Biofluid Mechanics: Blood Flow & Cardiovascular Devices

How blood moves and how devices change it. Learn blood as a non-Newtonian fluid (shear thinning, haematocrit, Fåhræus–Lindqvist), Poiseuille flow and the r⁴ law, stenosis and pressure drop, Reynolds and Womersley numbers for pulsatile flow, wall shear stress and atherosclerosis, pulse-wave velocity, and devices — stents, heart valves, VADs, dialysers, oxygenators — with Python analysing a stenosed artery.

Blood is a suspension of cells (45% red cells by volume) in plasma, pumped pulsatilely through elastic, branching vessels from 25 mm down to 5 µm. Its flow determines oxygen delivery, and disturbances in flow are linked to atherosclerosis, thrombosis and aneurysms.

Biofluid mechanics underpins the design of stents, artificial valves, ventricular assist devices, catheters, dialysers and oxygenators — and the interpretation of Doppler ultrasound and CT-FFR.

An analogy

Blood vessels are like a city's water mains that can stretch and pulse, feeding ever-thinner pipes. Narrow one pipe by half and, thanks to the r⁴ law, the pressure needed to push the same flow rises 16-fold.

Blood as a fluid

Plasma is Newtonian (~1.2 mPa·s), but whole blood is shear-thinning: red cells aggregate at low shear (high viscosity) and align at high shear (~3.5 mPa·s). Viscosity rises steeply with haematocrit. In vessels < 0.3 mm the apparent viscosity falls (Fåhræus–Lindqvist effect) as cells migrate to the centre.

Steady and pulsatile flow

Poiseuille's law gives the pressure–flow relation for laminar flow in straight rigid tubes — the r⁴ dependence explains why arteriolar constriction controls blood pressure. Real arterial flow is pulsatile; the Womersley number compares inertial to viscous forces: small α (< 1) gives quasi-steady parabolic profiles, large α (aorta ≈ 15–20) gives blunt profiles with phase lags.

Wall shear stress and disease

Endothelial cells sense wall shear stress (normal ~1–7 Pa in arteries). Low and oscillating WSS at bifurcations and curves correlates with atherosclerotic plaque; very high WSS in stenoses activates platelets. Aneurysms grow where wall stress and flow patterns combine unfavourably. CFD with patient-specific geometry now supports diagnosis (CT-FFR).

Cardiovascular devices

Design must respect haemodynamics and blood damage:

Formulas

Worked example: Wall shear stress

Given: μ = 3.5 mPa·s, Q = 5 mL/s, r = 1.5 mm

Find: τ_w.

Answer: ≈ 6.6 Pa

Worked example: Doppler stenosis gradient

Given: Peak jet velocity 4 m/s through an aortic valve

Find: Pressure gradient.

Answer: 64 mmHg (severe aortic stenosis > 40 mmHg mean)

Worked example: Pulse-wave velocity

Given: E = 0.5 MPa, h = 2 mm, r = 12 mm, ρ = 1060

Find: PWV.

Answer: ≈ 6.3 m/s (stiffer arteries → faster PWV)

Stenosis: pressure drop, velocity and Reynolds number

import math
# Poiseuille flow in a vessel and the effect of a stenosis
mu = 3.5e-3                      # blood viscosity Pa*s
L = 0.05                         # 5 cm segment
Q = 5e-6                         # 5 mL/s (coronary-like)
for r_mm in (1.5, 1.05, 0.75):   # healthy, 30% and 50% diameter reduction
    r = r_mm / 1000
    dP = 8 * mu * L * Q / (math.pi * r**4)
    v = Q / (math.pi * r * r)
    Re = 1060 * v * 2 * r / mu
    print(f"radius {r_mm:4.2f} mm: dP = {dP/133.3:6.2f} mmHg, mean v = {v:.2f} m/s, Re = {Re:.0f}")
# Womersley number for the aorta (r 12.5 mm, 1.2 Hz heart rate)
alpha = 0.0125 * math.sqrt(2 * math.pi * 1.2 * 1060 / mu)
print(f"aortic Womersley number = {alpha:.1f} (pulsatile, flat-ish profile)")
# radius 1.50 mm: dP =   3.30 mmHg, mean v = 0.71 m/s, Re = 643
# radius 1.05 mm: dP =  13.75 mmHg, mean v = 1.44 m/s, Re = 918
# radius 0.75 mm: dP =  52.83 mmHg, mean v = 2.83 m/s, Re = 1285
# aortic Womersley number = 18.9 (pulsatile, flat-ish profile)

Halving the radius over 5 cm raises the pressure drop 16× (r⁴). Real stenoses are short with extra losses from flow separation, and the body compensates by dilating downstream vessels — which is why clinicians measure FFR rather than rely on geometry alone. With α ≈ 19 the aortic flow is far from parabolic.

Common mistakes

Applications

Key takeaways

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