Which of the following assumptions is essential for applying Bernoulli’s theorem?

Explanation:

Bernoulli's theorem:

• It states that the total mechanical energy of the flowing fluid comprising the energy associated with fluid pressure the gravitational potential energy of elevation and the kinetic energy of fluid motion remains constant.
• It is based on conservation of energy.

Assumptions of Bernoulli's theorem i.e. Bernoulli's theorem is Valid for:

• Flow is ideal i.e inviscous.
• Flow is steady i.e. time variation is zero.
• Flow is incompressible i.e. ρ is constant.
• Flow is irrotaional i.e. ωx = ωy = ωz = 0.
• All the other external forces except gravity and pressure forces should be zero.
• The energy of the system is constant hence there should be no loss of energy.

Bernoulli's Equation:

Bernoulli's equation is obtained by integrating the Euler's equation of motion which is given by

Euler's Equation:

\(\frac{{dp}}{ρ } + gdz + vdv = 0\;\;\;\;\;(1)\)

In Euler's equation of motion the forces due to gravity and pressure are taken into consideration and which is derived considering the motion of a fluid element along a streamline.

Integrating the above equation (1):

\(\smallint \frac{{dp}}{ρ } + \smallint gdz + \smallint vdv = 0\)

\(\frac{p}{ρ } + gz + \frac{{{v^2}}}{2} = C\)

\(\frac {P}{ρ g}+ \frac {v^2}{2g}+ Z = Constant\)

This equation is called the Bernoulli's Equation.

where

 \(\frac{p}{ρg}\) = Pressure head or pressure energy per unit weight

 \(\frac{v^2}{2g}\) = Kinetic head or kinetic energy per unit weight

z = Potential head or potential energy by unit weight.

P = Pressure of fluid at given section V = Flow velocity at given section Z = Potential head at given section ρ = Density of fluid