# Bernoulli’s Theorem….

Ever thought of How can a plane fly?

In a storm how does a house lose its roof?

Why do rabbits not suffocate in the burrows?

Why does a cricket ball swing or a baseball curve?

Here the answer is given by the Bernoulli’s principle! Bernoulli’s Equation:

Let us focus our attention on the motion of the shaded region. This its our “system”. The lower cylindrical element of fluid of length Δl₁, A1 is at height y1 which moves with speed v1After some time, the leading section of our system fills the upper cylinder of fluid of length Δl₂ and area  A₂ at height y₂, and is then moving with speed v₂.

A pressure force F₁ acts on the lower part of the cylindrical tube towards right and pressure force F₂ acts on the upper part of the cylindrical tube towards left. The net work done on the system F₁ and F₂ is W = F₁Δl₁ – F₂Δl₂ = P₁Δl₁ – P₂Δl₂ = (P₁ – P₂) ΔV.

Where we have used the relations F = PA and ΔV = A₁Δl₁ = A₂Δl₂. The net effect of the motion of the system is to raise the height of the lower cylinder of mass and to change its speed. The changes in the potential and kinetic energies are

ΔU = Δmg (y₂ – y₁)

ΔK = ½ Δm (v₂² – v₁²)

W = ΔU + ΔK

(P₁ – P₂) ΔV = Δmg (y₂ – y₁) + ½ Δm (v₂² – v₁²)

Since the density is ρ = Δm/ΔV, we have P₁ + ρgy₁ + ½ ρv₁² = P₂ + ρgy₂ + ½ ρv₂².

Since the points 1 and 2 can be chosen arbitrarily, we can express this result as Bernoulli’s Equation p + ρgy + ½ ρv² = constant

It is applied to all points along a streamline in a nonviscous, incompressible and irrotational fluid.

In a storm how does a house lose its roof?

Air flow is disturbed by the house. The “streamlines” crowd around the top of the roof.

⇒ faster flow above house

⇒ reduced pressure above roof to that inside the house

⇒ roof lifted off because of pressure difference.

Why do rabbits not suffocate in the burrows?

Air must circulate. The burrows must have two entrances. Air flows across the two holes is usually slightly different

⇒ slight pressure difference

⇒ forces flow of air through burrow.

One hole is usually higher than the other and the a small mound is built around the holes to increase the pressure difference.