Bernoulli’s Principle

Bernoulli ’ s equation Consider a small time interval δt in which the fluid at X has moved to X’ and that at Y to Y’. At X, work done during δt by the fluid XY by p 1 A 1 pushing it into the tube =force x distance moved =force x velocity x time = p 1 A 1 x υ 1 x δt

At Y, work done during δt by the fluid XY emerging from the tube against p 2 A 2 = p 2 A 2 x υ 1 x δt Therefore, Net work W done on the fluid = (p 1 A 1 υ 1 – p 2 A 2 υ 2 )δt If the fluid is incompressible, volume between X and X’ equals volume between Y and Y’, i.e. A 1 x υ 1 δt = A 2 x υ 2 δt therefore, W=( p 1 – p 2 ) A 1 υ 1 δt

As a result of work done on it, the fluid gains p.e. and k.e. when XY moves X’Y’ Gain of p.e. = p.e. of X’Y’ – p.e. of XY = p.e. of X’Y + p.e. of YY’ – p.e. of XX’ – p.e.of X’Y’ = p.e. of YY’ – p.e. of XX’ = (A 2 υ 2 δt ρ) gh 2 – (A 1 υ 1 δt ρ)gh 1 (p.e.= mgh) =A 1 υ 1 δt ρ g( h 2 –h 1 ) (A 2 υ 2 δt ρ = A 1 υ 1 δt ρ)

Where h 1 and h 2 are the height of XX’ and YY’ above an arbitrary horizontal reference level and ρ is the density of the fluid. Similarly, gain of k.e. = k.e. of YY’- k.e. of XX’ = ½ (A 2 υ 2 δt ρ) υ 2 ² - ½ (A 1 υ 1 δt ρ) υ 1 ² = ½ A 2 υ 2 δt ρ ( υ 2 ² - υ 1 ² ) Net work done on fluid =gain of p.e. + gain of k.e. Therefore, (p 1 -p 2 ) A 1 υ 1 δt =A 1 υ 1 δt ρg( h 2 –h 1 ) + ½ A 2 υ 2 δt ρ ( υ 2 ² - υ 1 ² )

(p 1 -p 2 ) = ρg( h 2 –h 1 ) + ½ ρ ( υ 2 ² - υ 1 ² ) Or p 1 + h 1 ρg +½ ρ υ 1 ² = p 2 + h 2 ρg +½ ρ υ 2 ² p+ hρg +½ ρ υ² = constant

Example of Bernoulli ’ s Effect Airfoil An airfoil is a specially shaped surface which causes fluids to move at different speeds over its two surfaces. The faster moving air on the top surface of the airfoil has a lower pressure than the slower moving air underneath, so there is net upward force

In designing airfoils, computer modeling is often used to predict the behavior at different speeds and angles. As well as trying to maximize the lift, the designer has to minimize the lift the drag caused by the turbulent flow behind the airfoil where the airstream may break into eddies.

Spinning ball the rough surface of a spinning ball drags the air near it, setting up a circulation of air around it. If the spinning ball also has translational motion, the airflow round one side is faster than the other side.

The higher pressure on the low speed side pushes the ball to the other side. The ball curves in the direction of the deflection force as shown in Figure; this is called “magnus” affect.

Some movies about spinning ball

Paint Sprayer In a paint sprayer, air is blown along the tube from a pump. Speed of air is increased near the nozzle, where the air pressure is reduced. The liquid paint is sucked up through a small pipe inside. The paint disperses on leaving the nozzle.

The End