What is Bernaullie is theorem? Prove
this theorem?
Answers
Answer:
Bernoulli's theorem, in fluid dynamics, relation among the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar.
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
Explanation:
Answer:
Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. To prove Bernoulli's theorem, consider a fluid of negligible viscosity moving with laminar flow, as shown in Figure.
Explanation:
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
P + ½ ρv2 + ρgh is a constant
P + ½ ρv2 = a constant