In fully-developed laminar pipe flow, consider the rate of work done on an annulus of thickness dr (hint: consider, for each face of the annulus, rate of work = power = force * velocity).

(i) Find an expression for the power (per unit volume) dissipated by the flow in the fluid annulus, and show that it is equal to µ (du/dr)

(ii) By using u (r) from 3 (ii) above, and integrating this expression, show that the power dissipated across a length of pipe is Q∆P

Question

Engineering

Mckinley Townsend

5 September, 00:56

In fully-developed laminar pipe flow, consider the rate of work done on an annulus of thickness dr (hint: consider, for each face of the annulus, rate of work = power = force * velocity).

(i) Find an expression for the power (per unit volume) dissipated by the flow in the fluid annulus, and show that it is equal to µ (du/dr)

(ii) By using u (r) from 3 (ii) above, and integrating this expression, show that the power dissipated across a length of pipe is Q∆P

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Answers (1)

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Hadley Nolan · Answer 1

We write the share stress relationship for the pipe

t = u du/dy = u du/dr

Now calculating the force, we have

F = tA = udu/dr (A)

Cal. The power

∆P = Fdu

=udu/dr (Adu)

Now we cal. The power per unit volume

∆P = u (du/dr) ^2=Adr

Now power per unit length will be

Fdu = udu/dr (Adu)

=udu/dr (Adu)

=Q∆P