PULSATILE FLOW
Blood flow by heart is divided in two phases: Systole, during which blood is pumped
from the heart and diastole, during which no blood is pumped from the heart and the
ventricles fill with blood. This results in the variation of applied pressure gradient and
flow through the arteries with time over the period of the heart relaxation and contraction
(pulse).
Heart Beat Cycle:
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The Right Atrium, receives "used blood" from the body. Blood will be pushed
through the tricuspid valve to the
Right Ventricle, the chamber which will pump to the lungs through the pulmonic
valve to the
Pulmonary Arteries, providing blood to both lungs. Blood is circulated through
the lungs where carbon dioxide is removed and oxygen added. It returns through
the
Pulmonary Veins, which empty into the
Left Atrium, a chamber which will push the Mitral Valve open. Blood then passes
into the
Left Ventricle. Although it doesn't always look like it in drawings done from this
angle, this is the largest and most important chamber in the heart. It pumps to the
rest of the body. As it pumps, the pressure will close the mitral valve and open the
aortic valve, with blood passing through to the
Aorta, where it will be delivered to the rest of the body.
Example: Proximal aorta, femoral artery, coronary arteries, terminal arteries
Note that pressure pulses are almost nonexistent in the venous system and the flow is
steady.
Simplified governing equation with boundary conditions:
∂u
∂p µ ∂  ∂u z 
ρ z =− +
r

∂t
∂z r ∂r  ∂r 
∂u
r = 0: z = 0
∂r
r = R : uz = 0
Oscillatory/pulsatile pressure gradient (Pumping due to heart)
∂p ∆p
−
=
cos(ωt ) = Re[ A* e iωt ]
∂z
L
Womersley number
α = R ω / υ = Re Sr / 2
Separation of variables
u z = Ve iωt
−
iωVe iωt
(µ ρ )
+
e iωt d  dV
r
r dr  dr
A* iωt

e
=
−

µ

d V 1 dV iωV A*
+
−
+
=0
υ
µ
dr 2 r dr
2
Solution using Bessel’s functions
A*
+ C1 J 0 (i 3 / 2 ω / υ r ) + C 2Y0 (i 3 / 2 ω / υ r )
V =
iωρ
Apply B.C.s:
A*  J 0 (i 3 / 2αr / R)  iωt
1 −
e
uz =
iωρ 
J 0 (i 3 / 2α ) 
Bessel Function
Bessel functions are canonical solutions y(x) of Bessel's differential equation:
for an arbitrary real number α (the order). The most common and important special case
is where α is an integer, n.
Although α and −α produce the same differential equation, it is conventional to define
different Bessel functions for these two orders (e.g., so that the Bessel functions are
mostly smooth functions of α).
Bessel's equation arises when finding separable solutions to Laplace's equation and the
Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are
therefore especially important for many problems of wave propagation, static potentials,
and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n;
for spherical problems, one obtains half integer orders α = n+½.) For example:
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electromagnetic waves in a cylindrical waveguide
heat conduction in a cylindrical object.
modes of vibration of a thin circular (or annular) membrane.
Bessel functions also have useful properties for other problems, such as signal processing
(e.g., see FM synthesis or Kaiser window).
Bessel functions of the first kind:
Bessel functions of the first kind, denoted with Jα(x), are solutions of Bessel's differential
equation which are finite at x = 0 for α an integer or α non-negative. The specific choice
and normalization of Jα are defined by its properties below; another possibility is to
define it by its Taylor series expansion around x = 0 (or a more general power series for
non-integer α):
Here, Γ(z) is the gamma function, a generalization of the factorial to non-integer values.
The graphs of Bessel functions look roughly like oscillating sine or cosine functions that
decay proportionally to 1/√x (see also their asymptotic forms, below), although their roots
are not generally periodic except asymptotically for large x.
Plot of Jα(x) and Yα(x) for α = 0,1,2:
Orthogonal w.r.t. a weighting function. (Classical Sturm-Liouville Problem)
Flow-rate:
2πe iωt A*
Q = 2π ∫ u z rdr =
iωρ
0
R
∫ xJ
0
 J 0 (i 3 / 2αr / R) 
∫0 1 − J 0 (i 3 / 2α ) rdr
R
( x)dx = xJ 1 ( x)
∴Q =
πR 2 A*  2αi 3 / 2 J 1 (i 3 / 2αr / R)  iωt
e
1 −
iωρ 
α 2 i 3 J 0 (i 3 / 2α ) 
Viscous drag on the cylindrical surface:
∂u
F = −2πµr z | r = R
∂r
d
[J 0 (x )] = − J 1 (x )
dx
Explain phase lag between velocity and pressure gradient
Local maxima near the walls (inertia effects)
Limiting case of small Womersley number:
x2
x4
x6
J 0 ( x) = 1 − 2 + 4 − 6 2 + "
2
2 2! 2 (3!)
One can retrieve the parabolic profile for the velocity field by taking the limit aØ0.
Homework assignment: N/A
References:
1. Womersley, J.R. (1955). Method for the calculation of velocity, rate of the flow
and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127,
553-563.
2. Truskey, G.A., Yuan, F., Katz, D.F. (2004). Transport phenomena in biological
systems, Pearson Prentice Hall, 213-223.
3. http://en.wikipedia.org/wiki/Bessel_function
4. http://www.heartpoint.com/theheart.html