Bernoulli's equation

What is Bernoulli's equation?

Bernoulli's equation is a relationship between pressure and velocity at different parts of a moving incompressible fluid.

The following two assumptions must be met for this Bernoulli equation to apply:

1. the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline
2. friction by viscous forces has to be negligible.

Bernoulli's principle-

Bernoulli's principle states that for a incompressible, streamline and non viscous fluid, the work done by the pressure difference per unit volume plus the kinetic energy per unit volume plus the potential energy per unit volume is a constant.

Work done = Force  ∗ distance
Force = Pressure ∗ Area. Thus, Work = Presssure ∗ Area ∗ distance.

Thus, Work = Pressure ∗ Volume.

Therefore, the work done per unit volume = Pressure.

Kinetic energy = 1/2 ∗ mass ∗ velocity<sup>2

Therefore, kinetic energy per unit volume =</sup> 1/2 ∗ mass ∗ velocity²) / Volume.

Thus, kinetic energy per unit volume = 1/2 ∗ density ∗ velocity<sup>2

Similarly, potential energy = mass * gravitational acce. * height</sup>

Therefore, potential energy per unit volume =  (mass * gravitational acce. * height)/ volume = density * gravitational acce. * height.


If the density is ρ, pressure is p, height is h and velocity is v, from Bernoulli's principle,

P + 1/2 ρv² + ρgh = k , where k is a constant.

Thus for two instances where we apply Bernoulli's principle,

P₁ + 1/2 ρv₁² + ρgh₁    =    P₂ + 1/2 ρv₂² + ρgh₂

Applications of Bernoulli's theorem-

1. Airfoil lift

Consider the following shape of an airfoil.

(Photo credit: skybrary.aero)

The lift on an airfoil is primarily the result of its angle of attack and shape. An airfoil creates curved streamlines which results in lower pressure on one side and higher pressure on the other.

By Bernoulli's principle, 

P₁ + 1/2 ρv₁² + ρgh₁    =    P₁ + 1/2 ρv₁² + ρgh<sub>1

Since the effect from the gravitational acceleration can be ignored, the above equation, the above equation simplifies to</sub>


P₁ + 1/2 ρv₁²     =    P₂ + 1/2 ρv₂² 

Thus there is a pressure difference in the airfoil due to the difference of velocities. This pressure difference is proportional to a net force acting on the object, as the area is constant.

When the air above the airfoil moves at high speed than the lower part, due to the pressure difference, a force upwards acts on the airfoil. This results in the lifting off the airplane. 

(Photo credit: howstuffworks.com)

Similarly, when the velocity of the air upper is less than that in the lower parts, the net force is acting downwards. Thus, the airplane moves down.

When the velocities are equal, the net force is zero. Thus, the plane continues its motion horizontally.

2. Spinning ball

When a ball is spinning in the same direction as the air direction, the velocity of the ball will be increased. Similarly, when a ball is spinning in a direction opposite to the direction of the flowing air, it will decrease its velocity.

As described in the previous example, higher velocity will create lower pressures.

(Photo credit: http://cikguwong.blogspot.com)

Due to this, the pressure on one side of the ball is greater than the other side. This will create a varying force on the ball. 

(Photo credit: http://cikguwong.blogspot.com)

This results in a curved path of the spinning ball.