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At the right is shown a pipe with a uniform circular cross-section carrying a fluid at steady flow. The piece of the pipe we are considering has a length, \( L \), and a cross-sectional area, \( A \). The fluid is flowing at a rate of \( Q \). The pressure at the beginning of the segment is \( p_1 \), and at the end of the segment is \( p_2 \). The fluid has a viscosity, \( \mu \).
1. Starting with the original pipe, if the pressures are maintained at the same values at the ends of the pipe segment but the pipe is replaced by a narrower one (smaller radius), what do you expect will happen to \( Q \), the rate of flow through the pipe?
A. It will be greater than it was originally.
B. It will be less than it was originally.
C. It will stay the same as it was originally.
D. There is not enough information given to decide.
2. If the pressures are maintained at the same values at the ends of the pipe segment but the pipe is replaced by a longer one, what do you expect will happen to \( Q \), the rate of flow through the pipe?
A. It will be greater than it was originally.
B. It will be less than it was originally.
C. It will stay the same as it was originally.
D. There is not enough information given to decide.
3. In the figure at the right are given four equations. One of them is the correct Hagen-Poiseuille equation that describes the flow of fluid through a pipe. Which is it?
A. \(\Delta p = \left(\frac{8 \mu L}{\pi r^4}\right) Q\)
B. \(\Delta p = \left(\frac{8 \mu L}{A}\right) Q\)
C. \(\Delta p = \left(\frac{8 \mu A}{\pi r^4}\right) Q\)
D. \(\Delta p = \left(\frac{8 \mu L}{r^2}\right) Q\)
4. Explain how your choice for 3 implies the results for 1 and 2.
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