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Numerical solution of non-steady magnetohydrodynamic flow of blood through a porous channel
Numerical solution of non-steady magnetohydrodynamic flow of blood through a porous channel P. Sambasiva Department
Rao and J. Anand Rao
of Mathematics,
Osmania
University,
Received November 1986; revised and accepted July
Hyderabad
500 007, India
1987
ABSTRACT Masnetohydrodynamic (MHD) principles may be used to decelerate theflow of arterial blood and hence be of potential value in the treatment of cardiovascular disorders associated with an accelerated circulation. We examine the non-steady flow of blood in a porous parallel plate channel under the influence of a transverse magnetic field and different pressure gradients.
Keywords:
Blood flow, numerical methods
INTRODUCTION
2
Fung and Sobin’ studied the flow of blood between two endothelial layers using the ‘sheet model’; the morphometric basis for sheet flow was presented by Sobin’ and others. Tang and Fung3 studied the mass transfer problem in the alveolar sheet and Peskin4 applied numerical techniques to examine the flow of the blood in the heart. McCracken’ used Vorten-grid methods to investigate blood flow through the heart valves. In the lungs, blood can be visualized as flowing between the opposing layers of capillary endothelium, held apart by endothelium-covered ‘posts’ of special tissue. The capillary endothelium is, in turn, covered by a thin layer (interstitial space) lining the alveoli. In the lungs, the blood space can be considered as a two-dimensional channel and the interstitial tissue space as a porous medium. As the endothelial layer between the two regions is permeable to water and some solutes, it can be considered as a permeable membrane of negligible thickness.
FORMULATION PROBLEM
AND SOLUTION
OF THE
Although the blood circulation system consists of three-dimensional elastic tubes of varying crosssection, for mathematical convenience we consider the magnetic effect on the non-steady flow of blood through a two-dimensional, nonconducting, parallel plate channel. The geometry of the system is explained in Figure 1. Assume that the liquid is being injected into the flow region through the channel at z = 0 and is being sucked away through the channel at z = 6. At any time t, let u and w be the components of velocity at a point (x, z) of the flow region in the direction of x and t respectively. The non-steady flow of blood in the presence of a static magnetic field is governed by the following equations of motion and the continuity equations. Correspondence
and reprint
requests
0 1987 Butterworth & Co (Publishers) 0141-5425/88/030293-03 803.00
to Dr P. Samba&a
Rao
1 TTTiTTTiTlTTT BO
80
0000000000000
2=L5
.?=O
oooooolooooooo
TTTiTTTTlTl~T BO
80
Figure 1 Non-steady magnetohydrodynamic flow of blood in a parallel plate channel with uniform injection through the wall at z = 0 and suction through the wall at z = b
au
au
au
at+u~+w~=--gy+-
- i ap
CL* 4
i a2 (1)
aw aw -++-++-=-+--
aw
ax
a2
at
- lap
p
eat
e
(
+- azw a.2
1
and
au -+-= ax
aw aZ
azw
0
a2 (2)
(3)
Where P is the pressure at the point (x, z) in the flow region and B, is the applied magnetic field.
Ltd J, Biomed.
Eng.
1988, Vol. 10, May
293
Technical Records
Let the initial and boundary conditions be given by t
= 0
and w=OforO
(4)
t > 0, u = 0 and w = w0 = positive t=Oandz=b
constant
(5)
In view of conditions (4) and (5), the velocity distribution is independent of x, i.e.
-=au
field
,*=o a
0
ax
for
Substituting
0.25 -
=
g
0 in equation
(3)
condition
(5). We can write equations
au
au =--++--1
ap
p azu
e
ax
4
at + w”aZ
and
using
(1) and (‘2) as 4
a2
B&(7)
and
ap az
-=
0
(8)
Normalizing the above equations and applying finite difference formulae, taking the mesh size h in the zdirection and k in the time direction we get -rX(lIR
(h -
-
$)]
h).u(i+
1) + +
l,j+
*u(i,j + 1) -
[1 1
-
-I 0.8
0.2
0.4
Velocity
distribution
in the z direction
0.6
1.0
.?
