What radius needle should be used to inject a volume of 500...

What radius needle should be used to inject a volume of $500 \mathrm{~cm}^{3}$ of a solution into a patient in $30.0 \mathrm{~min}$ ? Assume the length of the needle is $2.50 \mathrm{~cm}$ and the solution is elevated $1.00 \mathrm{~m}$ above the point of injection. Further, assume the viscosity and density of the solution are those of pure water, and that the pressure inside the vein is atmospheric.

Key Concepts

Poiseuille's Law

A principle in fluid mechanics that governs the laminar flow of a viscous fluid through a cylindrical tube. According to Poiseuille's Law, the volumetric flow rate is directly proportional to the pressure difference across the tube and the fourth power of its radius, while it is inversely proportional to the fluid's viscosity and the length of the tube. This relationship is essential in determining how variations in tube dimensions and fluid properties affect flow rates.

Hydrostatic Pressure

The pressure difference created by a fluid due to gravity, which can be calculated as the product of the fluid’s density, gravitational acceleration, and the height of the fluid column. In many fluid flow problems, including those involving injection or infusion through a tube, the hydrostatic pressure provides the driving force needed to overcome resistance in the system.

Fluid Viscosity

A measure of a fluid's internal resistance to flow and deformation. In the context of laminar flow through a tube, higher viscosity results in lower flow rates for a given pressure difference. Understanding viscosity is crucial when applying Poiseuille's Law and designing systems where controlled flow is required.

Volumetric Flow Rate

The quantity of fluid that passes through a given section of a tube per unit time, usually expressed in terms such as cubic centimeters per minute. It is a key parameter in fluid dynamics as it directly connects the physical dimensions of the flow conduit, pressure differences, and fluid properties in systems governed by laminar flow.

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