Question

A drug is being delivered into a patient’s arm at a rate of 319 mL min–1. The drug is being delivered from a syringe through a needle with an internal diameter of 1 mm. This drug has a density of 1050 kg m–3. Is the flow laminar or turbulent? (ηdrug = 8.90×10–4 Pa s)

Answer 1

For the flow in a pipe (needle) the Reynold's number is what determines the laminar/turbulent flow. The Reynold's number is given as follows (see https://en.wikipedia.org/wiki/Reynolds_number#Flow_in_a_pipe):

$$Re = \dfrac{\rho Q D_H}{\mu A}$$

where $\rho = 1050kg/m^3$ is the density of the drug, $Q = 319\space mL/min = \dfrac{319}{60}\times 10^{-6}\space m^3/s$ is the volumetric flow rate, $D_H = 1mm = 10^{-3}m$ is the internal diameter of the needle, $A = \pi D_H^2/4$ is the cross-sectional area of the needle, and $\mu = 8.90\times 10^{-4}Pa\cdot s$ is the dynamic viscosity of the drug.

Thus, obtain:

$$Re = \dfrac{\rho Q D_H}{\mu\cdot \dfrac{\pi D_H^2}{4}} = \dfrac{4\rho Q}{\pi\mu D_H}\\ Re = \dfrac{4\cdot 1050\cdot \dfrac{319}{60}\times 10^{-6}}{\pi\cdot 8.90\times 10^{-4}\cdot 10^{-3}} \approx 7990$$

Since $Re>2900$, the flow is turbulent.

Answer. Turbulent.