0.6
A Figure
2
rXu(i - j + 1) I
=
r(1
-
X)(1/R-h)*u(i+l,j)
+ [l
_ &q2_K R
t + ~(1 -
X)(h + +)]
+ ~(1 -
X)*u(i-
k
where r = F,
1,j)
-
I
u(i, j) 1
(9)
R + KF(lg]
X (weighting
factor)
= ? and % dX - F(t). In equation (9) taking i = 2(1)10 and using boundary conditions, we obtain the following tridiagonal system of equations. 7
=
AU = B
(10)
Where A is a tridiagonal matrix of order nine, whose elements are defined by
ai,i
=
1
-
?A
a,-l,i
=
-r(---
U,j_l
=
-
R-hl
(h
-
+)
i
=
l(l)9
> i = 2(1)9
r
R
“0
0.2
0.4
0.8
1.2
t
and U, B components,
are column matrices having nine they are U(i,j + l), i = 2(1)10, which
294
Eng.
J. Biamed.
1988, Vol. 10, May
I.0
Figure
3
Velocity
against
time for t = 0.5.
at
t = 0.5
Technical Records
has to be calculated,
+
For an exponentially decreasing pressure gradient, the maximum velocity at z = 0.1 is obtained at t = 0.3 and the periodic pressure gradient maximum is at t = 1.2. For z = 0.9 the maximum velocity is obtained at t = 0.6 for an exponentially decreasing pressure gradient. The effect of a magnetic field is, therefore, to reduce the velocity of the fluid between the two plates.
and
r(l -A)-u(i+ R + KF(kj)
M.
l,]) ’
i
=
2(1)10
Equation (10) can be solved by a very efficient algorithm due to Evans’. Velocity has been computed for Harman numbers M = 0, 1, and 2 and the results are shown in Figures 2 and 3. The velocity increases rapidly between z = 0 and t = 0.6 in ail the three cases, i.e. a constant pressure gradient, an exponential pressure gradient and a periodic pressure gradient. In case of a consthe velocity is found to tant pressure gradient, decrease with an increase in the Hartmann number
REFERENCES Fung YC, Sobin SS. J A@1 Physiol 1969; 26: 472-88. Sobin SS, Tremer HM, Fung YC, Circulation Res 1970; 26: 397-414. Tang HT, Fung YC, J Appl Mech 1975; 42: 45-50 (Tram ASME 93, Series E). Peskin CS. Numerical analysis of blood flow in the heart. J Computational Physics 1977; 25: 220-52. McCracken MF. A vortex method for blood flow through heart valves. J Computational Physics 1980; 35: 183-205. Korohevaskii EM, Marochunic LS. Biophysics 1965; 10: 411. Evans DJ. Computer J 1972; 15: 356.
J. Biomed. Eng. 1988, Vol. 10, May
295
Rao and J. Anand Rao
of Mathematics,
Osmania
University,
Received November 1986; revised and accepted July
Hyderabad
500 007, India
1987
ABSTRACT Masnetohydrodynamic (MHD) principles may be used to decelerate theflow of arterial blood and hence be of potential value in the treatment of cardiovascular disorders associated with an accelerated circulation. We examine the non-steady flow of blood in a porous parallel plate channel under the influence of a transverse magnetic field and different pressure gradients.
Keywords:
Blood flow, numerical methods
INTRODUCTION
2
Fung and Sobin’ studied the flow of blood between two endothelial layers using the ‘sheet model’; the morphometric basis for sheet flow was presented by Sobin’ and others. Tang and Fung3 studied the mass transfer problem in the alveolar sheet and Peskin4 applied numerical techniques to examine the flow of the blood in the heart. McCracken’ used Vorten-grid methods to investigate blood flow through the heart valves. In the lungs, blood can be visualized as flowing between the opposing layers of capillary endothelium, held apart by endothelium-covered ‘posts’ of special tissue. The capillary endothelium is, in turn, covered by a thin layer (interstitial space) lining the alveoli. In the lungs, the blood space can be considered as a two-dimensional channel and the interstitial tissue space as a porous medium. As the endothelial layer between the two regions is permeable to water and some solutes, it can be considered as a permeable membrane of negligible thickness.
FORMULATION PROBLEM
AND SOLUTION
OF THE
Although the blood circulation system consists of three-dimensional elastic tubes of varying crosssection, for mathematical convenience we consider the magnetic effect on the non-steady flow of blood through a two-dimensional, nonconducting, parallel plate channel. The geometry of the system is explained in Figure 1. Assume that the liquid is being injected into the flow region through the channel at z = 0 and is being sucked away through the channel at z = 6. At any time t, let u and w be the components of velocity at a point (x, z) of the flow region in the direction of x and t respectively. The non-steady flow of blood in the presence of a static magnetic field is governed by the following equations of motion and the continuity equations. Correspondence
and reprint
requests
0 1987 Butterworth & Co (Publishers) 0141-5425/88/030293-03 803.00
to Dr P. Samba&a
Rao
1 TTTiTTTiTlTTT BO
80
0000000000000
2=L5
.?=O
oooooolooooooo
TTTiTTTTlTl~T BO
80
Figure 1 Non-steady magnetohydrodynamic flow of blood in a parallel plate channel with uniform injection through the wall at z = 0 and suction through the wall at z = b
au
au
au
at+u~+w~=--gy+-
- i ap
CL* 4
i a2 (1)
aw aw -++-++-=-+--
aw
ax
a2
at
- lap
p
eat
e
(
+- azw a.2
1
and
au -+-= ax
aw aZ
azw
0
a2 (2)
(3)
Where P is the pressure at the point (x, z) in the flow region and B, is the applied magnetic field.
