (blood) in the cross-section of a long pipe, and the axial velocity.
3 Application of DRBEM
The dual reciprocity boundary element method is used to transform Eqs. (14)- (18) and (24) into the boundary integral equations using the fundamental solution of Laplace equation (u∗= (1/2π)ln(1/r)) [13]. The main advantage of the method is the reduction of problem dimension. It enables one to discretize only the boundary of the problem domain. Taking all the terms other than Laplacian as inhomogeneity, weighting the equations with the fundamental solutionu∗ and using the Green’s first identity two times an integral equation
for the unknown is obtained ciRi+
Z
Γ
Rq∗dΓ− Z
Γ
u∗∂R
∂ndΓ =− Z
Ω
bRu∗dΩ (25) where R denotes u, v, w, T, p or Ψ and bR is the right-hand side of each corresponding equation for R. Γ = ∂Ω, q∗ =∂u∗/∂n and n is the outward unit normal to the boundary. ci = 1/2 for the boundary and ci = 1 for the interior nodes.
To eliminate the domain integral on the right-hand side of Eq. (25) bR is approximated by a linear radial basis functionfj(r) = 1+rjwhich is connected to the particular solution ˆujas∇2uˆj=fj, [13] andrjis the distance between the source and field points. Then,
bR=
N+L
X
j=1
(αR)j∇2uˆj (26)
where (αR)j’s are the undetermined coefficients for the approximation of the right-hand side bR. N and L are the number of boundary and the interior nodes, respectively.
Green’s first identity is applied to the right-hand side of Eq. (25) and the boundary is discretized withN constant elements to achieve a boundary inte- gral equation for the unknownR.
ciRi+
N
X
k=1
Z
Γk
Rq∗dΓ−
N
X
k=1
Z
Γk
u∗∂R
∂ndΓ =
N+L
X
j=1
(αR)j(ciuˆij+
N
X
k=1
Z
Γk
ˆ ujq∗dΓ
−
N
X
k=1
Z
Γk
u∗∂uˆj
∂ndΓ). (27) Eq. (26) is used to determine the unknown coefficients (αR)j’s in terms of the DRBEM coordinate matrixFwhich is constructed by taking radial basis functionsfj’s as columns
αR=F−1bR . (28)
Writing Eq. (27) for all boundary andL interior nodes and using Eq. (28) DRBEM discretized matrix-vector equations are obtained
Hu−G∂u
∂n = (H ˆU−G ˆQ)F−1{∂p
∂x+u∂u
∂x+v∂u
∂y−M n(Tc−T)H∂H
∂x} (29) Hv−G∂v
∂n= (H ˆU−G ˆQ)F−1{∂p
∂y+u∂v
∂x+v∂v
∂y−M n(Tc−T)H∂H
∂y −Ra P rT}
(30)
Hw−G∂w
∂n = (H ˆU−G ˆQ)F−1{Pz+u∂w
∂x +v∂w
∂y} (31) HT −G∂T
∂n = (H ˆU−G ˆQ)F−1{P r(u∂T
∂x +v∂T
∂y)
−M nEcP r(+T)H(u∂H
∂x +v∂H
∂y) (32)
−EcP r(2(∂u
∂x)2+ 2(∂v
∂y)2+ (∂v
∂x +∂u
∂y)2)}
Hp−G∂p
∂n = (H ˆU−G ˆQ)F−1{Ra P r
∂T
∂y −(∂u
∂x)2−(∂v
∂y)2−2∂v
∂x
∂u
∂y
−M n(∂T
∂xH∂H
∂x +∂T
∂yH∂H
∂y ) (33)
+M n(Tc−T)((∂H
∂x)2+ (∂H
∂y )2+H∇2H)}
HΨ−G∂Ψ
∂n = (H ˆU−G ˆQ)F−1{∂u
∂y −∂v
∂x} (34)
where
Hij=ciδij+ Z
Γj
q∗dΓj, Hii =ci, (35) Gij=
Z
Γj
u∗dΓj, Gii = l 2π
ln
2 l
+ 1
(36) lis the length of the element.
