| 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 convection
∫S 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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