The momentum equation for this flow can be written in terms of the flow
through a small control volume. The change in momentum per unit time is:
ρ S V (V+dV) - ρ S V2
= ρ
S V dV
This change in momentum arises from the pressures acting along the faces
of the control volume:
pressure force (ends) = pS - (p+dp)(S+dS) = -p dS - S dp
pressure force (sides) = (p + dp/2) dS = p dS (to first order)
pressure (total) = -S dp
Equating the force due to pressure with the force required to produce the
momentum change yields:
-S dp = ρ S V dV
or dp = -ρ V dV
This is a simple form of the Euler equation.
In the case that ρ = constant,
the above equation may be integrated to produce the incompressible form of the Bernoulli equation:
p2 + ρ/2 V22 = p1 + ρ/2
V12
or:
p + ρ/2 V2 = pt