Understanding Bernoulli’s Theorem: Definition, Formula, Derivation

Bernoulli’s theorem has a very special place in the world of physics. This theorem is generally based on the principle of conservation of energy applied to a liquid in motion. Bernoulli’s theorem is also known as Bernoulli’s principle.

It is defined as the sum of the pressure energy per unit volume, kinetic energy per unit volume, and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined flow that remains constant along the streamline. A fluid with less speed will use more force compared to a fluid that is flowing very fast.

Application of Bernoulli’s Theorem

Bernoulli’s theorem helps us to determine the relationship between pressure, density, and velocity at every point in a fluid. Bernoulli’s theorem has some important applications.

• It is applicable to ideal incompressible flow. 
• Bernoulli’s theorem is used in studying the wing lift of an aircraft. 
• It is observed in the blowing of a roof when a storm occurs, where only the roof moves, and there is no damage to the hut. 
• Bernoulli’s theorem is seen in the working of a Bunsen burner. 
• It is used during the motion of two parallel boats. 
• It is used in atomisers and filter pumps. 
• It is used in the pitot tube to find the velocity of a fluid in motion. 
• Most importantly, Bernoulli’s theorem is used in the Venturi metre to find the rate of flow of a liquid.
• When a rotating ball is thrown, it moves away from its actual position within the flight. This phenomenon is known as the Magnus effect, which is mostly seen in cricket, soccer, tennis, etc. Bernoulli’s theorem is also observed in this effect.

Limitations of the Bernoulli’s Theorem

There are some limitations to Bernoulli’s theorem.

1. There should be a steady flow. A steady flow is one with no change with time. 
2. If the liquid is flowing along a curved path, the energy due to centrifugal force should be taken into consideration. 
3. There should be no viscosity. 
4. There should be no loss of energy due to shear force. 
5. It is applicable to ideal incompressible flow. 
6. The heat transfer in or out of the fluid should be zero. 
7. The internal energy does not change as the temperature remains constant. 

The formula for Bernoulli’s theorem

Mathematically the formula for Bernoulli’s theorem is given as the equation:

P+12v2+gh=constant

Where P= static pressure of the fluid at the cross-section

ρ= density of the flowing fluid

g= acceleration due to gravity

v= mean velocity of fluid flow at the cross-section

h= elevation head of the centre of the cross-section with respect to a datum. 

Derivation

Bernoulli’s theorem is based on the conservation of energy, i.e., energy can neither be created nor destroyed, but it can change from one form to another.

Bernoulli’s theorem has a derivation that is as follows:

Consider a pipe with varying diameter and height through which an incompressible fluid is flowing. The relationship between the areas of cross-sections A, the flow speed v, height from the ground y, and pressure p at two different points 1 and 2 are given in the figure below.

Let us assume that the density of the incompressible fluid remains constant at both points, and the energy of the fluid is conserved as there are no viscous forces in the fluid.

So if we calculate the work done by the pressure on the liquid, then 

w=Fx

F=PA

Where A is the area of the cross-section of the pipe 

And dx = vΔt

Hence the energy associated with the pressure is 

W = PA ​⋅ vΔt  

       = P ​ΔV

where ∆V is the volume that passes through the region through the cross-section.

Then the pressure energy per unit volume is P

So as we discussed earlier, according to the conservation of the energy theorem, the summation of all energy remains constant.

Hence  P+12v2+gh=constant ,

         This is Bernoulli’s equation.

Conclusion

Bernoulli’s theorem is defined as the sum of the pressure energy per unit volume, kinetic energy per unit volume, and potential energy per unit volume of an incompressible, non-viscous fluid in a streamlined flow that remains constant along the streamline.

Bernoulli’s equation is represented as:

P+12v2+gh=constant

Bernoulli’s theorem has many applications: in pitot tubes to find the velocity of a fluid in motion, atomisers, filter pumps, aircraft wings, Bunsen burner, the motion of two parallel boats, Magnus effect and more. It also helps to find the rate of flow of a liquid.

Bernoulli’s theorem has many limitations: the flow of the liquid must be steady for Bernoulli’s principle to take place, the fluid must be incompressible, and the viscous effect must be negligible. Bernoulli’s equation can only be used with streamlined fluids, not with turbulent fluids. The fluid must be irrotational.