Euler Equation for Fluids

Euler's equation applies Newton's second law to an inviscid ideal fluid.

Conceptual context

Why does water accelerate when it passes through a narrow section of pipe, and why does pressure fall exactly where velocity is greatest? Euler's equation answers that question from first principles: it is Newton's second law (force equals mass times acceleration) applied to a fluid element instead of a rigid body, when viscosity can be neglected.

Instead of a point force, a pressure gradient appears; instead of the mass of a block, the density of the fluid appears; instead of the acceleration of a fixed point, the acceleration of a fluid particle travelling with the stream appears.

This leaf has two layers. The first is Euler's equation itself: a general differential law that describes the fluid at any point and at any instant, but which cannot be solved as a simple algebraic rearrangement without fixing a concrete geometry. The second is its integrated consequence for the most common case in practice (steady, incompressible, inviscid flow along a streamline): Bernoulli's equation, which is an algebraic relation between concrete numbers and is the one this leaf lets you calculate. The magnitudes that carry this story are $\rho$, $v$, $p$, $g$, $D\vec v/Dt$, $t$ and $\Phi$, in addition to the conserved quantity, $B$.

🟢 Essential level

The essential idea is simple to state, though deep in its consequences: in an ideal fluid, the only way to accelerate a fluid element is through an imbalance between the pressure pushing it and the body forces (such as gravity) acting on it. If those two influences balance exactly, the fluid element does not accelerate, although it may continue to move at constant velocity.

The most useful consequence of this idea, for a student who is starting out, is the one that connects velocity and pressure: in the same stream tube, where the fluid accelerates (for example, when entering a narrowing) the pressure must fall, and where the fluid slows down the pressure rises.

It is not magic or a memorised rule; it is the same Newton's second law read in a fluid instead of a solid. Before calculating anything, it is worth asking: in which section does the fluid gain speed? That is the clue to where the pressure will fall.

It is also worth distinguishing two different questions that are often confused: "how does the velocity of a fluid at a fixed point in space change as time passes?" and "how does the velocity of a particular fluid particle change while it travels?". The first is a photograph; the second follows the fluid on its journey. Euler's equation is about the second, and that is why it needs the concept of material acceleration, $D\vec v/Dt$.

🔵 Formal level

Basic formulas

Before using Euler's equation, confirm that the fluid is ideal, that is, without viscosity, because the equation contains no friction term. If there are boundary layers beside walls or wakes, the approximation stops being valid in those regions.

The formal level rests on four relations, each with a different role:

Material derivative of velocity

Euler's equation for ideal fluids

Continuity for incompressible flow

Bernoulli's equation (steady flow)

Real student questions

"Why does pressure fall where the fluid moves faster, if intuitively I would expect the opposite?" Because the total specific energy is conserved: the energy the fluid gains as kinetic energy must be lost as pressure energy; it cannot appear from nothing.

"Can I apply Bernoulli between a point inside a pipe and a point in open air?" Only if both belong to the same streamline (or the flow is irrotational) and the assumptions of incompressibility and absence of viscosity remain reasonable along the whole path between them.

Final synthesis

By the end of this leaf, the student should be able to explain why Euler's equation is Newton's second law applied to a fluid, recognise that its form that can be calculated in practice is Bernoulli's equation integrated along a streamline, and use it to relate pressure, velocity and height between two points whenever the assumptions of steady, incompressible and inviscid flow are reasonable between them.

The relevant final check is experimental: in a real Venturi tube, measure the pressure drop at the constriction and compare it with the Bernoulli prediction using the velocity increase expected from flow-rate continuity. The difference between what is measured and what is predicted is a direct measure of how far the real fluid departs from the ideal model.