Fluid mechanics
Bernoulli Equation
Cairo University
, Euler’s Equation
Consider an inviscid, steady flow of fluid
along a streamline.
Consider a small
cylindrical element of
sectional area dA, mass
m and length ds in
direction of streamline S.
Element mass,
m = ρ dA ds
, Euler’s Equation (cont.)
Consider forces acting on element in s-direction.
Fs = p.dA – (p + dp) dA – ρ.dA.ds.g.cos
But cos = dz/ds
Fs = p.dA – p.dA - dp.dA – ρ.dA.ds.g.dz/ds
= - dp.dA - ρ.g.dA.dz
From Newton’s 2nd law of motion:
Fs = m.dV/dt = m.V.dV/ds = - dp.dA - ρ.g.dA.dz
Bernoulli Equation
Cairo University
, Euler’s Equation
Consider an inviscid, steady flow of fluid
along a streamline.
Consider a small
cylindrical element of
sectional area dA, mass
m and length ds in
direction of streamline S.
Element mass,
m = ρ dA ds
, Euler’s Equation (cont.)
Consider forces acting on element in s-direction.
Fs = p.dA – (p + dp) dA – ρ.dA.ds.g.cos
But cos = dz/ds
Fs = p.dA – p.dA - dp.dA – ρ.dA.ds.g.dz/ds
= - dp.dA - ρ.g.dA.dz
From Newton’s 2nd law of motion:
Fs = m.dV/dt = m.V.dV/ds = - dp.dA - ρ.g.dA.dz
