Bernoulli’s Theorem—Derivation

Bernoulli’s principle states that the total mechanical energy of any fluid in motion remains the same. The total mechanical energy of the fluid is defined as the sum of its kinetic energy, potential energy, and pressure associated energies. The term ‘fluid’ encompasses both liquids and gases. Many applications are based on this idea. An aeroplane attempting to stay aloft, or even something as simple as a shower curtain blowing inward, are both good examples of this principle.

Bernoulli’s principle formula

As mentioned, the total mechanical energy of the fluid is defined as the sum of kinetic energy, potential energy, and pressure associated energies, according to which Bernoulli’s principle can be expressed as:

p + ½ ρ v² + ρgh = constant

where,

p denotes the pressure of the moving fluid,

v denotes the fluid’s velocity,

ρ is the fluid’s density, 

h is the height of the container, and

g is the gravitational acceleration.

Bernoulli’s equation elucidates the relationship between pressure, the velocity of the moving fluid, and potential energy due to elevation.

Deriving Bernoulli’s theorem:

Understanding that the flowing fluid has always been a challenge, the law of energy conservation is the only law that helps us understand various forces acting on a body. Streamlined flows are quite easy to get data from, but tracking goes haywire during turbulent flows. For example, we can imagine a pipe with variable cross-sections at different segments and of a variable height. 

Let us consider that the flow of an incompressible fluid flowing through the pipe is constant. As it has variable cross-sections and heights, the continuity equation varies throughout its length. Although for a fluid to flow in a pipe, we have to exert a force on it, pressure also varies at different sections. 

Bernoulli’s equation is a generic equation that defines the pressure difference between two places in a pipe in terms of the changing velocity (kinetic energy) and height (potential energy). In the year 1738, Swiss physicist and mathematician Bernoulli established this link.

The general expression of Bernoulli’s equation:

Draw four points along the length of the pipe and name them B, C, D and E, respectively. Now mark two areas BC and DE, where the fluid existed earlier for B as well as D. This fluid, on the other hand, will travel in a minute (infinitesimal) period (∆t).

Let us consider the speed at point B as v1 and the fluid speed at point D as v2. As a result, if the fluid starts at B and flows to C, the distance is v1∆t. On the other hand, v1∆t is quite small and may be considered constant across the cross-section in the area BC.

Finding the work done

To begin, we shall compute the work done (W1) on the fluid in area BC. 

Work done is W1 = P1A1 (v1∆t) = P1∆V

Furthermore, if we examine the continuity equation, the same volume of fluid will travel through BC and DE. As a result, the fluid’s work on the right side of the pipe or DE area is

W2 = P2A2 (v2∆t) = P2∆V

As a result, we may denote the work done on the fluid as –P2∆V. 

Therefore, the total work done on the fluid is W1–W2 = (P1−P2) ∆V

The overall effort contributes to the conversion of the fluid’s gravitational potential energy and kinetic energy. Consider the fluid density to be ρ and the mass going through the pipe to be ∆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.

∆U = pg∆V (h₂-h₁)

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

∆K = (½) ρ ∆V (v22-v12)

Limitations of the applications of Bernoulli’s equation

Limitations of the equation include velocity loss due to frictional force between the layers of water. A layer of water is referred to as a laminate and when two laminates glide on the surface of each other, the process gives birth to the loss of energy. Water also creates friction with the walls of the pipe which is left unaccounted for. 

Another significant restriction of this theory is the need for an incompressible fluid. As a result, the calculation ignores the fluid’s elastic energy, which is extremely significant for various applications. It also aids in our understanding of principles about low viscosity incompressible fluids.

Furthermore, in turbulent flows, Bernoulli’s principle is impossible to apply as pressure and velocity are continually changing.

Conclusion

We have now studied how Daniel Bernoulli invented Bernoulli’s equation. It helps us understand the flow of an incompressible fluid inside a pipe and gives a reason for the pressure decreasing when the velocity of the fluid increases. The total mechanical energy of the fluid is defined as the sum of the kinetic energy, potential energy, and pressure associated energies. There are a few limitations to Bernoulli’s equation as well. To understand more about the lift, we recommend you practice some problems on the same.