Bernoulli's Equation and Euler's Equation - Different terms

Bernoulli's Equation states that - 

For an incompressible liquid (density constant) which is flowing in a continuous stream; the total energy remains the same for all the particles, when the particle is moving from one point to another point.

Mathematical expression of Bernoulli's equation:

P1/w + (v2 )1/2g + z1 = P2/w +( v2)2 /2g + z1 = constant 

P1 , P2 = Pressure at points 1 and 2
V1,V1 = velocity at points 1 and 2
Z1 , Z2  = datum head at points 1 and 2
w= weight density of the liquid (also denoted by Greek letter gamma)

Different types of energies or head in Bernoulli's equation 

1. Potential energy or potential head: 

Potential energy is the energy stored in a body because of changing its position above the fixed datum head. It's denoted by Z. 

2. Kinetic energy or head: 

It is due to the velocity of the fluid. It is denoted by v2 /2g.  Here, v is the velocity and g is the gravitational acceleration. 

3. Pressure energy or pressure head:

 This energy is due to the pressure of the liquid. It is denoted by P/w . 

P has the unit Newton/square meter ,N/m
W has the unit Newton / cubmic meter , N/m3

Total Energy = Pressure energy + Kinetic energy + Potential energy 
Total Head = Pressure Head + Kinetic Head + Potential Head  

Applications of Bernoulli's Equation: 

1. A modified equation for the measurement of velocity in a pitot tube and pitot static can be got from Bernoulli's equation. 
2. The flow through venturi meter and orifice meter can be measured by this equation. 
3. Flow over notches and weirs can be calculated. 

Euler Equation or Bernoulli Differential Equation 

bernoullis equation

Euler's equation is expressed as 

dp/ρ + g.dx + v.dv = 0 

Euler's Equation is based on some assumptions: 

1. Non-viscous flow. 
2. Incompressible  and homogeneous flow. 
3. Steady flow. 
4. Continuous flow. 
5. Uniform velocity of the flow. 


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