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12.2 Bernoulli's Equation
A fluid's speed increases when it flows into a narrower channel.
That means it has more energy in it.
The net work done on the fluid pushes it into the channel and the work done on the fluid by the force of the air pushes it out of the horizontal position.
The channel narrows and there is a pressure difference.
The net work increases the fluid's power.
If the fluid is confined to a tube, the pressure will drop.
There are many examples of pressure dropping in fluids.
When the shower is on, shower curtains tend to bulge into the stall.
A region of lower pressure is created inside the shower by the high-velocity stream of water and air.
The curtain is pushed in by the net force of the pressure difference.
When you pass a truck on the highway, your car tends to go towards it.
The cars and trucks are pushed together by the higher pressure on the outside of the truck and the lower pressure on the inside of the car.
The effect was observed as far back as the mid- 1800s when it was found that trains in opposite directions tipped precariously towards one another.
A car passes a truck on a highway.
The air passing between the vehicles has to increase in speed in order to cause the pressure between them to drop.
The pressure on the outside pushes the car and truck together.
The total remains constant if we follow a small volume of fluid.
A form of the energy principle is Bernoulli's equation.
The second and third terms have the potential energy replaced by.
Each term in the equation has units of energy per volume.
The energy per unit volume is related to that.
The pressure has units of energy per unit volume as well.
Its units are.
We get energy per unit volume if we add these by m/m.
Bernoulli's equation is just a statement of how much energy is needed for an incompressible fluid.
Bernoulli's equation is based on the amount of energy applied to fluid flow.
Changes in the fluid's and per unit volume are caused by the net work done by the fluid.
Bernoulli's equation can be changed if other forms of energy are involved in fluid flow.
Thermal energy can be dissipated because of fluid viscosity.
There are three terms in the general form of Bernoulli's equation.
To understand it better, we will look at a number of situations that are easy to understand.
The situation where the fluid is static is very simple.
We can simplify the equation by taking all of the heights, just as we have done for other situations with the force of gravity.
The equation tells us that the pressure in static fluids increases with depth.
As we go from point 1 to point 2 in the fluid, the depth is greater than by an amount.
In the simplest case, zero at the top of the fluid is what we get.
Much of what we studied for static fluids in the preceding chapter is included in Bernoulli's equation for fluid flow.
There is a situation in which the fluid moves but its depth is unchanging.
It is Bernoulli's equation for fluids.
As we have discussed before, pressure drops as speed increases.
This can be seen from Bernoulli's principle.
If is greater than in the equation, then the equality to hold must be less than in the equation.
Bernoulli's principle applies because level flow means constant depth.
The subscript 1 is used for values in the hose and the subscript 2 is used for those in the nozzle.
The absolute pressure in the hose is greater than expected since it is greater in the nozzle.
The pressure in the nozzle must come from the atmosphere.
A number of devices and situations in which fluid flows at a constant height can be analyzed with Bernoulli's principle.
People have used the Bernoulli principle to move things around.
The high-velocity fluid pushes other fluids into the stream.
Entrainment is a process.
In draining swamps, fields, or other low-lying areas, pumps to raise water small heights have been used since ancient times.
Entrainment devices use increased fluid speed to create low pressures, and then entrain one fluid into another.
Both paint sprayers and carburetors use the same techniques to move liquids.
Aspirators can be used in a variety of ways, from producing a reduced pressure in a vessel to draining a flooded basement.
The airplane wing is an example of Bernoulli's principle.
The wing is tilted upward at a small angle and the upper surface is longer, which causes air to flow faster over it.
The net upward force is created when the pressure on the wing is reduced.
The shape of a wing is characteristic of sails.
The front side of the sail has less pressure than the back side.
You can sail into the wind if this results in a forward force.
For a good illustration of Bernoulli's principle, make two strips of paper, each about 15 cm long and 4 cm wide.
You can drape the strip over your finger if you hold the end up to your lips.
Take two strips of paper and put them up to your lips.
Two devices measure fluid velocity using Bernoulli's principle.
The manometer is small enough to not disturb the flow.
The tube facing the oncoming fluid creates a dead spot with zero velocity in front of it, while fluid passing the other tube has velocity.
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