Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 334 (Biological Flows)

Murray's law minimizes the sum of the power needed to pump a Newtonian fluid and the metabolic power needed to maintain blood volume. For a cylindrical vessel of radius $R$, length $\ell$, and prescribed volume flux $Q$, Hagen-Poiseuille flow givesIf maintenance costs $\alpha$ per unit volume,Setting the derivative of ${\mathcal{P}}_{\mathrm{pump}}+{\mathcal{P}}_{\mathrm{met}}$ with respect to $R$ to zero givesThus $Q\propto R^3$. Conservation of volume flux at a bifurcation gives

For plane Poiseuille flow between walls $y=\pm h$, let $G=-p_x>0$. Per unit out-of-plane depth,For vessel length $\ell$,The maintained cross-sectional area per unit depth is $2h$, soOptimization givesThe two-dimensional Murray law is therefore

The wall shear stress in the planar vessel isThe optimized relation $Q\propto h^2$ therefore makesthroughout an ideal network. Murray's optimization can equivalently be interpreted as selecting a uniform wall shear stress. The familiar three-dimensional law $Q\propto R^3$ has the same interpretation because cylindrical Poiseuille flow has ${\tau }_w=4\mu Q/(\pi R^3)$.

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.

The Zweifach--Fung effect, also called plasma skimming, is the tendency at an unequal microvascular bifurcation for the high-flow daughter to receive a disproportionately large fraction of the red blood cells. The low-flow daughter consequently has a lower discharge haematocrit than the parent.

Each daughter has the planar Poiseuille relationHenceThe higher-flow daughter therefore requires the larger pressure-gradient magnitude:

Across a cell of radius $a$, a pressure gradient $G_i$ changes the viscous shear stress by orderIn the two-dimensional model, the force per unit out-of-plane depth on one side is $O({\sigma }_{i}a)$ and its lever arm is $O(a)$. The net moment per unit depth is thereforeA fully three-dimensional force estimate adds one power of $a$. Since $Q_2>Q_1$, the stronger stress on the high-flow side gives a definite rotation that tips a deformable cell off the ideal point-particle separatrix and into daughter 2. This local finite-size mechanism biases cells toward the higher-flow branch and is consistent with the Zweifach--Fung effect.

For a unidirectional, $z$-independent velocityincompressibility holds automatically. The axial Stokes flow equation isNo external pressure gradient is imposed, so $p_z=0$ andThus the axial velocity is a harmonic function in the disk.

The boundary data are $U\mathrm{sgn}(\sin \theta )$, whose Fourier series isRegular harmonic modes in a disk are $(r/R{)}^n\sin (n\theta )$ and $(r/R{)}^n\cos (n\theta )$. Matching the odd boundary data givesEvery term is regular at $r=0$, and the series approaches the prescribed values at every boundary point away from the two jump discontinuities.

For $|z|<1$,Set $z=(r/R)e^{i\theta }$ and take the imaginary part. Since $r<R$, the real part of the relevant denominator is positive, and the result isAs $r\to R^{-}$ this tends to $U$ for $0<\theta <\pi$ and to $-U$ for $-\pi <\theta <0$.

The transverse and longitudinal Péclet numbers areThe Taylor dispersion regime requires transverse diffusion to act within a longitudinal advection time,while longitudinal molecular diffusion is slow on that advection time,Together,

The advection-diffusion equation isWritewhere the bar is the cross-gap average. In the long, late-time Taylor regime, $c^{\prime }$ adjusts rapidly across the gap while $\bar{c}$ varies slowly along the cell. The leading fluctuation balance isIts scaling isThis final inequality is precisely the transverse-equilibration condition from part i.

Put $c^{\prime }=\chi (y){\bar{c}}_x$. The balance and reflecting boundary conditions giveIntegration yieldsAveraging the full transport equation givesNowbecause $\overline{y^2}=h^2/3$ and $\overline{y^4}=h^4/5$. Thus the flow-induced Taylor dispersion coefficient and total effective diffusivity are