Question

bernolis theorum proof
​

Answer 1

Answer:

Proof of Bernoulli's theorem

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 fluid

Answer 2

Proof of Bernoulli's theorem

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.