In the past few post, we had seen enough fluid mechanics concepts in which we had seen various flow meters and there working principle, etc.

 Now we will see the basics of most important concepts putting a foundation of fluid mechanics, namely Bernoulli's principle.

 

 

Bernoulli's Principle

This principle describes the energy interconversions that occur in flowing fluid.

 Considering a volume element, applying differential momentum balance such as,

  

 

(Rate of momentum entering) - (Rate of momentum leaving) + (Sum of forces acting on system)

 =

(Rate of momentum accumulation.)

 

Momentum is the product of mass and velocity. Here velocity is a vector quantity, the above balance would be complicated.

So, considering momentum in all 3 directions,finally, we get equation relating Density, Velocity, and  Area known as Continuity Equation or Equation of Motion. And can be given as 

 

 

[Area of fluid flowing system  x   Velocity of fluid  x  Density of fluid
 = 
Constant]
 

 

Various forms of Continuity Equation

When considering Fluid of constant Density and constant Viscosity, the equation of motion is known as NAVIER-STOKES EQUATION.

 

When considering Fluid of Zero Density and Zero Viscosity , the equation of motion is known as EULER’S EQUATION.

 

Referring to Bernoulli’s principle, a general equation can be derived which state that the rate of increase of kinetic energy per unit mass equals the net rate of input of kinetic energy by following

• 1.     Rate of work done by Pressure of the surrounding.
• 2.     Rate of kinetic energy conversion to internal energy.
• 3.     Rate of work done by viscose forces.
• 4.     Rate of work done by gravity.

 

Assumption for Bernoulli’s Equation,

• 1.     Unidirectional Flow
• 2.     Incompressible fluid i.e Constant Density.
• 3.     Inviscid flow

Bernoulli's Equation without Friction.

Bernoulli's Equation without Friction can be derived by applying the momentum balance to steady flow of fluid in potential flow.

Momentum balance provides following expression,

When the same balance is applied between two specific points,following expression is obtained,

Above equation, interrelates pressure, velocity and position or height above datum.

It also shows that in absence of friction, when velocity is reduced, either height above datum or pressure or both must be increased.

Bernoulli's equation needs modification to cover practical fluid flow problems.

likewise, in case of friction ,friction factor must be consider.

In case pump is used ,friction in pump along with work done by pump must be consider.