CBSE Class 11-science Answered

Let us consider two different regions in the above diagram. Let us name the first region as BC and the second region as DE. Now consider the fluid was previously present in between B and D. However, this fluid will move in a minute (infinitesimal) interval of time (∆t).

If the speed of fluid at point B is v1 and at point D is v2. Therefore, if the fluid initially at B moves to C then the distance is v1∆t. However, v1∆t is very small and we can consider it constant across the cross-section in the region BC.

Similarly, during the same interval of time ∆t the fluid which was previously present in the point D is now at E. Thus, the distance covered is v2∆t. Pressures, P1 and P2, will act in the two regions, A1 and A2, thereby binding the two parts. The entire diagram will look something like the figure given below.

Finding the Work Done

First, we will calculate the work done (W1) on the fluid in the region BC. Work done is

W1 = P1A1 (v1∆t) = P1∆V

Moreover, if we consider the equation of continuity, the same volume of fluid will pass through BC and DE. Therefore, work done by the fluid on the right-hand side of the pipe or DE region is

W2 = P2A2 (v2∆t) = P2∆V

Thus, we can consider the work done on the fluid as – P2∆V. Therefore, the total work done on the fluid is

W1 – W2 = (P1 − P2) ∆V

The total work done helps to convert the gravitational potential energy and kinetic energy of the fluid. Now, consider the fluid density as ρ and the mass passing through the pipe as ∆m in the ∆t interval of time.

Hence, ∆m = ρA1 v1∆t = ρ∆V

Change in Gravitational Potential and Kinetic Energy

Now, we have to calculate the change in gravitational potential energy ∆U.

Similarly, the change in ∆K or kinetic energy can be written as

Calculation of Bernoulli’s Equation

Applying work-energy theorem in the volume of the fluid, the equation will be

Dividing each term by ∆V, we will obtain the equation

Rearranging the equation will yield

The above equation is the Bernoulli’s equation. However, the 1 and 2 of both the sides of the equation denotes two different points along the pipe. Thus, the general equation can be written as

Thus, we can state that Bernoulli’s equation state that the Pressure (P), potential energy (ρgh) per unit volume and the kinetic energy (ρv2/2)   per unit volume will remain constant