Poiseuille parabolic profile · Pulsatile flow · Stenosis · Wall shear stress
Interactive hemodynamics simulator showing Poiseuille's parabolic velocity profile, pulsatile flow from the heart, stenosis effects on flow velocity, and wall shear stress in blood vessels.
Laminar blood flow follows Poiseuille's law: Q = πr⁴ΔP / (8μL). Narrowing a vessel (stenosis) forces blood faster through a smaller cross-section, increasing shear stress on the vessel wall and risk of plaque rupture.
Adjust vessel radius and blood pressure to see the velocity profile change. Add stenosis to observe flow acceleration and turbulence. Watch wall shear stress (WSS) values update in real time.
Blood flow velocity at the vessel centre is about twice the average velocity. In a 50% stenosis, the velocity can increase 4× and wall shear stress 8× due to Poiseuille's fourth-power dependence on radius.
This simulation models laminar blood flow through a cylindrical vessel using the Hagen–Poiseuille law, Q = πr⁴ΔP / (8ηL), which gives volumetric flow as a function of radius, pressure gradient, viscosity and length. It renders the characteristic parabolic velocity profile u(r) ∝ (1 − r²/R²), where flow is fastest at the centreline and zero at the wall, alongside a longitudinal and cross-sectional view of the vessel.
Sliders set vessel radius R, stenosis percentage, length L, driving pressure ΔP, blood viscosity η, heart rate and pulse amplitude, while toggles switch between longitudinal and cross-section views and between pulsatile and steady flow. Results report flow Q, peak velocity, wall shear stress and Reynolds number in real time. Hemodynamics like this underpins how doctors assess arterial narrowing, atherosclerosis and the mechanical stresses that drive plaque rupture.
What does this blood flow simulation show?
It shows laminar blood flowing through a vessel, with the classic parabolic velocity profile where flow is fastest at the centre and slows to zero at the wall. You can view the vessel lengthwise or in cross-section, see flowing particle streaks, and read live values for flow rate, peak velocity, wall shear stress and Reynolds number.
What is Poiseuille's law and how is it used here?
Poiseuille's law states that volumetric flow Q = πr⁴ΔP / (8ηL), relating flow to vessel radius r, pressure difference ΔP, viscosity η and length L. The simulation evaluates this equation directly to compute the flow rate, then derives the parabolic velocity profile and mean velocity from it.
Why is the velocity profile parabolic?
In steady laminar pipe flow, the no-slip condition forces velocity to zero at the wall, while the pressure gradient pushes fluid hardest in the middle. Solving the Navier–Stokes equations for this balance gives u(r) = u_max(1 − r²/R²), a parabola whose centreline peak is exactly twice the cross-sectional mean velocity.
Radius R (0.5–10 mm) and length L set the vessel geometry; stenosis (0–90%) adds a narrowing; pressure ΔP (100–13330 Pa) is the driving force; viscosity η (0.001–0.010 Pa·s) reflects blood thickness; and heart rate plus pulse amplitude shape the pulsatile waveform when pulsatile mode is on.
Stenosis models a Gaussian narrowing at the vessel midpoint. Because flow depends on radius to the fourth power, even a modest narrowing sharply restricts flow, while the blood that does pass through speeds up to conserve volume. This acceleration raises wall shear stress and, beyond about 70%, the simulation flags severe stenosis.
Wall shear stress (WSS) is the tangential frictional force the moving blood exerts on the vessel wall, computed here as r·ΔP / (2L) in pascals. It matters clinically because abnormal WSS influences endothelial health, atherosclerotic plaque formation and the risk of plaque rupture or clot detachment.
The Reynolds number Re = ρ·u·2r / η compares inertial to viscous forces and predicts whether flow stays smooth or becomes turbulent. The model uses blood density ρ = 1060 kg/m³, and once Re exceeds roughly 2300 it warns of turbulent flow, which is more likely downstream of a tight stenosis.
In pulsatile mode the flow is multiplied by a time-varying pulse factor built from the heart rate, giving a systolic peak followed by a smaller dicrotic-notch bump, scaled by the pulse-amplitude slider. Switching to steady mode holds the pulse factor at one, showing constant Poiseuille flow for comparison.
It uses the correct Hagen–Poiseuille and wall-shear-stress equations and realistic blood parameters, so the trends and orders of magnitude are sound for idealised laminar flow. It is a teaching model, though: real arteries are elastic, tapering and branching, blood is non-Newtonian, and the pulse waveform here is a simplified approximation.
Hemodynamic principles like these guide the assessment of arterial disease: clinicians use flow velocity and pressure-gradient measurements to grade stenosis severity, and wall shear stress concepts help explain where atherosclerosis develops. The same physics also informs the design of stents, bypass grafts and vascular devices.