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28 July, 05:54

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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  1. 28 July, 06:11
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    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
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