Ideal Fluid Flow and Euler's Equation

Ideal fluid flow models a fluid with no viscosity, whose momentum balance is Euler's equation and whose steady flow along a streamline obeys Bernoulli's theorem.

Definition

Ideal fluid flow is the motion of a fluid with negligible viscosity, governed by Euler's equation derived from momentum conservation together with the continuity equation, and yielding Bernoulli's relation between pressure and speed.

Scope

This topic covers the dynamics of inviscid fluids: the continuity equation for mass conservation, Euler's equation of motion for a fluid element, Bernoulli's theorem relating pressure and velocity along streamlines, the description of irrotational potential flow, and the conservation of circulation expressed by Kelvin's theorem. It is the idealized core of fluid dynamics.

Core questions

• How does Euler's equation express momentum conservation for a fluid element?
• What does Bernoulli's theorem say about pressure and velocity in steady flow?
• When is a flow irrotational, and how does potential-flow theory describe it?

Key concepts

• Continuity equation
• Euler's equation
• Bernoulli's theorem
• Streamlines
• Irrotational (potential) flow
• Circulation and Kelvin's theorem

Key theories

Euler's equation of fluid motion

For an inviscid fluid, the acceleration of a fluid element equals the pressure-gradient and body forces per unit mass, the inviscid form of Newton's second law applied to a continuum.

Bernoulli's theorem

In steady inviscid flow the sum of pressure, kinetic, and potential energy per unit volume is constant along a streamline, so faster flow corresponds to lower pressure.

Clinical relevance

Ideal-flow theory gives the leading explanation of aerodynamic lift, the operation of venturi meters and flow nozzles, and the pressure-velocity relations used in piping and ventilation design, providing tractable models wherever viscous effects are confined to thin layers.

History

Daniel Bernoulli's 1738 Hydrodynamica introduced the energy relation now bearing his name, and Euler formulated the general equations of inviscid fluid motion in the 1750s. Helmholtz and Kelvin in the nineteenth century developed the theory of vorticity and circulation, completing the classical theory of ideal flow.

Key figures

• Leonhard Euler
• Daniel Bernoulli
• Hermann von Helmholtz
• Lord Kelvin

Seminal works

• landaufluid1987
• batchelor2000

Frequently asked questions

Why does pressure drop where a fluid speeds up?

Bernoulli's theorem says the total of pressure and kinetic energy per unit volume is constant along a streamline in steady inviscid flow, so an increase in speed must be balanced by a decrease in pressure.

Is any real fluid truly ideal?

No real fluid is perfectly inviscid, but the ideal-flow model is accurate away from boundaries where viscous effects are confined to thin layers, making it a powerful approximation for many high-speed and large-scale flows.