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The Euler momentum equation expresses Newton's second law for incompressible inviscid fluids: the acceleration of an infinitesimal fluid element equals the force acting on it (resulting from pressure and gravity).

Theorem:
In an incompressible inviscid fluid let ρ denote the density, u the velocity field, p the pressure and -F the body force (gravity). Then

ρDu/Dt = -p - F.

(D/Dt denotes the substantial derivative operator ∂/∂t + u.)

Proof:
Consider some fixed surface S enclosing a volume V. The total momentum of the fluid contained in V is ∫V ρu dV. This will change with time due to three factors: the force -∫S p dS exerted by the pressure on V, the body force -∫V F dV and the convectionS ρu(u.dS) of momentum across S. Thus

0 = ∂(∫V ρu dV)/∂t + ∫S p dS + ∫V F dV + ∫S ρu(u.dS)

This equation is the integral form of the Euler momentum equation. By applying the divergence theorem to the surface integrals we obtain

0 = ∫V (ρ∂u/∂t + p + F + ρ(u.)u) dV

Since this holds for any volume V the integrand must vanish everywhere, ie.

0 = ρ∂u/∂t + ρ(u.)u + p + F

QED

If we drop the assumption that the fluid is inviscid we must add a term proportional to 2u, which gives the Navier-Stokes equation (which is rather more difficult to derive).

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