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In: Physics

Starting with Bernoulli’s equation, explain how to derive Bernoulli’s principle

Starting with Bernoulli’s equation, explain how to derive Bernoulli’s principle

Solutions

Expert Solution

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.


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