Bernoulli's equation

Bernoulli's equation:

On observing a river, we notice that the speed of the water decreases in wider region whereas the speed of water increases in the regions where the river is narrow. From this we might think that the pressure in narrower regions is more than that in the wider region. However, the pressure within the fluid in the narrower parts is less while that in wider parts is more. Swiss scientist Daniel Bernoulli, while experimenting with fluid inside pipes led to the discovery of the concept mentioned above. He observed, in his experiment, that the speed of a fluid in a narrow region increases but the internal pressure of a fluid in the same narrow region decreases. This phenomenon is called Bernoulli’s principle. 

Figure shows flow of an ideal fluid through a tube of varying cross section and height. Consider an element of fluid that lies between cross sections P and R. Let,

• v1 and v2 be the speed the fluid at the lower end P and the upper end R respectively.

• A1 and A2 be the cross section area of the fluid at the lower end P and the upper end R respectively.

• P1 and P2 be the pressures of the fluid at the lower end P and the upper R respectively.

• d1 and d2 be the distances travelled by the fluid at the lower end P and the upper end R during the time interval dt with velocities v1 and v2 respectively.

• Now P1 A1 and P2 A2 are the forces acting on areas A1 at P and A2 at R respectively.

The volume dV of the fluid passing through any cross section during time interval dt is the same; i.e., dV = A1 d1 = A2 d2 net work, W, done on the element by the surrounding fluid during the flow from P to R is, W = P1 A1 d1 – P2 A2 d2 The second term in the above equation has a negative sign because the force at R opposes the displacement of the fluid. the above equation can be written as, 

W = P1 dV – P2 dV = (P1 - P2 ) dV 

As the work W is due to forces other than the conservative force of gravity, it equals the change in the total mechanical energy i.e., kinetic energy plus gravitational potential energy associated with the fluid element. i.e., W = 'K.E. + 'P.E. 

This is Bernoulli’s equation. It states that the work done per unit volume of a fluid by the surrounding fluid is equal to the sum of the changes in kinetic and potential energies per unit volume that occur during the flow.