Example 12.8 from College Physics, 12.4 Viscosity and Laminar Flow; Poiseuille’s Law

What Pressure Produces This Flow Rate?

An intravenous (IV) system is supplying saline solution to a patient at the rate of $0.120 {cm}^{3} /s$ through a needle of radius 0.150 mm and length 2.50 cm. What pressure is needed at the entrance of the needle to cause this flow, assuming the viscosity of the saline solution to be the same as that of water? The gauge pressure of the blood in the patient’s vein is 8.00 mm Hg. (Assume that the temperature is $20ºC$ .)

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Assuming laminar flow, Poiseuille’s law applies. This is given by

$Q = \frac{(P_{2} - P_{1}) π r^{4}}{8 η l},$
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where $P_{2}$ is the pressure at the entrance of the needle and $P_{1}$ is the pressure in the vein. The only unknown is $P_{2}$ .
Solution
Solving for $P_{2}$ yields

$P_{2} = \frac{8 η l}{{πr}^{4}} Q + P_{1 .}$
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$P_{1}$ is given as 8.00 mm Hg, which converts to $1 . 066 \times 10^{3}$ . Substituting this and the other known values yields
$\begin{array}{l} P_{2} & = & [\frac{8 (1 . 00 \times 10^{- 3} N \cdot {s/m}^{2}) (2 . 50 \times 10^{- 2} m)}{π (0 . 150 \times 10^{- 3} m)^{4}}] (1 . 20 \times 10^{- 7} m^{3} /s) + 1 . 066 \times 10^{3} {N/m}^{2} \\ = & 1 . 62 \times 10^{4} {N/m}^{2} . \end{array}$
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Discussion
This pressure could be supplied by an IV bottle with the surface of the saline solution 1.61 m above the entrance to the needle (this is left for you to solve in this chapter’s Problems and Exercises), assuming that there is negligible pressure drop in the tubing leading to the needle.

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Strategy

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