Question

prove Bernoulli's principle. ​

Answer 1

Answer: 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 p1, v1 and A1 at Q and p2, v2 and A2 at R. Let the volume bounded by Q and R move to S and T where QS =L1, and RT = L2.

Explanation:

Answer 2

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.

The density of the incompressible fluid remains constant at both points.

The energy of the fluid is conserved as there are no viscous forces in the fluid.

Therefore, the work done on the fluid is given as:

dW = F1dx1 – F2dx2

dW = p1A1dx1 – p2A2dx2

dW = p1dV – p2dV = (p1 – p2)dV

We know that the work done on the fluid was due to conservation of gravitational force and change in kinetic energy. The change in kinetic energy of the fluid is given as:

dK=12m2v22−12m1v21=12ρdV(v22−v21)

The change in potential energy is given as:

dU = mgy2 – mgy1 = ρdVg(y2 – y1)

Therefore, the energy equation is given as:

dW = dK + dU

(p1 – p2)dV = 12ρdV(v22−v21) + ρdVg(y2 – y1)

(p1 – p2) = 12ρ(v22−v21) + ρg(y2 – y1)

Rearranging the above equation, we get

p1+12ρv21+ρgy1=p2+12ρv22+ρgy2

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