In: Physics
Bernoulli’s principle can be derived from the principle of conservation of energy. This states that the total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant.
Consider a fluid of negligible viscosity moving with laminar flow, as shown in Figure 1.
Let the velocity, pressure and area of the fluid column be
v1, P1 and A1 at Q and
v2, P2 and A2 at R. Let the volume
bounded by Q and R move to S and T where QS = L1, and RT
= L2. If the fluid is incompressible:
A1L1 = A2L2
The work done by the pressure difference per unit volume = gain in
k.e. per unit volume + gain in p.e. per unit volume. Now:
Work done = force x distance = p x volume
Net work done per unit volume = P1 - P2
k.e. per unit volume = ½ mv2 = ½ Vρ v2 =
½ρv2 (V = 1 for unit volume)
Therefore:
k.e. gained per unit volume = ½ ρ(v22 -
v12)
p.e. gained per unit volume = ρg(h2 –
h1)
where h1 and h2 are the heights of Q and R
above some reference level. Therefore:
P1 - P2 = ½ ρ(v12 –
v22) + ρg(h2 -
h1)
P1 + ½ ρv12 + ρgh1 =
P2 + ½ ρv22 +
rgh2
Therefore:
P + ½ ρv2 + ρgh is a constant
For a horizontal tube h1 = h2 and so we
have:
P + ½ ρv2 = a constant
This is Bernoulli's theorem You can see that if there is a increase
in velocity there must be a decrease of pressure and vice
versa.
No fluid is totally incompressible but in practice the general
qualitative assumptions still hold for real fluids.