The matrices ˆU, ˆQare constructed by taking each vector ˆujand ˆqj=∂ˆuj/∂n as columns, respectively.
The spatial derivatives of the unknowns on the right-hand sides of Eqs. (29)- (34) are approximated by using DRBEM coordinate matrixFas
∂R
∂η =∂F
∂ηF−1R, ∂2R
∂ξ∂η = ∂F
∂ξF−1∂F
∂ηF−1R (37) withη,ξbeingxory.
The discretized system of Eqs. (29)-(34) are solved iteratively by taking ini- tially
u(0)=v(0) = 0 in Ω∪Γ (38)
and
T(0)=
(1 on the hot boundary
0 otherwise (39)
In the stagnant situation of the fluid the transverse pressure gradient forces are balanced by the effect of the magnetic and buoyancy forces hence, initial pressure gradients are
∂p
∂x
(0)
=M n(T−Tc)H(∂H
∂x) + 10−12
∂p
∂y
(0)
=M n(T−Tc)H(∂H
∂y) +Ra
P rT+ 10−12 . (40) The system of Eqs. (29)-(34) with the boundary conditions specified on Figure 1 are solved iteratively starting from Eq. (29) and relaxing temperature and pressure values with relaxation parameters in (0,1) to accelerate the conver- gence. The iteration continues until a preassigned tolerance is reached. The stopping criteria for the iteration is
||R(n+1)−R(n)||∞
||R(n)||∞ <10−3 (41) whereRdenotesu,v,w,p,T or Ψ andnis the iteration number.
4 Numerical Results and Discussions
Convection flow of blood in square and circular cavities under point magnetic source are investigated by using DRBEM. The physical properties of blood presented in [9] are considered to set the dimensionless parameters. P r= 20, Ec= 1.25×10−8,= 7 are taken. The pressure gradient given to the fluid in the axial direction isPz =−8000 [2] and c= 0.05. The proposed numerical scheme and results are validated with the natural convection flow of air in a square cavity without magnetization force by takingP r= 0.7 andRa= 103. The obtained results are in good agreement with the ones in [20]. The flow is studied in a square cavity with adiabatic vertical walls. Both in square and circular cavities the top wall is hot and the bottom wall is cold.
4.1 Convection of blood in a square cavity
The effect of buoyancy force only on the flow (M n= 0) is seen in Figures 2- and 3. The buoyancy force divides the flow into two vortices with centers on y= 0.5 line. Pressure is highly concentrated at the top of the cavity showing a drop at the center. Horizontal velocity consists of four loops emanating from the corners and vertical velocity spreads through the cavity. The buoyancy force shifts the isotherms through the hot wall. Thus, an increase in Rayleigh number accelerates the planar velocities, increases the pressure in magnitude and squeezes the isotherms through the hot wall.
Ψ
0.013932
−0.013932 0.0065992
−0.0065992
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
−2.7922
−1.5779 1.5792
−2.3065
−2.3065
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.031999 0.096532 0.29013
0.58053
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−0.030352 0.030352
0.033923
−0.033923 0.016069
−0.016069
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
0.27122
0.20341
0.081366
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 2: Viscous dissipation is neglected. N = 160,Ra= 103,M n= 0
Ψ
0.17278
0.081845
−0.17278
−0.081845
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
−36.6484
−14.2838
−6.8289
−6.8289
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.04 0.34
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−0.56257
0.56257
0.3257
−0.3257
−0.20726 0.20726
−0.14805 0.14805
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
2.8544
1.7983 3.0053
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 3: Viscous dissipation is neglected. N = 160,Ra= 105,M n= 0 Figures 4 and 5 display the effect of the magnetic source only on the flow
profiles and the temperature of the fluid forM n = 5,200 with Ra = 0. An increase in Magnetic number causes an increase in the magnitude of the pla- nar velocities and the pressure. The axial velocity shows a flattening tendency around the point magnetic source and the pressure around the magnetic source extends through the cavity. Flow on the transverse plane is divided into two vortices rotating in opposite directions. A further increase inM n moves the center of vortices through the magnetic source. The main effect of the point magnetic source below the cold wall is the cooling of the channel starting from the bottom wall. For values M n≥80 the fluid in the channel is completely cooled except a thin layer near the upper wall and secondary flows show up through bottom and top corners. v−velocity shows the pushing effect of the magnetic point source and a thin boundary layer occurs just above and around the source.
