Bernoulli’s Theorem and It’s Application | Physics Grade 11 Notes

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Bernoulli’s Theorem and It’s Application | Physics Grade 11 Notes

Bernoulli’s Theorem | Application of Bernoulli’s Theorem

Bernoulli’s Theorem
It states that, "when an ideal gas is flowing in a streamline flow through a non-uniform horizontal tube, then the sum of pressure energy per unit volume, Potential energy per unit volume and Kinetic energy per unit volume remain constant".

Let us consider an incompressible and non-viscous liquid of density 'd' is flowing through a non-viscous uniform tube XY. Let the cross-sectional areas of tube at X and Y are A1 and A2 respectively. Let v1 p1 and v2 p2 are velocities and pressure at cross-section X and Y. The axes of the tube are at height h1 and h2 respectively.

Bernoulli' Theorem DiagramNow, the work done per second on the liquid by pressure force at the end X is given by,

W1 = f * d
     = P1A1d
     = P1A1d/t
     = P1A1v1
 
Similarly, the work done per second by the liquid against pressure force at the end Y,
W2 = -P2A2v2
 
.: The total amount of work done on the liquid is given by,
W = W1 + W2
   = P1A1v1 – P2A2v2 …...... (i)
 
But from equation of continuity,
A1v1 = A2v2 = m/ϱ
Where, m is the mass per second of liquid flowing through the tube.
 
So, equation (i) becomes
W = (P1 – P2)m/ϱ
 
Also, the increase in potential energy
= (mgh2 – mgh1)
= mg (h2 – h1)
 
Similarly, change in Kinetic energy
= ½ m (v22 – v12)
 
According to work energy theorem,
W = Increase in energy
or, (P1 – P2)m/ϱ   = mg (h2 – h1) + ½ m (v22 – v12)
or, (P1 – P2)m/ϱ  = m {g (h2 – h1) + ½ (v22 – v12)}
or, P1/ϱ - P2/ϱ   = gh2 – gh1 + ½ v22 – ½ v12
or, P1/ϱ + gh1 + ½ v12 = gh2 +  + ½ v22
or, P1 + ϱgh1 + ½ ϱv12 = P2 + ϱgh2 + ½ ϱv22
 
In general,
P + ϱgh + ½ ϱv2 = constant
 
Application of Bernoulli’s Theorem
Lifting of aircraft: The shape of the wings of airplane is made convex upward and concave downwards. Due to this, there is more speed of air above the wings than in below the wings. Therefore, there is less pressure in the upper part than in the lower part. Due to this pressure difference, it experiences an upward lift known as a dynamic lift.

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