Question

proof for bernauli's theorem

Answer 1

Hey mate here is your answer

Proof of Bernoulli's theorem. Consider a fluid of negligible viscosity moving with laminar flow, this is Bernoulli's theorem You can see that if there is a increase in velocity there must be a decrease of pressure and vice versa.

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Answer 2

HELLO MATE

Here is ur answer

okk see

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.

HERE IS THE PROOF :)

We will prove the Bernoulli’s theorem here.

Let the velocity, pressure and area of a fluid column at a point X be v1, p1 and A1 and at another point Y be v2, p2 and A2. Let the volume that is bounded by X and Y be moved to M and N. let XM = L1 and YN = L2.

Now if we can compress the fluid then we have,

A1 × L1

= A2 × 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 done

= force × distance

⇒ Work done

= p × volume

Therefore, net work done per volume = p1 – p2

Also, kinetic energy per unit volume = 12

m v2 = 12 ρ v2

Therefore, we have,

Kinetic energy gained per volume of unit = 12

ρ ((v2)2 – (v1)2)

And potential energy gained per volume of unit

= p g (h2 – h1)

Here, h1

and h2 are heights of X and Y

above the reference level taken in common.

Finally we have

p1 – p2

= 12 ρ ((v2)2 – (v1)2) + ρ g (h2 – h1)

⇒ p1

+ 12 ρ (v1)2 + ρ g h1 = p2 + 12 ρ (v2)2 + ρ g h2

⇒ p

+ 12 ρ v2 + ρ g h

is a constant

When we have h1

= h2

Then we have, p

+ 12 ρ v2

is a constant.

This proves the Bernoulli’s Theorem

Thanks :)

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