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 . Assume that the cross-sectional areas of the pipe are at its entrance on the left and at its exit on the right.
Part A
Find , 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 of fluid entering the pipe in time ?
Part A.1.a Part not displayed
Express your answer in terms of any or all of the following quantities: , , and .
 = A_1 * v_1 * Deltat
The volume of fluid entering the pipe per unit time is .
Express the volumetric flow rate in terms of any of the quantities given in the problem introduction.
 = 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 of the fluid flowing out of the right end of the pipe.
Part B.1 Part not displayed
Part B.2 Part not displayed
Express your answer in terms of any of the quantities given in the problem introduction.