Bernoulli's Theorem

Statement :

           When an incompressible and non-viscous liquid (or gas) flows in streamlined motion from one place to another, then at every point of its path the total energy per unit volume

            is constant.

           That is= constant.

Bernoulli’s Theorem:

=constant

 Thus, Bernoulli’s theorem is in one way the principle of conservation of energy for a flowing liquid (or gas).

Bernoulli’s Equation:

           Let us focus our attention on the motion of the shaded region. This its our “system”. The lower cylindrical element of fluid of length A1 is at height y1 which moves with speed v1After some time, the leading section of our system fills the upper cylinder of fluid of length and area  A2 at height y2 and is then moving with speed v2.

           A pressure force F1 acts on the lower part of the cylindrical tube towards right and pressure force F2 acts on the upper part of the cylindrical tube towards left. The net work done on the system F1 and F2 is

           

           Where we have used the relations and . The net effect of the motion of the system is to raise the height of the lower cylinder of mass  and to change its speed. The changes in the potential and kinetic energies are

                                                                   

           Since the density is, we have  

    Since the points 1 and 2 can be chosen arbitrarily, we can express this result as Bernoulli’s                                  Equation 

= constant               

           It is applied to all points along a streamline in a nonviscous, incompressible and irrotational fluid.

Example  :                 Figure shows a liquid of density 1200 kg/m3 flowing steadily in a tube of varying cross-section. The cross-section at a point A is 1.0 cm2 and that at B is 20 mm2, the points A and B are in the same horizontal plane. The speed of the liquid at A is 10 cm/s. Calculate the difference in pressures at A and B.

Solution :                   From equation of continuity, the speed n2 at B is given by,

                                  A1n1 = A2n2 

                             

                               

                                  By Bernoulli’s equation,

                             

                                  Here h1 = h2. Thus,