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PES 3950/PHYS 6950
Spendier Spring 2015
Lecture 12
Announcement: Exam 1 graded and solutions posted
Laminar Flow:
A relatively simple case is given by laminar flow (fluid flows in parallel layers). Here,
there is no turbulence, the fluid flows in ‘layers’ and there are no currents perpendicular
to the flow direction.
Within the human body, understanding laminar flow is important to describe blood flow.
As the heart pumps the blood is circulating the body through tubes (blood vessels 1cm to
2 μm).
Arteries deliver blood to a vast network of capillaries that provide nourishment to tissue
through the body. Veins return blood from the capillaries back to the heart and lungs.
Our goal: Derive fluid velocity in a pipe from which we will estimate the average
blood flow in a blood vessel
Simplest model of blood flowing through vessel is laminar flow through a pipe where the
blood is described as a Newtonian Fluid. For laminar flow, it is possible to calculate the
fluid flow rate Φ through a cylindrical pipe of radius R and length L with a pressure
difference ΔP between the ends:
In a tube of radius R and length L >> R. The flow is produced because of a difference in
pressure P(0) > P(L). The problem is invariant through rotation around the cylindrical
axis. We will use cylindrical coordinates. We thus consider a laminar flow, such that :
 
v = v z  r , z  zˆ
where z-hat is the unitary basis vector for direction of the cylindrical axis. In this
problem we forget about any other forces than pressure and the viscosity.
 v



r    v  v   P  h2 v
 t

With

 v  0
Radius of pipe R and length of pipe L
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PES 3950/PHYS 6950
Spendier Spring 2015
Simplifications:
1) rotational symmetry along the axis :
Lecture 12
v q
0
q
2) no spatial variation of the velocity field in the direction of the flow
v z
0
z
Therefore we conclude that
 
v = v z  r  zˆ
3) We are looking for the steady state solution

v
 0 b/c fluid flow in the pipe is incompressible
t
 
And since v = v z  r  zˆ therefore


 v  v=v z
v z (r )
zˆ  0
z
This means that the convective term and net acceleration are both zero and the NavierStokes equation reduces to

 
0  P  h2 v

 
P  h 2 v
Thus, only the force terms remain.
Project Equation on the cylindrical basis
The Laplacian in cylindrical coordinates is (applied to a vector, it acts on projected
components individually)
 2  1   v  1  2v  2v
 r  
 v

r r  r  r 2 q 2 z 2
 
Since v = v z  r  zˆ
  1   v 
 r 
2 v 
r r  r 
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PES 3950/PHYS 6950
Spendier Spring 2015
Lecture 12
Next, the gradient is, using the that the pressure varies only linearly in z (independent of r
and θ)
 P

0
 r


1 P
P  
0
 r q

 P
 z
rˆ
qˆ
zˆ
Therefore, the Navier-Stokes equation becomes
P
1   v z 
h

r
z
r r  r 
So pressure only depends on z. In the z-projection of Navier-Stokes, the left term depends
on z only, and the right term depends on r only. The only possibility for this to be true is
that actually neither depends on z nor r. Both terms are constant. The pressure depends
linearly with z.
 
Determine v = v z  r  zˆ
You will show in HW 4 that after solving the above equation you will get:
r 2P
 C1 ln(r )  C2  h v z
4L
Now We will need two boundary condition on vz in order to finally get its expression:
v z
1) and
at r = 0 must be bounded.  C1 = 0
r
2) continuity of the velocity field along the edges (vz(R) = 0)
and the second BC:
R 2P
 C2  h v z ( R)  0
4L
R 2 P
 C2
4L
Hence
r 2 P R 2 P

 h v z (r )
4L
4L
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PES 3950/PHYS 6950
v z (r ) 
Spendier Spring 2015
Lecture 12
P 2
R r2

