Bernoulli's Principle

Have you ever stood in the shower and noticed that the shower curtain tends to wrap itself against you? Why does your seem to get pulled towards a large truck on the highway when it passes you? These are examples of a phenomenon described by Daniel Bernoulli: When the velocity of a fluid is high, the pressure is low, and when the velocity of a fluid is low, the pressure is high. We can use this principle to show why airplanes fly, but first, let's consider some principles of fluid flow.

Fluid Dynamics is the area of physics that deals with the flow of fluids through and around an object. We use fluid dynamics to describe the flow principles of liquids in a pipe, air around an airplane wing, and even the motion of a boat hull through the water. How fluids flow are as important as the amount of fluids that can flow. For example, when fluids flow smoothly along a surface without creating bubbles or turbulence, we call this laminar flow. Laminar flow provides a smooth path - it is the fluid equivalent of removing or minimizing friction. The opposite of this is turbulent flow. Turbulent flow occurs when there are currents, eddys, whirlpools, and rough spots along the flow path. Turbulence on an airplane can lead to a rough ride. On the hull of a ship, turbulent flow can slow the ship down. On a submarine,  it can create unwanted noise which could allow the submarine to be detected. In critical fluid systems, such as the cooling channels in a nuclear reactor vessel, turbulent flow can cause poor heat transfer between the fuel cells and the cooling medium and lead to possible overheating or meltdown of the core.
Fluid engineers spend a lot of time trying to minimize turbulence in fluid systems.
When a fluid passes through a closed system, such as a pipe, the speed of the fluid changes depending on the size of the pipe. we use a term called mass flow rate to define how much fluid is flowing through a pipe at a given time. Mass flow rate is defined as the mass per unit time and is measured in kg/sec.  The formula is:   

Let's take a look at a  system:

The mass flow rate through the pipe must be constant even though the pipe narrows down. So we can say that

but from our definition of density, mass is equal to density times volume, or m = rV. Thus,

m1= r1V1 and m2= r2V2.

And from the drawing, we can see that the volume is the cross-sectional area times the change in length for each section of pipe. So V1 = A1DL1 and V2 = A2DL2. Substituting these values in to the above equation, we get:

We also know that the change in length per unit time is our definition of velocity (v). Substituting again,

We need to note that here, the v is the velocity of the fluid, not the volume. This is called The Equation of Continuity. For incompressible or nearly incompressible fluids, such as liquids in a pipe, r1= r2. So our equation may also be written:

So how does Bernoulli fit into this? Through his work, he found a formula that allows us to account for fluid flows in systems with different pressures and heights in the system, such as in the diagram to the right. Bernoulli's equation states:

or in a system,

where P is the pressure of the system, r is the density of the fluid, y is the height of the fluid, g is the gravitational
constant, and v is the velocity of the fluid at a point.
Evangelista Torricelli, a student of Galileo, had actually provided a special case of Bernoulli's equation nearly a century before Bernoulli. When both points are at the same pressure, such as when the system is opened to atmospheric pressure (for example, a draining tank), then the velocity may be found from:

This is known as Torricelli's Theorem.

So why does an airplane fly? The wind flowing over the wing flows faster than the wind other the wing, which
creates a low pressure condition on the top of the wing. With higher pressure on the bottom, there is more force pushing up on the wing. This force is called lift. Similarly, in your shower, the water flowing past the shower
curtain causes the air to speed up on one side of the curtain creating a low pressure area. The curtain is blown in towards you by the higher pressure and higher force on the other side of the curtain.
Finally, we can take the work of Bernoulli, Torricelli, Venturi, and others, and make a simple statement:

Nothing in Physics Sucks!

For more on these subjects, try:

http://www.allstar.fiu.edu/aerojava/pic3-2.htm

http://www.ce.utexas.edu/prof/KINNAS/319LAB/Applets/Venturi/venturi.html (Venturi Tube Simulation)

For Practice Problems, Try: Giancoli Multiple Choice Practice Questions