5.3 Euler’s equation

With the acceleration of a fluid parcel written correctly as a material derivative, Newton’s second law can be applied to the parcel directly. For a fluid with no viscosity — an inviscid fluid — the only surface force on a parcel comes from pressure, and the resulting equation of motion is Euler’s.

The pressure force on a fluid element

Consider a small rectangular element of volume $dV = dx\,dy\,dz$ . Pressure pushes inward on every face. On the two faces perpendicular to $x$ , the left face at $x$ is pushed in the $+x$ direction with force $p(x)\,dy\,dz$ , and the right face at $x + dx$ is pushed in the $-x$ direction with force $p(x+dx)\,dy\,dz$ . The net $x$ -force is

\big[p(x) - p(x+dx)\big]\,dy\,dz \;=\; -\frac{\partial p}{\partial x}\,dx\,dy\,dz \;=\; -\frac{\partial p}{\partial x}\,dV.

∂p/∂x-2.00 Pa/m

p_L − p_R0.800 Pa

a = −∇p/ρ (ρ = 1)2.000 m/s²

Pressure gradient ∂p/∂x = -2.00 Pa/m Slab thickness dx = 0.40 m

Pressure pushes a slab from both sides. When the pressures are equal the slab is in static equilibrium; when they differ, the imbalance is a force per unit area = (p_L − p_R). For a slab of thickness dx that's −(∂p/∂x)·dx — equivalently, −∇p per unit volume. Newton's second law on this volume element is exactly Euler's equation: ρ·Du/Dt = −∇p.

The element is pushed not by pressure itself but by the difference in pressure across it. A uniform pressure squeezes from all sides and produces no net force; only a pressure gradient accelerates the fluid. Repeating the argument on all three axes, the net pressure force per unit volume is $-\nabla p$ .

Newton’s second law for the parcel

Force per unit volume equals mass per unit volume times acceleration, $\rho\,D\mathbf{u}/Dt$ . Adding gravity (or any body force) $\rho\mathbf{g}$ to the pressure force gives Euler’s equation:

\rho\,\frac{D\mathbf{u}}{Dt} \;=\; -\nabla p \;+\; \rho\mathbf{g}.

▶ Written out with the convective term Derivation

Expanding the material derivative from the previous lesson,

\rho\left(\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u}\right) \;=\; -\nabla p \;+\; \rho\mathbf{g}.

The left side is mass density times the parcel acceleration; the convective term $(\mathbf{u}\cdot\nabla)\mathbf{u}$ is the nonlinearity inherited from the kinematics of the previous lesson. The right side is the sum of surface force (pressure) and body force (gravity) per unit volume.

p_left

0.50

p_right

0.50

∂p/∂x = 0 → no net force → slab coasts at constant velocity.

A fluid parcel accelerates exactly as a particle does — down the pressure gradient, and under gravity — but it is continuously deformed by its neighbours as it goes, which is what the field description captures and a single particle trajectory cannot.

A closed system

Euler’s equation has three scalar components but four unknown fields: the three components of $\mathbf{u}$ and the pressure $p$ (the density $\rho$ is the fifth in a compressible flow). Continuity from the previous lesson supplies one more equation. For an incompressible flow the count closes: continuity becomes $\nabla\cdot\mathbf{u}=0$ , and pressure adjusts instantaneously to enforce it. For a compressible flow one more relation — an equation of state linking $p$ and $\rho$ — is needed to close the system. Euler’s equation governs inviscid flow exactly; the next refinement, viscosity, is what makes the equations describe real fluids near walls.