Question

how do you derive the equation for Bernauli's principle?? (any easy way ,like short one , also need the real one) <3​

Answer 1

Answer:

To derive Bernoulli's equation, we first calculate the work that was done on the fluid: dW=F1dx1−F2dx2=p1A1dx1−p2A2dx2=p1dV−p2dV=(p1−p2)dV.

Answer 2

Answer:

Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.

To prove Bernoulli's theorem, consider a fluid of negligible viscosity moving with laminar flow, as shown in Figure.

Let the velocity, pressure and area of the fluid column be p1p1, v1v1 and A1A1 at Q and p2p2, v2v2 and A2A2 at R. Let the volume bounded by Q and R move to S and T where QS =L1L1, and RT = L2L2.

If the fluid is incompressible:

The work done by the pressure difference per unit volume = gain in kinetic energy per unit volume + gain in potential energy per unit volume. Now:

A1L1=A2L2A1L1=A2L2

Work done is given by:

W=F×d=p×volume⇒Wnet=p1−p2W=F×d=p×volume⇒Wnet=p1−p2

⇒K.E=12mv2=12Vρv2=12ρv2(∵V=1)⇒K.Egained=12ρ(v22−v12)⇒K.E=12mv2=12Vρv2=12ρv2(∵V=1)⇒K.Egained=12ρ(v22−v12)

P1+12ρv12+ρgh1=P2+12ρv22+ρgh2∴P+12ρv2+ρgh=const.P1+12ρv12+ρgh1=P2+12ρv22+ρgh2∴P+12ρv2+ρgh=const.

For a horizontal tube

∵h1=h2∴P+12ρv2=const.∵h1=h2∴P+12ρv2=const.

Therefore, this proves Bernoulli's theorem. Here we can see that if there is an increase in velocity there must be a decrease in pressure and vice versa.

Explanation:

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