Ltd J, Biomed.
Eng.
1988, Vol. 10, May
293
Technical Records
Let the initial and boundary conditions be given by t
= 0
and w=OforO
(4)
t > 0, u = 0 and w = w0 = positive t=Oandz=b
constant
(5)
In view of conditions (4) and (5), the velocity distribution is independent of x, i.e.
-=au
field
,*=o a
0
ax
for
Substituting
0.25 -
=
g
0 in equation
(3)
condition
(5). We can write equations
au
au =--++--1
ap
p azu
e
ax
4
at + w”aZ
and
using
(1) and (‘2) as 4
a2
B&(7)
and
ap az
-=
0
(8)
Normalizing the above equations and applying finite difference formulae, taking the mesh size h in the zdirection and k in the time direction we get -rX(lIR
(h -
-
$)]
h).u(i+
1) + +
l,j+
*u(i,j + 1) -
[1 1
-
-I 0.8
0.2
0.4
Velocity
distribution
in the z direction
0.6
1.0
.?
0.6
A Figure
2
rXu(i - j + 1) I
=
r(1
-
X)(1/R-h)*u(i+l,j)
+ [l
_ &q2_K R
t + ~(1 -
X)(h + +)]
+ ~(1 -
X)*u(i-
k
where r = F,
1,j)
-
I
u(i, j) 1
(9)
R + KF(lg]
X (weighting
factor)
= ? and % dX - F(t). In equation (9) taking i = 2(1)10 and using boundary conditions, we obtain the following tridiagonal system of equations. 7
=
AU = B
(10)
Where A is a tridiagonal matrix of order nine, whose elements are defined by
ai,i
=
1
-
?A
a,-l,i
=
-r(---
U,j_l
=
-
R-hl
(h
-
+)
i
=
l(l)9
> i = 2(1)9
r
R
“0
0.2
0.4
0.8
1.2
t
and U, B components,
are column matrices having nine they are U(i,j + l), i = 2(1)10, which
294
Eng.
J. Biamed.
1988, Vol. 10, May
I.0
Figure
3
Velocity
against
time for t = 0.5.
at
t = 0.5
Technical Records
has to be calculated,
+
For an exponentially decreasing pressure gradient, the maximum velocity at z = 0.1 is obtained at t = 0.3 and the periodic pressure gradient maximum is at t = 1.2. For z = 0.9 the maximum velocity is obtained at t = 0.6 for an exponentially decreasing pressure gradient. The effect of a magnetic field is, therefore, to reduce the velocity of the fluid between the two plates.
and
r(l -A)-u(i+ R + KF(kj)
M.
l,]) ’
i
=
2(1)10
Equation (10) can be solved by a very efficient algorithm due to Evans’. Velocity has been computed for Harman numbers M = 0, 1, and 2 and the results are shown in Figures 2 and 3. The velocity increases rapidly between z = 0 and t = 0.6 in ail the three cases, i.e. a constant pressure gradient, an exponential pressure gradient and a periodic pressure gradient. In case of a consthe velocity is found to tant pressure gradient, decrease with an increase in the Hartmann number
REFERENCES Fung YC, Sobin SS. J A@1 Physiol 1969; 26: 472-88. Sobin SS, Tremer HM, Fung YC, Circulation Res 1970; 26: 397-414. Tang HT, Fung YC, J Appl Mech 1975; 42: 45-50 (Tram ASME 93, Series E). Peskin CS. Numerical analysis of blood flow in the heart. J Computational Physics 1977; 25: 220-52. McCracken MF. A vortex method for blood flow through heart valves. J Computational Physics 1980; 35: 183-205. Korohevaskii EM, Marochunic LS. Biophysics 1965; 10: 411. Evans DJ. Computer J 1972; 15: 356.
J. Biomed. Eng. 1988, Vol. 10, May
295