Answer to Question #109417 in Physics for Laura

Question

A needle with an internal diameter of 0.38 mm and length 4.1 cm is used to deliver an intravenous fluid of density 1050 kg m−3 and viscosity 12×10−4 Pa s into a vein.

If the pressure difference across the needle is 8.0 kPa then how many hours will it take to empty 1 litre of IV fluid through the needle? (to 2 s.f)

Answer 1

Assuming that the needle is of form of cylindrical pipe, according to Poiseuille law, the laminar flow through the needle is $Q = \frac{\pi d^4 \Delta p }{128 \eta l}$, where:

$d$ is the diameter of the needle, $\Delta p$ is the pressure difference across the needle,

$\eta$ is viscosity of the fluid, $l$ is the length of the needle.

Substituting the values into Poiseuille formula, obtain $Q = 8.32 \cdot 10^{-8} \frac{m^3}{s}$.

One litre is equal to $10^{-3} m^3$, hence the time to empty this volume through the needle is $t = \frac{V}{Q} = \frac{10^{-3}m^3}{8.32 \cdot 10^{-8} \frac{m^3}{s}} = 12023.23 s \approx 3.34 h$.