Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 352 / 1 / c

Solution

Put

G = Δ p / L > 0

, with flow in the positive

z

direction, so

d p / d z = - G

. Balancing pressure forces and wall shear on a cylindrical fluid volume of radius

R

L

gives

Δ p π R^{2} = 2 π R L τ_{R}, τ_{R} = \frac{Δ p R}{2 L} = \frac{GR}{2} .

Thus the wall stress is fixed before any constitutive equation is specified.

For fully developed pipe flow, the axial Cauchy equation reduces to

- G = \frac{1}{r} \frac{d}{d r} (r τ_{rz}) .

Regularity at the axis removes the

1/ r

integration constant, so

τ_{rz} (r) = - \frac{G r}{2} = - τ_{R} \frac{r}{R} .

Its magnitude rises linearly from zero at the axis to

τ_{R}

at the wall.

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