SME
Biotransport · Problem type
Wall shear stress in a vessel
Use this for “What shear stress does 108 mL/min put on the wall of a 2 mm-radius artery?”
Diagram
vₘₐₓ = 2 vavg = 2 × 0.1432 = 0.2865 m/s at the centre, 0 at the wall.
Not to scale: the vessel is far longer than it is wide.
This is an example. Type your own values to start a new problem.
τw (Pa)
=
4
×
μ (mPa·s)
×
Q (mL/min)
60π
×
(
r (mm)
)
3
Formulas
τw
=
4 ×
μ
×
Q
60π ×
r
3
1
=
4 × 3.5 × 108
60π ×
2
3
Assumptions
•
Poiseuille flow: the slope of the parabolic profile at the wall sets the shear on the endothelium.
•
In SI, τw = 4μQ ÷ (πr³). With μ in mPa·s, Q in mL/min and r in mm, that number is 60 times τw in Pa: hence 60π.
•
1 Pa is 10 dyn/cm², the unit many physiology papers use for wall shear.
Step-by-step
WE KNOW
μ = 3.5 mPa·s, Q = 108 mL/min, r = 2 mm
FIND
wall shear stress (τw)
Step 1 · Find wall shear stress (τw)
Use τw = (4 × μ × Q) ÷ (60π × r³)
The
wall
shear
of
Poiseuille
flow,
4
μ
Q
πr³
;
the
60
turns
mPa·s,
mL/min
and
mm
into
pascals.
τw
=
4 × 3.5 × 108
60π ×
2
3
τw = 1 Pa
Check
Put the numbers back in. Both sides should match.
1
=
4 × 3.5 × 108
60π ×
2
3
✓
Topic check
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