How does the conservation of energy apply to fluid flow?

In fluid dynamics, the conservation of energy manifests through principles like Bernoulli's principle and the compressible Euler equations. For steady fluid flow, the total of all forms of energy remains constant, meaning kinetic, potential, and internal energies balance each other. Leonhard Euler derived the foundational equations describing inviscid flow, which correspond to the Navier-Stokes equations with zero viscosity and thermal conductivity.

What we know

• The compressible Euler equations include equations for the conservation of mass, balance of momentum, and balance of energy. [2]
• Bernoulli's principle can be derived directly from the principle of conservation of energy, stating that in a steady fluid flow the total of all forms of energy is conserved. [3]
• The conservation of energy in steady fluid flow requires that the sum of kinetic energy, potential energy, and internal energy remains constant. [3]
• Leonhard Euler derived Bernoulli's equation in its usual form in 1752, expanding upon Daniel Bernoulli's earlier deductions. [3]

Where to go next

• the bigger pictureWhat are the Euler equations in fluid dynamics?
• how it workedHow does Bernoulli's principle relate pressure and speed?
• whoWho was Leonhard Euler?
• compared withHow do Navier-Stokes equations differ from Euler equations?
• what followedWhat causes turbulence in fluid flows?
• an unexpected connectionHow does fluid dynamics model fission weapon detonation?Fluid dynamics is not just for everyday liquids and gases like water or air, but also provides the systematic structure used to model extreme events like nuclear weapon detonations.

Sources

• [1]Euler equations (Wikidata Q375175) · CC0 1.0
• [2]Euler equations (fluid dynamics) · CC BY-SA 4.0
• [3]Bernoulli's principle · CC BY-SA 4.0
• [4]Fluid dynamics · CC BY-SA 4.0
• [5]Turbulence · CC BY-SA 4.0