4h L
This is the final expression for the fluid velocity in the pipe. This is a parabolic velocity
profile, with maximum in the center of the tube, and null in contact with the outside wall
From this expression we can calculate the volumetric flow rate Q of the pipe.
R
Q
0
Q
R
R
R 2P
P
2p R 4P 2p R 4P
2
vz (r )2prdr 
2
p
rdr

r
2
p
rdr


4h L 0
4h L 0
8h L
16h L
p R 4P
8h L
This is the well known Hagen-Poiseuille law for laminar pipe flow. Because of this R4
dependence, a small change in the area of the blood vessel will result in a large change in
the volume flow rate. This expression is also biologically relevant for blood flow through
the cardiovascular system.
Estimate Blood Flow through a capillary:
To estimate the blood flow it would be useful to know the average fluid velocity across
the cross-section of the capillary.
volumetric flow rate Q p R 4P R 2P
v 
 

pipe cross section
A 8h Lp R 2
8h L
You can look up
∆P ~ 3000 Pa over the length of a capillary
L ~ 1cm
R ~ 2.5 μm
η ~ 10-3 Pa s
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PES 3950/PHYS 6950
Spendier Spring 2015
Lecture 12
6
2
R 2P 2.510  m 3000 Pa
v 

 0.02 cm / s
8h L
8103 Pa  s102 m
2
This is very close to the measured flow which is roughly 0.05 cm/s
Application: Blood Vessel Constriction and Aneurysm
Typically, we use sound waves to measure blood flow, so called Doppler Flow. The
sound of moving blood produces wave-forms that reflect the speed and amount of the
blood as it moves through a blood vessel. Due to local changes on the blood flow speed
one can determine if the blood vessels are constricted by some sediment or widened.
Constriction of blood vessel:
The constriction of a blood vessel due to vulnerable plaque can result in a transition from
laminar to turbulent flow.
The constriction of a blood vessel due to vulnerable plaque can result in a transition from
laminar to turbulent flow.
Aside:
Laminar flow proceeds with the smooth motion of adjacent fluid layers sliding
continuously past each other without breaking into whirlpools or vortices.
Streamline flow though a pipe
with a greater density of
streamlines in a constriction
with higher fluid speed and
reduced pressure.
Turbulent flow occurs with a breakup of adjacent fluid layers resulting in whirlpools and
vortices. The fluid velocity at a given point in a turbulent region changes rapidly with
time. A flow may transition from laminar to turbulent as the fluid speed is increased.
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PES 3950/PHYS 6950
Spendier Spring 2015
Lecture 12
Fluid speed:
The steady state flow rate through a given pipe or blood vessel is
dV
 const.
dt
Since the volume V of fluid entering the blood vessel of varying cross sectional area A
must be equal the volume leaving the vessel in the same time interval
A1v1  A2 v2
A2
v2
Or
Dx2
Av  const.
You should recognize this expression – it is again
the equation of continuity!
Fluid will flow faster through a constriction in a
blood vessel
A1
v1
Dx1
A  v 
Consequences:
The fluid speed is higher in the constricted region so that the pressure inside the vessel is
lower than the external pressure. The vessel can momentarily collapse as a result of this
pressure drop. The fluid quickly pushes the vessel open again, and the cycle repeats.
Distinctive sounds of both fluid turbulence and vascular fluttering can be heard with a
stethoscope.
Enlarged Vessel: Aneurysm
An aneurysm can form in a blood vessel where a portion of the vessel is enlarged with a
balloon-like bulge.
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PES 3950/PHYS 6950
Spendier Spring 2015
Lecture 12
Let’s derive an expression for the change in average blood pressure in a capillary that
increases in area A due to an enlarged vessel.
We start from Bernoulli’s Equation. We may derive Bernoulli’s equation from NavierStokes equation neglecting viscous drag and acceleration of the fluid.
Or
The sum of the pressure kinetic energy density and potential energy density is a constant
for any two points along a streamline flow.
For simplicity, we can consider constant height, y1 = y2 (no change of gravitational
potential. Comparing two points along the streamline flow, we can calculate the pressure:
P1  P2  12 r v22  v12 
Or from the continuity equation: A1 v1 = A2 v2
 A2 v 2

DP  P1  P2  12 r  1 21  v12 
 A2

 2 
2  A1
1
DP  P1  P2  2 rv1  2 1
A

2
So a dilated vessel A2 > A1 results in an increase in pressure where
A  v  P 
We call this effect “Venturi” effect.
Consequence:
In case of an enlarged vessel, the blood pressure is higher in the bulge compared to the
normal vessel pressure. Higher pressure in the enlarged vessel tends to expand the vessel
even farther until the blood vessel can eventually burst.
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