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Streamlines and Fluid Flow
Learning Goal: To understand the continuity equation.
Streamlines represent the path of the flow of a fluid. You can imagine that they represent a time-exposure photograph that shows the paths of small particles carried by the flowing fluid. The figure shows streamlines for the flow of an incompressible fluid in a tapered pipe of circular cross section. The speed of the fluid as it enters the pipe on the left is v_1. Assume that the cross-sectional areas of the pipe are A_1 at its entrance on the left and A_2 at its exit on the right.
Part A  
Find F_1, the volume of fluid flowing into the pipe per unit of time. This quantity is also known as the volumetric flow rate.
Part A.1 Find the volume of fluid entering the pipe
The volumetric flow rate has units of volume per unit time (cubic meters per second). What is the volume DeltaV of fluid entering the pipe in time Deltat?
Part A.1.a  

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Express your answer in terms of any or all of the following quantities: v_1, A_1, and Deltat.
ANSWER:
  DeltaV =  A_1 * v_1 * Deltat 
The volume of fluid entering the pipe per unit time is \Delta V /\Delta t.
Express the volumetric flow rate in terms of any of the quantities given in the problem introduction.
ANSWER:
  F_1 =  v_1 * A_1 
Part B  
Because the fluid is assumed to be incompressible and mass is conserved, at a particular moment in time, the amount of fluid that flows into the pipe must equal the amount of fluid that flows out. This fact is embodied in the continuity equation. Using the continuity equation, find the velocity v_2 of the fluid flowing out of the right end of the pipe.
Part B.1  

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Part B.2  

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Express your answer in terms of any of the quantities given in the problem introduction.
ANSWER:
  v_2 =  v_1*A_1/A_2 
Part C  
If you are shown a picture of streamlines in a flowing fluid, you can conclude that the __________ of the fluid is greater where the streamlines are closer together.
Enter a one-word answer.
ANSWER:
 speed 
Thus the velocity of the flow increases with increasing density (number per unit area) of streamlines.
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