A renal artery is modelled as a thin-walled

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Biophysics & Medical Physics · Bioelectricity, the senses and the physics of the body

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

Problem

A renal artery is modelled as a thin-walled, incompressible, inviscid elastic tube of radius

r = 2.5\,\mathrm{mm}

, wall thickness

h_{\rm w} = 1.0\,\mathrm{mm}

, carrying blood of density

\rho = 1060\,\mathrm{kg\,m^{-3}}

. Here

h_{\rm w}/r = 0.4

is not small, so the thin-wall results below give an estimate, not an exact speed. The wall obeys Hooke's law with Young's modulus

E

, and wave reflections are ignored. ((a)) From the hoop stress

\sigma_\theta = pr/h_{\rm w}

and Hookean strain

\Delta r/r = \Delta\sigma_\theta/E

, derive

\mathrm{d}A/\mathrm{d}p = 2\pi r^3/(Eh_{\rm w})

for the cross-sectional area

A=\pi r^2

. ((b)) Starting from linearised continuity,

\partial A/\partial t + A_0\,\partial u/\partial x = 0

, and momentum,

\rho\,\partial u/\partial t = -\partial p/\partial x

, combine them into a wave equation for

p(x,t)

and show that the pulse-wave speed is

c_{\rm pw} = \sqrt{Eh_{\rm w}/2\rho r}

. ((c)) In a healthy young wall,

E = 0.50\,\mathrm{MPa}

. Evaluate

c_{\rm pw}

. ((d)) With age the wall stiffens to

E = 2.0\,\mathrm{MPa}

, with

h_{\rm w}

and

r

unchanged. Find the new

c_{\rm pw}

and state, in words, how

c_{\rm pw}

scales with

E

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