Bernoulli's Equation

• Section 5.1 - Understanding Bernoulli's Equation

Section 5.1 - Understanding Bernoulli's Equation

It is quite remarkable to understand how an airplane flies. The lift forces the wings generate by slicing through the air can be calculated from Bernoulli's equation. Essentially the equation states that the total energy of a fluid is a constant. Before understanding Bernoulli's equation let us first understand what defines a fluid.

A fluid is a continuous and shapeless substance. It has the unique property of assuming the shape of whatever container it is put in. Its molecules can move freely past one another. Additionally fluids are incompressible, i.e their volume does not change under increasing pressure. Under low speeds, gases also behave like fluids and can be treated as incompressible.

As fluids travel through a container, their energy comes from three sources:

1. Kinetic energy from the movement of the fluid particles
2. Potential energy from the fluid under pressure
3. Potential energy from the height of the fluid, i.e gravitaional potential energy

To make things simple, let us assume we have a fluid moving through a straight level pipe. So there is no change in height of the fluid. So we can ignore the the third source of energy, gravitational potential energy. This reduces the total energy of a moving fluid to just:

1. Kinetic energy from the movement of the fluid particles
2. Potential energy from the fluid under pressure

The first , kinetic energy is easiest to understand. We have already learned that the kinetic energy of a mass in motion is simply 1/2 mv^2.

This leaves us with the last source, potential energy from a fluid under pressure. This source is the most difficult to grasp. One analogy is a spring being compressed with some force. Since a fluid is incompressible, then it has no stored energy like a spring. So how is this potential energy stored?

The easiest way to understand this is to think of fluid in a piston or pump under pressure. If the pump is closed, then any pressure you apply results in no movement. However, if you open the end of the pump then the pressure you apply will drive all the liguid or air out of the pump. The work done by this pressure to push the fluid out of the piston is simply equal to the pressure times the distance.

Since pressure is force per unit area, we can rewrite the work done as force times volume of the liquid, where volume is area time distance travelled perpendicular to the force:

Therefore the volume of fluid under pressure has a potential energy equal to its volume times the pressure. Now we can move on to consider a fluid moving through a pipe with a change in cross-section. From the principle of the conservation of mass we know that the volume of fluid flowing through any cross-section must be the same. So if we slice the pipe at the large cross section we can calculate the volume flowing through it to be, density * velocity * cross-section area. At the smaller cross section we have a similar equation. Setting these two equal to each other yields:

As the cross-sectional area decreases, the velocity of the fluid increases.

Now from the law of conservation of energy, the energy of the fluid at these two cross-sections must also be equal. When the velocity increases in the narrow cross section, the fluid's kinetic energy increases. To compensate for this increase, the potential energy due to fluid pressure must drop. So we have:

We can note that mass equals the density times the volume of the liquid. Also the volume of the fluid going through each section of the pipe is equal to each other. So after we divide out volume from the equation, the equation then simplifies to:

From the law of conservation of mass we have shown that:

Substituting this into our previous equation results in:

Therefore, as the cross-section changes the pressure in the fluid changes proportionally to the square of the difference in the ratio of the areas. In conclusion, we have shown that:

• As the cross-section changes, the velocity changes ( conservation of mass )
• As the cross-section changes, the pressure changes ( conservation of energy )

We assume the volume was constant in the two cross section changes in our derivation. However, if the volumes are not equal then the equation changes to include the length of the fluid under pressure. The longer the pipe is, the more energy is needed to pressurize it.

Now we can look at a simple airplane wing to understand how the Bernoulli equation can explain why they fly.