The main effect of the magnetic source (and its increasing intensity) is to ac- celerate the fluid flow and increase the pressure on the fluid, and thus to cool down the channel through the upper wall. This is also the idea of cooling the head capsules of nuclear reactors.
Ψ
0.014155 0.006705
−0.014155
−0.006705
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
6.4458 1.6472 0.047664
0.047664
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.032258 0.096774 0.32258 0.6129
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−0.11653
−0.042934 0.042934
0.11653
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
0.32716 0.24537 0.11451
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 4: Viscous dissipation is neglected. N = 160,Ra= 0,M n= 5
Ψ
−0.020441 0.020441
0.47015
−0.47015 0.26573
−0.26573
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
193.8241 130.8778
319.7166
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T 0.017945
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−5.8685
−4.0153
5.8685 4.0153
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
9.2369
7.7228
−0.35233
−0.35233
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 5: Viscous dissipation is neglected. N= 200, Ra= 0, M n= 200
Ψ
0.025671 0.014862
−0.025671
−0.014862
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
7.5924 1.5163 1.5163
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.028251 0.2226 0.57891
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−0.21558
−0.079424 0.079424
0.21558
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
0.5991 0.44932
0.23964
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 6: Viscous dissipation is neglected. N = 160,M n= 10,Ra= 103
Both the buoyancy force and magnetic source effects are seen (M n = 10 andRa= 103, 105) on Figures 6, 7. For smallM nthe buoyancy effect starts to dominate the flow asRaincreases. WhenRa≥104the thermal convection in the flow reduces the pushing effect of the magnetic source and viscous effect is reduced. AsRaincreases the behavior of axial velocity stays the same since theu−andv−velocities are not altered significantly.
Ψ
0.18457 0.067999
−0.18457
−0.087427
−0.048571
−0.029142 0.029142
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
4.1681
−34.6093
−44.3037
−44.3037
−24.9149
4.1681
4.1681
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.02 0.37
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−0.073548 0.073548
−0.12258 0.12258
−0.71097 0.71097
0.12258
−0.12258
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
3.293
2.6315
1.3086
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 7: Viscous dissipation is neglected. N = 160,M n= 10,Ra= 105 WhenM n= 80 is reached (Figures 8-9) the magnetization force dominates the buoyancy force up to Ra = 104. The influence of buoyancy force is ob- served in streamlines and pressure as the center of vortices move upwards and pressure starts to concentrate close to the top heated wall. But with an in- crease in magnetic number the cooling of the channel is much faster compared to an increase in Rayleigh number.
Ψ
0.2369
−0.2369 0.11221
−0.11221
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
104.0925 53.5161 28.2279
28.2279
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.02 0.3
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−2.4808 2.4808
−1.6974 1.6974
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
5.0695
4.0357
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 8: Viscous dissipation is neglected. N = 160,M n= 80,Ra= 104
Ψ
0.2357 0.086837
−0.012405 0.012405
−0.2357
−0.086837
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
66.4569 41.1696
41.1696
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.02 0.16 0.44
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−2.4229
−1.6578
2.4229 1.6578
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
4.8342 3.8557 1.4094
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 9: Viscous dissipation is neglected. N = 160,M n= 80,Ra= 105
Ψ
0.18451 −0.18451
0.067978 −0.067978
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
−44.3009
−44.3009
4.1556
−24.9183
4.1556
4.1556
−34.6096
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.02 0.3
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−0.71033 0.71033
0.12247
−0.073482 0.073482
−0.12247
0.073482
−0.073482
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
3.2918
2.4653 1.3082 v
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 10 100 200 300 400 500 600
x w
y
Figure 10: Viscous dissipation is added. N = 160,M n= 10,Ra= 105
Ψ
0.24018 0.11377
−0.24018
−0.11377 0.012641
−0.012641
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
89.3212 38.5937 38.5937 13.23
13.23
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T
0.02 0.23
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
u
−2.4242
−1.4035
2.4242 1.6586
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
4.949 2.6956 0.44222
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.2 0.4 0.6 0.8 1
0 0.5 1 0 100 200 300 400 500 600
x w
y
Figure 11: Viscous dissipation is added. N = 160,M n= 80,Ra= 105 Figures 10 and 11 show the behavior and heat exchange of a biomagnetic
dissipative viscous fluid. It is observed that the viscous dissipation does not influence the flow and the heat transfer characteristics for small Rayleigh num- bers. A slight difference is observed only when Ra= 105 comparing Figures 7,10 and 9,11.
