Bernoulli's theorem

Bernoulli's theorem

Bernoulli's Principle is one of the most versatile principle ever.
This is an important principle involving the movement of a fluid through a pressure difference. Suppose a fluid is moving in a horizontal direction and encounters a pressure difference. This pressure difference will result in a net force, which by Newton's 2nd law will cause an acceleration of the fluid. The fundamental relation,

in this situation can be written as

which furthermore can be expressed as

In other words,

which is known as Bernoulli's principle. This is very similar to the statement we encountered before for a freely falling object, where the gravitational potential energy plus the kinetic energy was constant (i. e., was conserved).
Bernoulli's principle thus says that a rise (fall) in pressure in a flowing fluid must always be accompanied by a decrease (increase) in the speed, and conversely, if an increase (decrease) in , the speed of the fluid results in a decrease (increase) in the pressure. This is at the heart of a number of everyday phenomena. As a very trivial example, Bernouilli's principle is responsible for the fact that a shower curtain gets ``sucked inwards'' when the water is first turned on. What happens is that the increased water/air velocity inside the curtain (relative to the still air on the other side) causes a pressure drop. The pressure difference between the outside and inside causes a net force on the shower curtain which sucks it inward. A more useful example is provided by the functioning of a perfume bottle: squeezing the bulb over the fluid creates a low pressure area due to the higher speed of the air, which subsequently draws the fluid up. This is illustrated in the following figure.

Bernouilli's principle also tells us why windows tend to explode, rather than implode in hurricanes: the very high speed of the air just outside the window causes the pressure just outside to be much less than the pressure inside, where the air is still. The difference in force pushes the windows outward, and hence explode. If you know that a hurricane is coming it is therefore better to open as many windows as possible, to equalize the pressure inside and out.
Another example of Bernoulli's principle at work is in the lift of aircraft wings and the motion of ``curve balls'' in baseball. In both cases the design is such as to create a speed differential of the flowing air past the object on the top and the bottom - for aircraft wings this comes from the movement of the flaps, and for the baseball it is the presence of ridges. Such a speed differential leads to a pressure difference between the top and bottom of the object, resulting in a net force being exerted, either upwards or downwards. This is illustrated in the following figure.

Bernoulli's Principle
When I was a kid, one way that I could torment my siblings was with the garden hose. This simple piece of equipment provided hours of fun for me because I could use it to spray them and soak them with water. This couldn't be achieved by simply holding the hose out toward them, though, because the water would only run out of the end and onto the ground. I had to use my thumb to block a portion of the hose's opening to make the water come out faster.
At such a young age, I had no idea who Daniel Bernoulli was or why partially blocking the end of the hose made the water come out faster. All that mattered to me was that I had a powerful weapon in a regular, everyday piece of garden equipment.
I no longer spray my siblings with a garden hose, but I do now know the physics behind why this was possible (and so much fun!). The reason the water spray increased when the hose opening was blocked is because a fluid increases speed when it flows through a narrower space. The water coming out of the hose traveled faster when part of the opening was blocked because it was flowing through a narrower space than when the entire end of the hose was open.
You can see same thing happen in rivers as they change width. Water running through the wider regions travels more slowly but speeds up as it passes through the narrower parts. But what you may not believe is that the pressure in these narrower regions decreases within the fluid as the water speeds up. Likewise, the pressure in the fluid increases as the water slows down in the wider regions.
This discovery was made by Swiss scientist Daniel Bernoulli and is called Bernoulli's principle. Bernoulli studied fluids in pipes and found that where the speed of a fluid increases, the internal pressure in the fluid decreases.
This is not an easy concept to grasp. In fact, you may be thinking that if the water is in a tighter space, the pressure should increase. Well, it does, but not the pressure within the fluid. The pressure increase is experienced by whatever is surrounding the fluid. In fact, it's this change in pressure that actually causes the fluid to change speed, not the other way around.