SME
Biotransport · Problem type
Is blood flow laminar? (Reynolds number)
Use this for “Is blood flowing at 108 mL/min through a 4 mm vessel laminar?”
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.
v (m/s)
=
Q (mL/min)
60π
×
(
r (mm)
)
2
Re
=
2
×
ρ (kg/m³)
×
v (m/s)
×
r (mm)
μ (mPa·s)
Formulas
v
=
Q
60π ×
r
2
0.143
=
108
60π ×
2
2
Re
=
2 ×
ρ
×
v
×
r
μ
174
=
2 × 1,060 × 0.143 × 2
3.5
Assumptions
•
Re compares inertia with viscosity on the diameter 2r; below about 2000 the flow is laminar.
•
In SI, v = Q ÷ (πr²). 1 mL/min is 10⁻⁶ ÷ 60 m³/s and 1 mm² is 10⁻⁶ m², so v = Q ÷ (60πr²) in m/s.
•
In Re the 10⁻³ of r in mm and of μ in mPa·s cancel.
•
Blood: ρ = 1060 kg/m³ and μ = 3.5 mPa·s.
Step-by-step
WE KNOW
Q = 108 mL/min, r = 2 mm, ρ = 1,060 kg/m³, μ = 3.5 mPa·s
FIND
mean speed (v), Reynolds number (Re)
Step 1 · Find mean speed (v)
Use v = Q ÷ (60π × r²)
The flow over the cross-section πr²; the 60 turns mL/min over mm² into m/s.
v
=
108
60π ×
2
2
v
=
0.14324
m/s
Step 2 · Find Reynolds number (Re)
Use Re = (2 × ρ × v × r) ÷ μ
Inertia
over
viscosity,
measured
on
the
diameter
2
r
.
Re
=
2 × 1,060 × 0.14324 × 2
3.5
Re
=
2,120
×
0.1432
×
2
3.5
Re
=
303.6688
×
2
3.5
Re
=
607.3376
3.5
Re = 174 → Re < 2000: the flow is laminar.
Check
Put the numbers back in. Both sides should match.
0.14324
=
108
60π ×
2
2
✓
174
=
2 × 1,060 × 0.14324 × 2
3.5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.