BERNOULLI PRINCIPLE, understanding the basic concepts and it’s mathematical derivation//

BERNOULLI PRINCIPLES

So we are going to talk about a very important and interesting concept of hydrodynamics that is the BERNOULLI PRINCIPLE.
Daniel Bernoulli a Swiss physicist developed this concept in 1738.
See how old this principle are but it’s usefulness and importance has never been fade till now.
This principle plays a important role in finding the speed of efflux in Torricelli law,
It is used in venturimeter , in sprayer, flying of aeroplane, blowing off of roofs in Strom, flying of birds etc, all these phenomena are explained with the help of Bernoulli principle.

The BERNOULLI PRINCIPLE explaination and it's mathematical derivation//

So everyone should understand the underlying concept of this principle. So read it carefully and keep patience.
This principle is strictly based on the conservation of energy. However, we are talking about the topic of hydrodynamics. Then which type of energy are involved in it. When Bernoulli was doing his research then he observed that, if any ideal fluid that is non viscous , incompressible, streamlined and irrotational, then at every cross section throughout the liquid flow, the sum of  pressure energy , kinetic energy and potential energy at per unit volume is constant.
Then he expressed this mathematically as follows:
P + (1/2)ρv^2 + ρgh = constant
Where P stand for pressure energy , (1/2)ρv^2  stands for kinetic energy and ρgh
Stands for potential energy.

Now we are going to see, that how did he come to this relationship, so derivation is as follows:

DERIVATION

Let’s take p1 , a1, h1, v1 and p2, a2 , h2 , v2, be the pressure , area of cross section, height and velocity of flow of liquid at point A and B respectively.
Then the force acting on the fluid at point A is F1 = p1a1
See the diagram below from more precise understanding.

The BERNOULLI PRINCIPLE explaination and it's mathematical derivation//

Then distance travel by the fluid in one second at point A is given as = v1×1=v1
Then work done per second on the fluid at point A is :
W1 = p1a1v1
similarly we can say that work done on second point B is , W2 = p2a2v2
Then net work done by the pressure is given as pressure energy difference.
W = W1 -W2 = p1a1v1 -p2a2v2
But from the equation of continuity, we know that ,
V = a1v1= a2v2 = m/ρ
So work done can be written as
W = p1m/ρ – p2m/ρ
This pressure energy of the fluid is further converted into the kinetic energy and potential energy energy , because due to this pressure difference liquid flow from point A to another point B. From doing this liquid attain different heights ie. From h1 to h2

It means this pressure energy is further balanced by the sum of change of kinetic energy and potential energy.

p1m/ρ -p2m/ρ = (mgh2 – mgh1) +(mv2^2/2 – mv1^2/2)
Cancelling m by taking common on both side we get,
(p1 -p2)/ρ = (gh2 – gh1) + v2^2/2 – v1^2/2
Multiplying ρ on RHS we get,
p1 -p2 = ρgh2 – ρgh1 + ρv2^2/2 – ρv1^2/2

Taking 1 marked on LHS and 2 marked on RHS , we get
p1 + ρgh1 + ρv1^2/2 = p2 + ρgh2 + ρv2^2/2
So both sides are Equal then we can conclude that
p + ρgh + (1/2)ρv^2 = constant
This mathematical modelling is done by Bernoulli in 1738.
But if both side are divided by the ρg , then we get
p/ρg + h + v^2/2g = constant/ρg = new constant
Then , p/ρg is called pressure head , h is gravitational head or potential head and
v^2/2g is called velocity head.

This suggests that for a ideal fluid which are flowing in a pipe, then sum of pressure head, gravitational head and velocity head is always a constant.

Here a video is given to you for a better understanding.

FLUID IN HORIZONTAL PIPE

If fluid is flowing in a horizontal pipe then , one element of Bernoulli principle has been eliminated that is its height , because pipe is lying horizontally to the ground level so height is zero . Then for this type of conditions Bernoulli principle as follows:
  p1 + (1/2)ρv1^2 = p2 + (1/2)ρv2^2
So it becomes
    p + (1/2)ρv^2 = constant

FLUID AT REST

If fluid are in rest, then it’s velocity is zero, it means in Bernoulli equation , kinetic energy element is zero, in such conditions, this principle is as follows:
p1 + ρgh1 = p2 + ρgh2
p1- p2 = ρg( h2 – h1)
Then , in constant form it is written as:
p + ρgh = constant.
This is all about THE BERNOULLI PRINCIPLE.

Suchit prajapati

Suchit Prajapati is the Founder and CEO of Laws Of Nature. He is also an Entrepreneur, motivational speaker, spiritual thinker, affiliate marketer and physics enthusiast.

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