Question

prove that Bernoulli's theorem

Answer 1

Here is your answer :

Bernoulli's Theorem proves that if the fluid flows horizontally so that no change in gravitational potential energy occurs, then a decrease in fluid pressure is associated with an increase in fluid velocity. If the fluid is flowing through a horizontal pipe of varying cross-sectional area, for example, the fluid speeds up in constricted areas so that the pressure the fluid exerts is least where the  cross - section is smallest. This phenomenon is sometimes called the Venturi effect, after the Italian scientist G.B. Venturi (1746–1822), who first noted the effects of constricted channels on fluid flow.



Hope this will help you...........

Answer 2

Bernoulli's Theorem



Definition

According to Bernoulli’s theorem in physics, whenever there is an increase in the speed of the liquid, there is a simultaneous decrease in the potential energy of the fluid or we can say that there is a decrease in the pressure of the fluid. Basically, it is a principle of conservation of energy in the case of ideal fluids. If the fluid flows horizontally such that there is no change in the gravitational potential energy of the fluid then increase in velocity of the fluid results in a decrease in pressure of the fluid and vice versa.

Derivation (Proof)

We will prove the Bernoulli’s theorem here.


Let the velocity, pressure and area of a fluid column at a point XX be v1, p1v1, p1 and A1A1 and at another point YY be v2, p2v2, p2 and A2A2. Let the volume that is bounded by XX and YY be moved to MM and NN. let XMXM = L1L1 and YNYN= L2L2. Now if we can compress the fluid then we have,


A1 × L1A1 × L1 = A2 × L2A2 × L2


We know that that the work done by the pressure difference per volume of the unit is equal to the sum of the gain in kinetic energy and gain in potential energy per volume of the unit.

This implies


Work doneWork done = force × distanceforce × distance


⇒ Work done⇒ Work done = p × volumep × volume

Therefore, net work done per volume = p1 – p2p1 – p2


Also, kinetic energy per unit volume = 1212 m v2m v2 = 1212 ρ v2ρ v2

Therefore, we have,


Kinetic energy gained per volume of unit = 1212 ρ ((v2)2 – (v1)2)ρ ((v2)2 – (v1)2)


And potential energy gained per volume of unit = p g (h2 – h1)p g (h2 – h1)


Here, h1h1 and h2h2 are heights of XX and YYabove the reference level taken in common. 


Finally we have


p1 – p2p1 – p2 = 1212 ρ ((v2)2 – (v1)2) + ρ g (h2 – h1)ρ ((v2)2 – (v1)2) + ρ g (h2 – h1)


⇒ p1⇒ p1 + 1212 ρ (v1)2 + ρ g h1ρ (v1)2 + ρ g h1 = p2p2 + 1212 ρ (v2)2 + ρ g h2ρ (v2)2 + ρ g h2


⇒ p⇒ p + 1212 ρ v2ρ v2 + ρ g hρ g h is a constant


When we have h1h1 = h2h2


Then we have, pp + 1212 ρ v2ρ v2 is a constant.


This proves the Bernoulli’s Theorem