Bernoulli’s principal
Bernoulli's principle is equivalent to the principle of conservation of energy. This states that in a steady flow the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline.
This requires that the sum of kinetic energy and potential energy remain constant.
If the fluid is flowing out of a reservoir the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit mass (the sum of pressure and gravitational potential ρgh) is the same everywhere.
Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation.
In fact, there are different forms of the Bernoulli equation for different types of flow.
The simple form of Bernoulli's principle is valid forincompressible flows (e.g. most liquid flows and gases moving at low Mach number).
More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).
Applications of Bernoulli's Principle
It might help to think of a traveling fluid in terms of streamlines. These are imaginary lines that represent the path of fluid particles. Streamlines are far apart when the area surrounding the fluid is wide.
But when the area becomes narrow, the streamlines are pushed together, decreasing the pressure in the fluid and increasing its speed.
We can apply the idea of fluid streamlines to all sorts of situations. Since both liquids and gases are fluids, we can apply Bernoulli's principle to things like air as well as water.
Bernoulli’s principle Formula
p + 1/2 ρ v2 + ρgh =constant
Where
p is the pressure exerted by the fluid
v is the velocity of the fluid
ρ is the density of the fluid
h is the height of the container
Relation between Conservation of Energy and Bernoulli’s Equation
The conservation of energy is applied to the fluid flow for produceing Bernoulli’s equation. The net work done is the result of a change in fluid’s kinetic energy and gravitational potential energy.
The Bernoulli’s equation can be modified depending on the form of energy that is involved.
Any other forms of energy include the dissipation of thermal energy due to fluid viscosity.
Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow.
Some facts about Daniel Bernoulli
Born : 8 Feb. 1700, Groningen,
Died : March 17, 1782, Basel, Switzerland
The most distinguished of the second generation of the Bernoulli family of Swiss mathematicians. He investigated not only mathematics but also such fields as medicine, biology, physiology, mechanics, physics, astronomy, and oceanography. Bernoulli’s theorem (q.v.) which he derived, is named after him.
Limitations of Bernoulli's equation
1. The Bernoulli's equation has been derived by assuming that the velocity of every element of the liquid across any cross-section of the peipeis uniform. It is not true in practically.
The elements of the liquid in the innermost layer have the maximum velocity.
The velocity of the liquid decreases towards the walls of the pipe. Therefore, we should take into account the mean velocity of the liquid.
2. While deriving equation, the visous drag of the liquid has not been taken into consideration. The viscous drag comes into play, when a liquid is in motion.
3. Bernoulli's equation has been derived on the assumption that there appears no loss of energy, when a liquid is in motion.
In fact some kinetic energy is converted into heat energy and a part of it is lost due to shear force.
4. In the case, the liquid is flowing along a curved path the energy due to centrifugal force should also be taken into consideration.