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A glucose solution being administered with an IV has a flow rate of $3.00 \textrm{ cm}^3 \textrm{/min}$. What will the new flow rate be if the glucose is replaced by whole blood having the same density but a viscosity 2.50 times that of the glucose? All other factors remain constant.
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Final Answer

$1.60 \textrm{ cm}^3\textrm{min}$

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OpenStax College Physics Solution, Chapter 12, Problem 31 (Problems & Exercises) (2:04)

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Video Transcript

This is College Physics Answers with Shaun Dychko. Poiseuille's Law says that the volume flow rate is equal to the difference in pressure along a horizontal tube, times pi times its radius to the power of four, divided by eight times the viscosity of the fluid in it times the length of the tube. Now most of these terms don't matter because the only thing that's going to change in this question is the viscosity because we're exchanging whole blood for glucose and they have different viscosities. So, the volume flow rate in case two with the whole blood divided by the volume flow rate in case one of glucose is going to be the change in pressure times pi <i>r</i> to the four over eight, viscosity two, viscosity of whole blood times the length of the tube. Then divide by <i>Q one</i> but I'm going to multiply by the reciprocal of <i>Q one</i> to make things a little easier here. So we're multiplying by eight <i>nu one l</i> over <i>delta p</i> pi <i>r</i> to the fourth. We see that most of these things cancel with the exception of the viscosities. So this works out to the ratio of the viscosities. So multiply both sides by <i>Q one</i> to solve for <i>Q two</i>. So <i>Q two</i> is <i>nu one</i> over <i>nu two</i> times <i>Q one</i>. This letter here, looks like the letter n but it's the Greek letter nu. Okay. So we're told that the viscosity of whole blood is two and a half times that of glucose and so we'll substitute 2.50 <i>nu one</i> in place of <i>nu two</i>. The viscosity of glucose cancels. So <i>Q two</i> is <i>Q one</i> divided by two and a half. So that's four cubic centimeters per minute, divided by two and a half which is 1.60 cubic centimeters per minute will be the volume flow rate of the whole blood.