Name: Capillary Rise
Uploaded: 2019-07-19
Duration: 10 min 29 s
Description: Have you ever seen what happens to a celery stalk dipped in red-coloured water? We notice that after a day or two, the leaves turn reddish! Let's learn the r...

Question

State Bernoulli's principle. From the conservation of energy in a fluids flow through a tube, arrive at Bernoulli's equation. Give an application of Bernoulli's theorem.

Accepted Answer

Bernoulli's principle states that as we move along a streamline the sum of the pressure, the kinetic energy per unit volume, and the potential energy per unit volume remain constant. Bernoulli's equation gives the relation between the pressure difference between two points in a pipe to both velocity changes(kinetic energy change) and elevation change(potential energy change).

$P + \frac{1}{2} ρ v^{2} + ρ g h = constant$

Consider the flow of ideal fluids at two regions 1 and 2 as shown in the below figure.

Fig: The flow of ideal fluids in a pipe of varying cross-section.

The work done on the fluids at the left end is

$W_{1} = P_{1} A_{1} (v_{1} △ t) = P_{1} △ V$

The work done on the fluids on the right side,

$W_{2} = P_{2} A_{2} (v_{2} △ t) = P_{2} △ V$

Now, the total work done on the fluids is

$W_{1} - W_{2} = (P_{1} - P_{2}) △ V$

The change in gravitational potential energy is,

$△ U = ρ g △ V (h_{2} - h_{1})$

The change in kinetic energy is

$△ K = \frac{1}{2} ρ △ V ({v_{2}}^{2} - {v_{1}}^{1})$

By applying the work-energy theorem, we get

$(P_{1} - P_{2}) △ V = \frac{1}{2} ρ △ V ({v_{2}}^{2} - {v_{1}}^{2}) + ρ g △ V (h_{2} - h_{1})$

Divided by $△ V$ by both side, we get

$(P_{1} - P_{2}) = \frac{1}{2} ρ ({v_{2}}^{2} - {v_{1}}^{2}) + ρ g (h_{2} - h_{1})$

In other words, Bernoulli's relation may be stated as follows: As we move along a streamline the sum of the pressure $(P)$ , the kinetic energy per unit volume $(\frac{ρ v^{2}}{2})$ , and the potential energy per unit volume $(ρ g h)$ remain a constant.

Application of Bernoulli's theorem:

1. Venturi meter which is base on the principle of Bernoulli's equation is used to measure the speed of incompressible fluids.

2. The speed of the free-falling body at a certain height.

3. This principle also helps to explain the blood flow in the artery.

State Bernoulli's principle. From the conservation of energy in a fluids flow through a tube, arrive at Bernoulli's equation. Give an application of Bernoulli's theorem.

Important Questions on Mechanical Properties of Fluids

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