In the figure below, a liquid is shown flowing through a horizontal pipe in steady-state fashion. The cross-sectional area of the pipe at location A is larger than at location B. Using the Bernoulli equation and the concept of mass conservation, explain why there is a difference in the kinetic energy of the liquid between locations A and B.
Answers
According to the Bernoulli equation, (1/2)VB2 − (1/2)VA2 = (PA − PB)/ρ. Since the cross-sectional area at location B is less than at location A, then VB > VA, by mass conservation. Therefore, according to the Bernoulli equation, the kinetic energy, per unit mass of liquid, at location B is greater than at location A. This means that the fluid pressure at location A is greater than the fluid pressure at location B.
When a fluid flows into a narrower channel, its speed increases. That means its kinetic energy also increases. Where does that change in kinetic energy come from? The increased kinetic energy comes from the net work done on the fluid to push it into the channel and the work done on the fluid by the gravitational force, if the fluid changes vertical position. Recall the work-energy theorem,
W
net
=
1
2
m
v
2
−
1
2
m
v
0
2
.
There is a pressure difference when the channel narrows. This pressure difference results in a net force on the fluid: recall that pressure times area equals force. The net work done increases the fluid’s kinetic energy. As a result, the pressure will drop in a rapidly-moving fluid, whether or not the fluid is confined to a tube.
There are a number of common examples of pressure dropping in rapidly-moving fluids. Shower curtains have a disagreeable habit of bulging into the shower stall when the shower is on. The high-velocity stream of water and air creates a region of lower pressure inside the shower, and standard atmospheric pressure on the other side. The pressure difference results in a net force inward pushing the curtain in. You may also have noticed that when passing a truck on the highway, your car tends to veer toward it. The reason is the same—the high velocity of the air between the car and the truck creates a region of lower pressure, and the vehicles are pushed together by greater pressure on the outside. (See Figure 1.) This effect was observed as far back as the mid-1800s, when it was found that trains passing in opposite directions tipped precariously toward one another.
An overhead view of a car passing by a truck on a highway toward left is shown. The air passing through the vehicles is shown using lines along the length of both the vehicles. The lines representing the air movement has a velocity v one outside the area between the vehicles and velocity v two between the vehicles. v two is shown to be greater than v one with the help of a longer arrow toward right. The pressure between the car and the truck is represented by P i and the pressure at the other ends of both the vehicles is represented as P zero. The pressure P i is shown to be less than P zero by shorter length of the arrow. The direction of P i is shown as pushing the car and truck apart, and the direction of P zero is shown as pushing the car and truck toward each other.
Figure 1. An overhead view of a car passing a truck on a highway. Air passing between the vehicles flows in a narrower channel and must increase its speed (v2 is greater than v1), causing the pressure between them to drop (Pi is less than Po). Greater pressure on the outside pushes the car and truck together.
MAKING CONNECTIONS: TAKE-HOME INVESTIGATION WITH A SHEET OF PAPER
Hold the short edge of a sheet of paper parallel to your mouth with one hand on each side of your mouth. The page should slant downward over your hands. Blow over the top of the page. Describe what happens and explain the reason for this behavior.
Bernoulli’s Equation
The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli’s equation, named after its discoverer, the Swiss scientist Daniel Bernoulli (1700–1782). Bernoulli’s equation states that for an incompressible, frictionless fluid, the following sum is constant:
P
+
1
2
ρ
v
2
+
ρ
g
h
=
constant
,
where P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli’s equation becomes
P
1
+
1
2
ρ
v
1
2
+
ρ
g
h
1
=
P
2
+
1
2
ρ
v
2
2
+
ρ
g
h
2
.