Before we begin here’s a few definitions to make following this post easier.

1. Kinetic energy – noun. the energy of a body or a system with respect to the motion of the body or of theparticles in the system.

2. Potential Energy – noun. the energy of a body or a system with respect to the position of the body or thearrangement of the particles of the system

Birds, planes, and everything else with wings are able to fly because of lift, the phenomenon explained by Bernoulli’s Theorem and Equation.

They are as follows:

**Bernoulli’s Theorem:** “…the total mechanical energy of the flowing fluid, comprising the energy associated with fluid pressure, the gravitational potential energy of elevation, and the kinetic energy of fluid motion, remains constant.”1

**Bernoulli’s Equation:**

** **

This equation may look difficult and confusing but if we break it down its extremely manageable. Lets begin by looking at the underlying principle.

Essentially what Bernoulli found was that the sum of all of the energy in a system, from all sources ie. potential energy, kinetic energy, and the internal energy from fluid pressure, adds up to a constant. This can be seen in the equation above.

The formula for kinetic energy of a fluid is 1/2 times the density of the fluid times the velocity its moving squared. Velocity is given by V and density by the Greek letter rho ρ. This is the first part of Bernoulli’s equation.

The formula for potential energy of a fluid is the density times the acceleration due to gravity times the height of the fluid above the source of the gravity (normally earth). Acceleration from gravity is given by g and the height is given by z.

The final part is simply the pressure of the fluid given by p.

Now, why is this important at all? Who really cares if they add up to some constant that seems to mean nothing? Well its important because the identical fluids will always have the same constant. So if you can calculate how changing one variable effects the others. For example, we can change the velocity at which a fluid flows, and calculate from the expression what the change in pressure would be if we kept the other variables constant.

This allows us to get back to flight, and the applications of the Bernoulli Equation. Airfoils (wings are essentially big airfoils), cause the air above the wing to be at a lower pressure than below the wing because of their shape. Lower pressure means the air above the wing moves faster. Recall that as you decrease pressure you need to increase one of the other variables to still equal the constant. If the fluid is moving horizontally (or the wing is moving through the fluid horizontally) the height and acceleration due to gravity wont change. The fluids density wont change either so the only way for it to equal the same value is if the velocity of the fluid increases. This continues to keep the pressure low and air speed fast on the top of the wing. On the bottom it is the exact opposite, higher pressure means lower fluid velocity.

Now, Bernoulli’s equation only shows that there will be a difference of pressure between the top and the bottom of a wing. The fact that this generates an upward force is an other topic all together, but the point is that it does. The pressure difference leads to a force that counters the weight of the object enabling it to stay in the air with out falling.

Here is a video that explains how Bernoulli’s Principle works with examples other than wings and flight.

And that is how Bernoulli’s theorem works and why it is important.

### References:

1. “Bernoulli’s Theorem (physics).” *Encyclopedia – Britannica Online Encyclopedia*. Britannica Online Encyclopedia. 25 Sept. 2011.

2. “Bernoulli’s Principle.” *Wikipedia, the Free Encyclopedia*. Wikipedia. 25 Sept. 2011.

3. Fishbane, Paul M., Stephen Gasiorowicz, and Stephen T. Thornton. “16-7 Bernoulli’s Equation.” *Physics for Scientists and Engineers*. Upper Saddle River, NJ: Pearson/Prentice Hall, 2005. Print.

## 3 Comments

Lorena Barba posted on September 25, 2011 at 7:24 pm

Hi Ryan,

Can you point us to the tool you used to generate the equation?

Ryan Erf posted on September 25, 2011 at 9:31 pm

Sure, I found the equation on Wikipedia, highlighted it, right clicked the highlighted section, and clicked “view source selection” in the menu. This brings up the HTML source code for that element of the web page. You simply copy and paste it into the HTML editor on the blog and it generates the same image.

Lorena Barba posted on October 3, 2011 at 6:04 pm

One important detail to keep in mind is that the equation applies *along a streamline*, that is: for a particular parcel of fluid, as it moves along, Bernoulli’s equation applies. But it would not apply to unrelated parcels of fluid.