Apply the master energy equation to a fluid flowing in a pipe.

U₁ + K₁ + W_nc = U₂ + K₂

This relates the energy at point 1 in the fluid to the energy at some other point 2. The potential energy we'll consider is gravitational. Any work done can be written as a force multiplied by a distance:

W = F_net Δx = ΔPAΔx

so:

mgy₁ + ½mv₁² + ΔPAΔx = mgy₂ + ½mv₂²

ΔP is the pressure difference between points 1 and 2:

ΔP = P₁ - P₂.

The energy expression becomes:

mgy₁ + ½mv₁² + P₁AΔx = mgy₂ + ½mv₂² + P₂AΔx

If we divide through by volume we get energy/volume, which is energy density. Mass over volume is mass density, and AΔx = volume, so the energy density relationship is:

ρgy₁ + ½ρv₁² + P₁ = ρgy₂ + ½ρv₂² +P₂

This is Bernoulli's equation. Combining this with the continuity equation allows us to relate pressures, speeds, and heights at any two points in a flowing fluid.