Bernoullis equation

bernoullis and its some applications

1. Bernoulli’s Equation and its Applications 1 Topic

2. Bernoulli’s Equation Theory - Introduction Bernoulli's principle  states that an increase in the speed of a fluid , decrease in  pressure. The principle is named after  Daniel Bernoulli   who published it in his book  Hydrodynamica  in 1738.   Leonhard Euler  who derived  Bernoulli's equation  in its usual form in 1752. Leonhard Euler (1707   - 1783 ) Daniel Bernoulli (1700 – 1782)

3. Bernoulli’s Principle Theory - Statement 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 . Mathematical form: P + +mgh = constant pressure velocity height Applicable : I ncompressible Steady Non viscous   3

4. Bernoulli’s Equation Explanation 4 Consider the following diagram where water flows from left to right in a pipe that changes both area and height. When fluid move upward, the water will be gaining gravitational potential energy  U g  as well as kinetic energy  K .

5. Derivation Work done on the fluid: W 1 = F 1 x 1   As P = F = PA   Then W 1 = P 1 A 1 x 1   V 1 =   In terms of velocity x 1   = V 1 t W 1 = P 1 A 1 V 1 t W 2 = -F 2 x 2   W 2 = -P 2 A 2 x 2   W 2 = -P 2 A 2 V 2 t The water at P 2 will do negative work on our system since it pushes in the opposite direction as the motion of the fluid.

6. Derivation Net Work done on the fluid: W net = W 1 + W 2 W net = P 1 A 1 V 1 t - P 2 A 2 V 2 t The Volume of both sections are equal A 1 V 1 t = A 2 V 2 t = V So W = P 1 V - P 2 V W = (P 1 - P 2 )V As we know (Density) V =   W = (P 1 - P 2 )V W = (P 1 - P 2 )  

7. Derivation Work Energy Principle: Work done = change in energy W = (K.E) + (P.E)   Changing in (K.E) :   Changing in (P.E ) :   (K.E) = mv 2   (K.E) = mv 2 2   - mv 1 2   (P.E) = mgh   (P.E) = mgh 2 – mgh 1   (1) Put the values in equ (1) (P 1 - P 2 ) = mv 2 2 - mv 1 2 + mgh 2 – mgh 1   (P 1 - P 2 ) = m ( v 2 2 - v 1 2 + gh 2 – gh 1 )   (P 1 - P 2 ) = v 2 2 - v 1 2 + gh 2 – gh 1 )   This is Bernoulli's equation! Generalize: P+ +mgh = constant   This constant will be different for different fluid systems, but the value of  P+ +mgh  will be the same at any point along the flowing fluid.  

8. Bernoulli’s Principle APPLICATIONS

9. Application - LIFT 9 The wings of plane have what is called an aerofoil shape . The aerofoil shape helps us overcome weight which is the effect of gravity pulling down on the mass of the aircraft. The aerofoil shape gives us something called lift . This is the upward force required to overcome gravity. Something that slows us down is drag , which is the resistance to airflow through the air . The drag force is opposite to the flight path. Thrust is the forward force required to move an aircraft through the air . This must be provided by an engine .

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11. 11 Bernoulli’s principle helps to explain that an aircraft can achieve lift because of the shape of its wings . They are shaped so that that air flows faster over the top of the wing and slower underneath. Fast moving air = low air pressure while slow moving air = high air pressure. The high air pressure underneath the wings will therefore push the aircraft up through the lower air pressure.  

12. Application - Base b all 12 One side will experience more pressure than the other thus having more air turbulence and a slower air speed over the ball. The other side would accelerate and move faster, because of lesser pressure . This example explains the path of a baseball that’s thrown with a clockwise spin. If a ball is thrown with a counter-clockwise spin, it will curve towards the left.

13. Application –Atomizer 13 Atomizer is a device that is used to emit liquid droplets as fine spray. 'Atomize' here means  splitting up a large body  into small particles.   It works on Bernoulli's principle.  When high speed horizontal air passes over a vertical tube, it creates a low pressure and draws the air and liquid inside the vertical tube upward. Atomizer has a nozzle at the end of the horizontal tube which causes the liquid to break up into small drops and mixes it with the air.

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