Blood Rheology: Why Blood Refuses to Be a Simple Newtonian Fluid

1. What "Newtonian" actually means

A Newtonian fluid obeys Newton's law of viscosity: shear stress is directly proportional to shear rate, with a single constant of proportionality — the viscosity μ, independent of how hard you shear it.

Water, air, and glycerin are essentially Newtonian across the shear rates you'll ever encounter in a syringe or a pipe. Blood is not — plot τ against shear rate for blood and you get a curved, not a straight, line.

2. Why blood isn't Newtonian

Blood is roughly 55% plasma (which is Newtonian, close to water) and ~45% red blood cells (RBCs) by volume (the hematocrit), plus much smaller amounts of white cells and platelets. At low shear rates the RBCs are not being pulled apart by flow, so they clump into stacks called rouleaux — held together by plasma proteins like fibrinogen — which raises the effective viscosity a lot. As shear rate increases, three things happen in sequence:

• Rouleaux breakup — the stacks are torn apart by the flow, reducing effective particle size.
• RBC alignment — individual cells rotate to align with the flow, reducing drag.
• RBC deformation — the biconcave discs elongate into ellipsoid "bullet" shapes that slip past each other more easily.

All three reduce resistance to flow as shear rate rises — this is shear-thinning (pseudoplastic) behaviour: apparent viscosity decreases as shear rate increases. It's the opposite of shear-thickening fluids like cornflour-and-water suspensions.

3. The Casson model and yield stress

The most widely used clinical model for blood is the Casson equation. It captures a striking feature: blood appears to need a minimum stress before it flows at all — a yield stress τ_y, caused by the rouleaux network needing to be broken before flow can start.

τ_y depends strongly on hematocrit and fibrinogen concentration — it rises in conditions like polycythemia or hyperfibrinogenemia, and can contribute to the "sluggish flow" seen clinically in those states.

4. The Carreau-Yasuda model

For CFD simulations, the Casson model's √ singularity at low shear is numerically awkward, so the smoother Carreau-Yasuda model is often preferred. It interpolates continuously between a high, near-constant viscosity at very low shear rate and a low, near-constant viscosity at very high shear rate:

At γ̇ → 0, μ → μ₀ (thick, near-yield behaviour). At γ̇ → ∞, μ → μ_∞ (blood behaves almost like plasma alone, since RBCs are fully aligned and deformed).

5. The Fåhræus-Lindqvist effect

There's a second, geometric non-Newtonian effect specific to microcirculation. In vessels below roughly 300 μm in diameter, the apparent viscosity of blood decreases as the tube gets narrower — the opposite of what a simple viscosity model would predict.

The mechanism: RBCs migrate away from the vessel wall toward the centreline (axial migration), leaving a thin, RBC-poor, low-viscosity plasma skimming layer next to the wall. In very narrow capillaries this cell-free layer becomes a significant fraction of the tube radius, effectively lubricating the flow and lowering the pressure drop needed to push blood through — this is the Fåhræus-Lindqvist effect (Fåhræus & Lindqvist, 1931), and it reverses again below ~10 μm as individual RBCs (≈7–8 μm) must squeeze through single-file.

6. Pseudocode

Computing local viscosity with the Carreau-Yasuda model inside a flow solver:

function carreauYasudaViscosity(shearRate):
  const mu0    = 0.056   // Pa·s, zero-shear plateau
  const muInf  = 0.0035  // Pa·s, infinite-shear plateau
  const lambda = 3.31    // s, transition time constant
  const n      = 0.3568 // power-law index
  const a      = 2.0     // transition sharpness

  term = Math.pow(1 + Math.pow(lambda * shearRate, a), (n - 1) / a)
  return muInf + (mu0 - muInf) * term

function stepBloodFlow(velocityField, dt):
  for each grid cell:
    shearRate = localVelocityGradient(velocityField, cell)
    mu = carreauYasudaViscosity(shearRate)
    // use mu(x) in place of constant viscosity in the Navier-Stokes solve
    applyViscousStress(cell, mu, dt)

The key difference from a plain Navier-Stokes / SPH solver is that μ is recomputed per cell, per step from the local shear rate, rather than being a single global constant — this is what lets the same solver reproduce both thick, near-stationary flow in small vessels and thin, fast-sliding flow in large arteries.