A mesenteric artery of radius carrying steady Poiseuille flow divides into two…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Biophysics & Medical Physics · Viscous flow and dynamical similarity

Forces, Machines and Signals, Ch Bones, blood and body size

Problem

A mesenteric artery of radius

r_0

carrying steady Poiseuille flow

Q_0

divides into two daughter vessels of radii

r_1

and

r_2

carrying flows

Q_1

and

Q_2

, with

Q_0=Q_1+Q_2

. Blood has viscosity

\eta

, and the metabolic cost of maintaining a unit length of vessel filled with blood is

b_V

per unit cross-sectional area, both the same throughout the tree. The total cost per unit length of a vessel of radius

r

carrying flow

Q

is the sum of the pumping power per unit length (from Poiseuille's law) and the metabolic cost of its blood volume:

\mathcal{C}(r) = \frac{8\eta Q^2}{\pi r^4} + b_V\pi r^2.

((a)) Minimise

\mathcal{C}

with respect to

r

at fixed

Q

and show that the optimum satisfies

Q = k r^3

for a constant

k

that is the same for every vessel in the tree. ((b)) Use conservation of flow at the branch together with the result of part (a) to derive Murray's law,

r_0^3 = r_1^3+r_2^3

. ((c)) The parent radius is

r_0 = 1.80\,\mathrm{mm}

and one daughter has radius

r_1 = 1.50\,\mathrm{mm}

. Find the radius

r_2

of the other daughter. ((d)) Show, using

Q=kr^3

, that the Poiseuille wall shear stress

\tau_{\rm w} = 4\eta Q/(\pi r^3)

is the same in the parent and both daughters, and compute the ratio of the daughters' total cross-sectional area to the parent's cross-sectional area for this branch.

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