Bernoulli's Equation

Bernoulli's Equation

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Bernoulli's EquationBernoulli's Equation

As the fluid moves through a pipe of varying cross section and height , the pressure will change along the pipe. Bernoulli's equation is a fundamental equation in fluid dynamics that relates pressure to fluid speed & height.

In deriving Bernoulli's equation , we assume that the fluid is incompressible, non viscous and flows in a steady state manner.

Derivation Derivation Let us consider the flow of fluid through the pipe in time t

as shown in the fig. The force on the upper end of the fluid is P1A1 . The work

done on the fluid in moving it through the distance x1 is

W1 = F1x1 = P1A1 x1 ……1

P=F/AF=PA

Similarly the work done on the lower end is

W2 = -F 2x2 = -P2A2x2 ……..2

The work done in this case is –ve because the force on fluid is opposite to that of F1

The net work done is

W= W1 + W2

W= P1A1x1 – P2A2x2 ……….3

If v1&v2 are the velocities of the fluid at upper & lower end respectively.

W= P1A1v1t –P2A2v2t …………...4

S=v tx1=v1t

From the equation of continuity

A1v1= A2 v2

Hence A1v1t = A2v2t = V

From equation 4 we get

W=P1A1v1t-P2A2v2t W= (P1 – P2 )V ………5

If m is mass & is density then V= m / then

W= (P1-P2) m /.............6

Part of this work is utilized by the fluid in changing its K.E & a part is used in changing its gravitational P.E.

change in K.E = ½ mv22- ½ mv1

1 ………7

Change in P.E = mgh2-mgh1 ……………8

Applying the law of conservation of energy(P1-P2) m /½ mv2

2- ½ mv11 + mgh2-mgh1 …9

Rearranging P1 + ½ v1

2 +gh1 = P2+ ½ v22 + gh2 …..10

In a general form P + ½ v2 +g h=constant

An interesting exampleAn interesting example A stream of air passing

over a tube dipped in a liquid will force the liquid to rise in the tube as shown. This effect is used in perfume bottles & paint sprayers.

P + ½ v2 g h=constant

Chimney works Chimney works

A chimney works best when it is tall and exposed to air currents , which reduces the pressure at the top and force the upward flow of smoke.

Applications of Bernoulli's EquationApplications of Bernoulli's Equation

Suppose a large tank of fluid has two small orifices A & B on it. Let us find the speed with which the water flows from the orifice A .

TERRCELLI,S THEOREM

v1

As the orifices are so small efflux speeds v2 and v3 will be much larger than the speed v1 of the top surface of the water. Due to this reason, we can take v1 approximately equal to zero. Now Bernoulli's equation can be written as .

P1+gh1 = P2+1/2 v22 +gh2

v1

1. But P1 = P2 = atmospheric pressure. So

2. gh1 = 1/2 v22 +gh2

3. 1/2 v22 = gh1 - gh2

4. v2= 2g(h1 – h2 )…………….1

This is TERRICELLI,S THEOREM which states that the speed of efflux depends upon the height (h1-h2) & the action of gravity.

v1

P1

P2

The top level of the tank has moved down a little and the P.E has been transferred into K.E of the efflux of fluid. If the level of orifice B has the level same as the water of the tank , the water will rise to the level of the water tank.

Relation b/w speed & pressure of the fluidRelation b/w speed & pressure of the fluid

A result of Bernoulli's equation is that the pressure will be low where the speed of the fluid is high.

Suppose that the water flows through a pipe of system as shown in the fig.

It is clear that the water will flow faster at point B than A or C. Assume that the speed at point A is 0.20ms-1 & at B 20ms-1. In order to compare the pressure at B with A, we use Bernoulli, s equation .

PA + ½ vA2 = PB + ½ vB

2 ( P.E is constant at both points)

Putting v A = 0.20 ms-1 v B = 2.0ms-1 & = 1000kgm-3

PA + ½ vA2 = PB + ½ vB

2

PA – PB = 1980 Nm-2

It means that the pressure in narrow pipe where streamlines are closer together is much smaller than in wider pipe.

RESULTRESULT

Where the speed is high , the pressure will be low.

Example1Example1 The wing of aero plane

is designed to deflect the air so that the streamlines are closer together above the wing than below it. Where the streamlines are closer to each other, the speed is faster. The air is traveling faster on the upper side of the wing than the lower.

Example2Example2

When a tennis ball is hit by a racket in such a way that it spins as well as moves forward , the velocity of the air on one side of the ball increases due it spins and air pressure decreases. This gives extra curvature to the ball & as a result ball swing which deceives an opponent player.