4.2 Convection of blood in a circular cavity
In this problem, the blood flow and its temperature disturbance is studied in a circular cavity heated on the upper half wall which is a more realistic case.
The influence of buoyancy force is the acceleration of planar velocities as in the case of square cavity whenM n= 0. The pressure is concentrated around the hot wall and the discontinuity points since there is no adiabatic wall. The cavity is nearly cooled down forRa≈104(Figure 12).
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Ψ
−0.20654
0.20654
0.11957
−0.097834
0.054352
−0.054352
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
p
0.53338 24.565
0.53338
−11.4824
−11.4824
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
T
0.047619 0.19048 0.61905
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
u
1.0408
−1.0408
0.60258
−0.60258 0.38346
0.2739
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
v
0.20449
0.5453
0.74979 0.34081
1.3633
1.2951
0.95428
0.95428
−1 0
1
−1
−0.5 0 0.5 1 0 500 1000 1500 2000
x w
y
Figure 12: Viscous dissipation is neglected. N= 120, M n= 0,Ra= 104 Figures 13-14 display the velocity, pressure and the temperature variations for increasing magnetic effect M n = 10 and 30 when Ra = 0. The flow behavior is similar to the square cavity case. Increasing magnetic field intensity accelerates the flow in the cavity and decelerates it in the axial direction. The heat transfers directly between the hot and cold walls. Pressure increases and center of vortices move through the source point.
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Ψ
0.51832
−0.51832 0.24552
−0.24552
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
p
86.2099 54.0118
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
T 0.047619
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
u
−2.0542
2.0542
−0.97306
0.97306
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
v
5.9923
4.7939
3.5954
−1 0
1
−1
−0.5 0 0.5 1 0 500 1000 1500 2000
x w
y
Figure 13: Viscous dissipation is neglected. N = 120,M n= 10,Ra= 0
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Ψ
0.6735
−0.6735 0.31903
−0.31903
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
p
173.3453 145.5529 117.7605
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
T
0.047619 0.095238
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
u
−5.83
−2.1479
5.83 2.1479
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
v
8.8559
7.0847
5.7563
−1 0
1
−1
−0.5 0 0.5 1 0 500 1000 1500 2000
x w
y
Figure 14: Viscous dissipation is neglected. N = 160,M n= 30,Ra= 0
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
Ψ
−0.01784 0.01784
0.33897 −0.33897
0.16056 −0.16056
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
p
66.1534 45.8099
45.8099 55.9817
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
T
0.047619 0.28571
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
u
−2.1167
2.1167
−1.2254
1.2254
0.1114 −0.1114
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
v
3.6138 2.5296 0.90345
−1 0
1
−1
−0.5 0 0.5 1 0 500 1000 1500 2000
x w
y
Figure 15: Viscous dissipation is neglected. N = 120,M n= 10,Ra= 104
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8
1 Ψ
0.70348
−0.70348 0.33323
−0.33323
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
p
535.8615 505.632
505.632
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
T
0.095238 0.095238
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
u
−5.4911
5.4911
−2.6011
2.6011
−1 −0.5 0 0.5 1
−1
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8 1
v
9.7852
7.8281
5.8711
−1 0
1
−1
−0.5 0 0.5 1 0 500 1000 1500 2000
x w
y
Figure 16: Viscous dissipation is neglected. N = 120,M n= 50,Ra= 104 When both magnetic source and buoyancy force are present the buoyancy
