Question

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

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