**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 A_{1} and A_{2} respectively. Let v_{1} p_{1} and v_{2} p_{2} are velocities and pressure at cross-section X and Y. The axes of the tube are at height h_{1} and h_{2} respectively.

Now, the work done per second on the liquid by pressure force at the end X is given by,

W

_{1}= f * d = P

_{1}A_{1}d = P

_{1}A_{1}d/t = P

_{1}A_{1}v_{1}Similarly, the work done per second by the liquid against pressure force at the end Y,

W

_{2}= -P_{2}A_{2}v_{2}.: The total amount of work done on the liquid is given by,

W = W

_{1}+ W_{2} = P

_{1}A_{1}v_{1}– P_{2}A_{2}v_{2}…...... (i)But from equation of continuity,

A

_{1}v_{1}= A_{2}v_{2}= m/ϱWhere, m is the mass per second of liquid flowing through the tube.

So, equation (i) becomes

W = (P

_{1}– P_{2})m/ϱAlso, the increase in potential energy

= (mgh

_{2}– mgh_{1})= mg (h

_{2}– h_{1})Similarly, change in Kinetic energy

= ½ m (v

_{2}^{2}– v_{1}^{2})According to work energy theorem,

W = Increase in energy

or, (P

_{1}– P_{2})m/ϱ = mg (h_{2}– h_{1}) + ½ m (v_{2}^{2}– v_{1}^{2})or, (P

_{1}– P_{2})m/ϱ = m {g (h_{2}– h_{1}) + ½ (v_{2}^{2}– v_{1}^{2})}or, P

_{1}/ϱ - P_{2}/ϱ = gh_{2}– gh_{1}+ ½ v_{2}^{2}– ½ v_{1}^{2}or, P

_{1}/ϱ + gh_{1}+ ½ v_{1}^{2}= gh_{2}+ + ½ v_{2}^{2}or, P

_{1}+ ϱgh_{1}+ ½ ϱv_{1}^{2}= P_{2}+ ϱgh_{2}+ ½ ϱv_{2}^{2}In general,

P + ϱgh + ½ ϱv

^{2}= 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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