   Chapter 9, Problem 58P

Chapter
Section
Textbook Problem

A hypodermic needle is 3.0 era in length and 0.30 mm in diameter. What pressure difference between the input and output of the needle is required so that the flow rate of water through it will be 1 g/s? (Use 1.0 × 10−3 Pa · s as the viscosity of water).

To determine
The required pressure difference between the input and output of the needle.

Explanation
According to the principle of Poiseuille, the pressure difference required producing volume flow rate of the given fluid with viscosity η through a tube of radius R and length L is ΔP=[8ηL(ΔV/Δt)]/πR2 . Now the volume flow rate is defined as ΔV/Δt=[(Δm/Δt)]/ρ . Using all these together, the pressure difference is calculated.

Given info: The water flow rate is 1g/s , density of water is 103kg/m3 , viscosity of water is 1.0×103Pas , length of the needle is 3.0cm , and the radius of the needle is 0.30mm .

The formula for the pressure difference is,

ΔP=[8ηL[(Δm/Δt)]/ρ]πR2

• η is viscosity of water.
• L is length of the needle.
• R is radius of the needle.
• Δm/Δt is mass flow rate.
• ρ is density of water.

Substitute 1g/s for Δm/Δt , 103kg/m3 for ρ , 1

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