Question

Do any one know about Bernoulli' s theorem. pls with derivation.​

Answer 1

Answer:

The total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant. Bernoulli's principle can be derived from the principle of conservation of energy.

Answer 2

Answer:

Explanation:

Bernoulli's equation simply states that total energy per unit mass of flowing fluid, at any point in the subsurface, is the sum of the kinetic, potential, and fluid-pressure energies and is equal to a constant value.

General Form:

$P+\frac{1}{2}+pv^2+pgh=constant$

where P is static pressure, ρ is fluid density, v is fluid speed, h is height above some datum, and g is the acceleration of gravity.

A number of variations of the basic equation have practical advantages. For example, if we are interested in comparing two specific locations along a streamline, the constant drops out and we just look at differences between location 1 and location 2:

$(P_{2} -P_{1} )+\frac{1}{2} p(v_{2} ^2-v_{1} ^2)+pg(h_{2} -h_{1} )=0$

If net changes in height are negligible, h2 – h1 ≈ 0 so the gravitational term can be neglected:

$P_{2} -P_{1} +\frac{1}{2} p v_{2} ^2-v_{1} ^2$

In this form, Bernoulli's equation illustrates that pressure varies inversely with the square of speed along a streamline: doubling the speed will produce a four-fold drop in pressure.

Finally, if we set v = 0 (the fluid is at rest), we get the standard manometric pressure equation:

$P_{2} -P_{1} =pg(h_{2} -h_{1} )$

NOW ONTO YOUR QUESTION:

The Bernoulli theorem expresses the law of flow in conduits. For a constant discharge in an open conduit, the theorem states that the energy head at any cross section must equal that at any other downstream section plus the intervening losses. Thus above any datum:

$Z_{1} +\frac{V_{1}^2 }{2g} =Z_{2} +\frac{V_{2}^2 }{2g} +h_{c}$

n Figure 4, Z is the elevation of a free water surface above datum whether it be in a piezometer tube or a quiescent or moving surface of a stream, V the mean velocity, hc the conduit losses between the two sections considered, and e the energy head above the chosen datum. Obviously, Z may be made up of a number of elements such as elevation of streambed and depth of water in an open channel y.

I have attached the figure in the answer.

I hope you understood!