A small arteriole has radius and carries blood with mean speed

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 small arteriole has radius r=50 μmr = 50\,\mathrm{\mu m} and carries blood with mean speed v=5.0 mm s−1v = 5.0\,\mathrm{mm\,s^{-1}}, pulsing at the heart rate f=1.5 Hzf = 1.5\,\mathrm{Hz}. Take blood density ρ=1060 kg m−3\rho = 1060\,\mathrm{kg\,m^{-3}} and viscosity η=3.5 mPa s\eta = 3.5\,\mathrm{mPa\,s}. ((a)) Find the Womersley number α=rρω/η\alpha = r\sqrt{\rho\omega/\eta} for this vessel and say whether the flow profile is quasi-steady or flattened. ((b)) Find the Reynolds number Re=ρv(2r)/η\mathrm{Re} = \rho v (2r)/\eta for the same vessel and decide, using the criterion for the Stokes limit met in low-Reynolds-number flow, whether inertia matters anywhere in this vessel, including where it branches or bends. ((c)) State whether Poiseuille's law can be used for this arteriole, and say what each of the two numbers separately contributes to that verdict.

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