Inflow 0 nL/sCapillary speed 0 mm/sO₂ extracted 0 %Hypoxic tissue 0 %readyFPS 0

Click a capillary to block it (an occlusion) and click again to reopen it; flow re-routes through the neighbours. Drag to orbit, wheel or pinch to zoom. Vessel colour = haemoglobin saturation (red high, blue low); cells turn blue where oxygen runs out.

Capillary Bed: Poiseuille Flow, Blood Viscosity and Oxygen Delivery

Blood enters a branching capillary network from an arteriole, leaves through a venule, and feeds the surrounding cells with oxygen. Change pressure, vessel radius, haematocrit or tissue demand and block capillaries to see where flow and oxygen go. Educational model, not a medical tool.

Driving pressure

Vessels and blood

Tissue

Display

Live readouts

Selected capillary

noneclick one

Flow vs arteriole radius

Krogh tissue cylinder: PO₂ vs distance

Solid: closed form. Dashed: numerical integration of the diffusion equation. They overlap.

Independent numerical checks

Controls

Drag / one finger: orbit; wheel / pinch: zoom.
Click or tap a capillary: block or reopen it (vessel turns grey).
Sliders change the model live; red cells and oxygen dots move in slow motion (factor shown in the display section).
Quality: High is the default on every device; Low (also ?q=low) uses fewer cells.

The science

Poiseuille flow. For a round tube of radius r and length L the laminar flow is Q = π r⁴ ΔP / (8 μ L), so conductance scales with r⁴: halving the radius cuts the flow 16-fold. A vessel of conductance g between nodes i and j carries Q = g (P_i − P_j); at every junction the flows sum to zero (Kirchhoff). The network is solved for all node pressures by Gauss-Seidel iteration.

Blood is not water. Apparent viscosity depends on tube diameter D and haematocrit Ht (Fåhræus-Lindqvist effect). The model uses the empirical in-vitro fit of Pries and co-workers: μ_0.45 = 220 e^−1.3D + 3.2 − 2.44 e^{−0.06 D^0.645} with a haematocrit correction, times a plasma viscosity of 1.2 mPa·s. Haematocrit is the same in every vessel (no phase separation at bifurcations), and red-cell speed equals the mean blood speed.

Oxygen carriage. Saturation follows the Hill curve S = Pⁿ/(Pⁿ + P₅₀ⁿ) with n = 2.7 and P₅₀ = 26.8 mmHg; blood content is 1.34·Hb·S + 0.00003·PO₂ mL O₂ per mL. Content is mixed at junctions weighted by flow and reduced along each capillary by the tissue uptake.

Krogh cylinder. Each capillary of radius r_c feeds a coaxial tissue cylinder of radius R consuming oxygen at a uniform rate M. Steady diffusion gives PO₂(r) = P_c + M/(4Dα) [ r² − r_c² − 2R² ln(r/r_c) ], lowest at r = R. If the capillary PO₂ is below the drop to the cylinder edge, only a fraction of the tissue is supplied; the rest counts as hypoxic.

Simplifications and model values. Rigid round tubes, steady flow, no pulsatility, no vessel wall mechanics, no autoregulation, uniform consumption, the Krogh constant, plasma viscosity, vessel lengths and the upstream arteriole length (4 mm, drawn shortened) are typical round numbers chosen for a plausible capillary speed of roughly 0.5 to 1 mm/s. This is a teaching model and gives no diagnosis or medical advice. Models built with Blender, scene by Three.js.