Bernoulli’s equation is obtained from the Mechanical Energy Conservation Theorem and the relationship between mechanical work and the energy of bodies.
Representation of the flow of a fluid in a pipe
We consider for this figure an ideal fluid that has the following characteristics:
- Linear flow – constant velocity at any point in the fluid;
- Incompressible – with constant density;
- No viscosity ;
- irrotational flow.
In this case, the factors that interfere with the fluid flow are the pressure difference at the ends of the tube, the cross-sectional area and the height.
Since the liquid is moving at a certain height, it has gravitational potential energy and kinetic energy . Thus, the energy of each portion of fluid is given by the equations:
E 1 = mgh 1 + mv 1 2 and E 2 = mgh 2 + mv 2 2 2 2
Since the volumes and density of the two portions of the fluid are the same, we can replace the mass m in the above expression with:
m = ρ.V
The above equations can be rewritten as follows:
E 1 = ρ.V (gh 1 + 1v 1 2 ) and E 2 = ρ.V(gh 2 + 1v 2 2 ) 2 2
E 2 – E 1 = F 1 .S 1 – F 2 .S 2
The force can be obtained by the expression:
F = PA
Thus, the above equation can be rewritten as:
ρ.V(gh 2 + 1v 2 2 ) – ρ.V (gh 1 + 1v 1 2 ) = (P 1 – P 2 ) . V 2 2
By grouping the factors that have subscript 1 on the left side of the equality and those that have subscript 2, we can rearrange the expression above and obtain Bernoulli’s equation:
ρ.Vgh 1 + ρ.V. v 1 2 + P 1 .V = ρ.Vgh 2 + ρ.V. v 2 2 + P 2 .V
This equation can also be rewritten as follows:
ρ.Vgh + ρ.V. v 2 + PV = Constant
Bernoulli’s equation is the main equation in fluid mechanics studies and explains, for example, how airplanes stay in the air. The pressure exerted by the air passing through the plane’s wings is less than the pressure on its underside. This pressure difference creates an upward force, holding the plane in the air.