Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 / 1 / b

The Fahraeus--Lindqvist effect is the decrease of blood's apparent or effective viscosity as a microvessel narrows through much of the physiological small-vessel range. Deformable red blood cells migrate away from the wall and concentrate near the centre, creating a cell-free layer of relatively low-viscosity plasma beside the vessel wall. Because the largest shear occurs near the wall, replacing cell-rich blood there by plasma reduces hydraulic resistance particularly effectively. At diameters comparable with a red blood cell, confinement eventually invalidates this decreasing trend.

Let $b=h-\delta$ be the half-width of the cell-rich core. In fully developed pressure-driven flow the shear stress is fixed by momentum balance, independently of the local viscosity:With no slip at $y=h$,Interchanging the order of integration gives the total fluxBy definition, the homogeneous effective fluid hasWriting $\epsilon =\delta /h$ therefore givesor the reciprocal of the right-hand side.

If $\delta =0$, then $\epsilon =0$, the cell-rich material fills the gap, andIf $\delta =h$, then the core disappears andFor the physical ordering ${\mu }_w<\mu$, increasing the cell-free layer thickness monotonically lowers ${\mu }_{\mathrm{eff}}$ between these limits, exactly as intuition and the Fahraeus--Lindqvist effect suggest.