A barometer, that is, a device that measures pressure, can be constructed from a tube which is open at both ends and shaped like a U. A fluid, like mercury, is placed inside, filling the bottom portion of the U. If the height of the mercury column at one end is greater than at the other end, then the pressure above the liquid on that end is less.
A modification of a barometer can be used to measure flow speed. Consider a pipe which carries an inviscid, incompressible fluid moving a speed v and pressure P1 far upstream. Barometer 1 does not interrupt the flow, so it measures P1. If Barometer 2 is placed in the flow as shown in the figure below, then the tip of the barometer forms an obstruction in the flow.
The flow just in front of the tip comes to stop at what is called a stagnation point. Nevertheless, we can consider the line shown in the figure, which comes to an end at the stagnation point, to be a streamline.
The figure below shows a flow with a constriction. The flow far upstream has a speed v3, pressure P3, and a cross-sectional area A3. In the constriction the flow has a velocity v4, pressure P4, and a cross-sectional area A4.
For the following problems, letρbe the density of mercury.
How does the pressure P4 compare with the pressure P3?