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

What is Bernoulli&rsquo;s principle? Explain any one application of it.

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).

Venturimeter is a device based on Bernoulli’s theorem by which we can know the rate of flow of water in a tube.

It consists of a tube $X Y$ , whose middle part $Z$ is narrow. It is joined with two vertical tubes at $X$ and $Z$ . This meter is connected with the tube in which the rate of flow of water is to be determined.
When water flows in the tube $X Y$ , the velocity of flow at $Z$ is larger than at $X$ . Hence, the pressure at Z is less than that at $X$ . This pressure-difference is measured by the vertical tubes. From this, the rate of flow of water in the tube can be calculated.

Let $A_{1}$ and $A_{2}$ be the areas of cross-section of the tube, $v_{1}$ and $v_{2}$ the velocities of flow of water and $p_{1}$ and $p_{2}$ the pressures at $X$ and $Z$ respectively.
If we assume that the flow of water in the tube is streamlined, then by the principle of continuity, we have,

$A_{1} v_{1} = A_{2} v_{2}$ or

$\frac{A_{1}}{A_{2}} = \frac{v_{2}}{v_{1}} . . . . . (i)$

According to Bernoulli's theorem,

$p_{1} - p_{2} = \frac{1}{2} ρ ({v_{2}}^{2} - {v_{1}}^{2}) = \frac{1}{2} ρ {v_{1}}^{2} (\frac{{v_{2}}^{2}}{{v_{1}}^{2}} - 1)$

From equation $(i)$ , we get,

$p_{1} - p_{2} = \frac{1}{2} ρ {v_{1}}^{2} (\frac{{A_{1}}^{2}}{{A_{2}}^{2}} - 1) = \frac{1}{2} ρ {v_{1}}^{2} (\frac{{A_{1}}^{2} - {A_{2}}^{2}}{{A_{2}}^{2}})$

If $h$ be the difference in levels of water in the vertical tubes, then $p_{1} - p_{2} = h ρ g$ .

Thus, $h ρ g = \frac{1}{2} ρ {v_{1}}^{2} (\frac{{A_{1}}^{2} - {A_{2}}^{2}}{{A_{2}}^{2}})$ or

${v_{1}}^{2} = {A_{2}}^{2} \frac{2 g h}{{A_{1}}^{2} - {A_{2}}^{2}}$ or

$v_{1} = A_{2} \sqrt{\frac{2 g h}{{A_{1}}^{2} - {A_{2}}^{2}}}$

If $Q$ be the volume of water flowing per second in the tube $X Y$ , then

$Q = A_{1} v_{1} = A_{1} A_{2} \sqrt{\frac{2 g h}{{A_{1}}^{2} - {A_{2}}^{2}}}$

What is Bernoulli’s principle? Explain any one application of it.

Important Questions on Motion in Fluids : Viscosity

Learn and Prepare for any exam you want !