Name: Fluid Mechanics
Uploaded: 2019-07-19
Duration: 6 min 9 s
Description: This video enables students to understand that fluid mechanics comprises fluid statics and fluid dynamics. Students also learn about steady and turbulent flo...

Question

Write Bernoulli’s theorem for the flow of an ideal liquid. Use it to prove that the velocity of efflux of a liquid emerging from a hole in the wall of a vessel is

\sqrt{2 g h}

, where

h

is the height of the liquid level above the hole.

Accepted Answer

When an incompressible and non-viscous liquid (or gas) flows in streamlined motion from one place to another, then at every point of its path the total energy per unit volume (Pressure energy $+$ Kinetic energy $+$ Potential energy) is constant. That is,

$P + \frac{1}{2} ρ v^{2} + ρ g h = c o n s \tan t$ .

Thus, Bernoulli’s theorem is in one way the principle of conservation of energy for a flowing liquid (or gas).

Let a vessel be filled with a liquid up to a height $H$ and let there be an orifice at a depth $h$ below the free surface of the liquid. The pressure at the free surface of the liquid and also at the orifice is atmospheric, and so there will be no effect of atmospheric pressure on the flow of liquid from the orifice. The liquid on the free surface has no kinetic energy, but only potential energy, while the liquid coming out of the orifice has both the kinetic and potential energies.

Let $P$ be the atmospheric pressure, $ρ$ the density of the liquid and $v$ the velocity of efflux of the liquid coming out from the orifice. According to Bernoulli’s theorem, the sum of the pressure and the total energy per unit volume of the liquid must be the same at the surface of the liquid and at every point of the orifice.

Thus, $P + 0 + ρ g H = P + \frac{1}{2} ρ v^{2} + ρ g (H - h)$ or

$\frac{1}{2} ρ v^{2} = ρ g h$ or

$v = \sqrt{2 g h}$

Write Bernoulli’s theorem for the flow of an ideal liquid. Use it to prove that the velocity of efflux of a liquid emerging from a hole in the wall of a vessel is $\sqrt{2 g h}$ , where $h$ is the height of the liquid level above the hole.

Important Questions on Motion in Fluids : Viscosity

Learn and Prepare for any exam you want !

Write Bernoulli’s theorem for the flow of an ideal liquid. Use it to prove that the velocity of efflux of a liquid emerging from a hole in the wall of a vessel is 2gh, where h is the height of the liquid level above the hole.

Important Questions on Motion in Fluids : Viscosity

Learn and Prepare for any exam you want !

Write Bernoulli’s theorem for the flow of an ideal liquid. Use it to prove that the velocity of efflux of a liquid emerging from a hole in the wall of a vessel is $\sqrt{2 g h}$ , where $h$ is the height of the liquid level above the hole.