Question

According to Bernoulli's equation the expression which remains constant is:
A. \[P + \dfrac{{\rho {v^2}}}{2}\]
B. \[P + \dfrac{{\rho {v^2}}}{2} - \rho gh\]
C. \[P + \rho gh\]
D. \[P + \rho gh + \dfrac{{\rho {v^2}}}{2}\]

Answer 1

Hint: First of all, we will write Bernoulli's equation and then we will multiply both sides by the density of the fluid. Conservation of energy is explained by it.

Complete step by step answer:
According to Bernoulli's theorem, in any cross section, i.e. during flow, for the streamlined flow of an ideal liquid the total energy per unit mass is constant (sum of pressure energy, potential energy and kinetic energy).
Where,
\[\dfrac{P}{\rho } + gh + \dfrac{1}{2}{v^2} = C\] is constant.
We multiply both sides by \[\rho \],
We have,
$P + \rho gh + \dfrac{{\rho {v^2}}}{2} \\
= C \times \rho \\
= C' \\$
This is another constant.
In the above expression,
\[P\] indicates pressure energy.
\[\rho gh\] indicates potential energy per unit volume.
\[\dfrac{{\rho {v^2}}}{2}\] indicates kinetic energy per unit volume.
Hence, the required answer is \[P + \rho gh + \dfrac{{\rho {v^2}}}{2}\].

So, the correct answer is “Option D”.

Additional Information:
Bernoulli's theorem: The velocity and the elevation of fluid mobile (liquid or gas) are insignificant and the fluid of which is steady or laminar compressible and viscosity (internal friction). The first derivation is that the theorem taken from the Swiss mathematician Daniel Bernoulli notes that the total mechanical energy of a fluid, which includes fluid pressure energy, elevation gravitational potential energy and fluid kinetic movement energy, is constant. Bernoulli’s theorem for steady or streamline flows is the concept of energy savings for ideal fluids, and for many engineering applications. The theorem of Bernoulli this means that a reduction in fluid pressure is connected with a rise in fluid velocity if the fluid fluctuates horizontally so that no change in the gravitational potential energy occurs. For example, when the fluid flows through a horizontal pipe with different cross-sections, fluid speeds up in narrow areas to a lesser degree than the pressures exerted by the fluid. Often after the Italian scientist G.B., this phenomenon is called the Venturi effect. Venturi, who noticed the first impact on fluid flow by the restricted channels.

Note:
It is important to note that Bernoulli’s equation is based on the principle of conservation of energy. The fluid velocity increases when it passes through a narrow path of smaller cross section than the velocity at larger cross section. In every case, energy is always conserved.