Bernoulli's equation

Bernoulli’s equation is the application of the conservation of energy to a fluid in motion, under the assumptions that it has constant density, there is no viscosity and the flow is steady. Although these conditions could seem very restrictive, in fact Bernoulli’s equation has many applications both in science and engineering.

Consider the incompressible fluid that flows with no viscosity at a steady flow through the pipe shown in the figure below:

A certain amount of fluid Δm (usually called mass element) flows through the left side of the pipe in a time Δt (variables written in blue). Since the continuity equation must hold, the same mass element Δm will have to flow through the right side of the pipe (variables in red).

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The work done by the forces acting on a particle can be related to the kinetic energy by:

The following forces are acting on the mass element Δm:

    • Gravitational force (weight)
    • The force due to the pressure difference at both sides of the pipe.

Therefore, the works done on Δm are Wg (due to gravity) and Wp (due to the pressure difference). Substituting into the equation above:

Since weight is a conservative force, we can write:

As for the work due to the pressure difference:

Where the minus sign is due to the fact that Δm moves against the force exerted by the fluid on the right side of the pipe.

Substituting both expressions for the works as well as the kinetic energy:

Dividing both members by ΔV and rearranging terms (assuming that ρ1 = ρ2 = ρ):

This is Bernoulli’s equation. Its unit is Pascal.

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