Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 352 / 1 / c

Solution

The shear stress decreases monotonically from the fibre to zero at the free surface, so its maximum is

τ_{rz} (a) = \frac{Δ p}{2 L} \frac{b ^{2} - a ^{2}}{a} .

Flow occurs precisely when this exceeds the yield stress:

Δ p > \frac{2 L a σ _{y}}{b ^{2} - a ^{2}} .

When this condition holds, there is one yield surface

r = r_{y}

determined by

\frac{G}{2} (\frac{b ^{2}}{r _{y}} - r_{y}) = σ_{y} .

The region

a \leq r < r_{y}

is yielded and the outer region

r_{y} \leq r \leq b

is an unyielded plug flow of a yield-stress fluid. In the yielded region,

d u / d r = (τ_{rz} - σ_{y}) / η

. The no-slip condition

u (a) = 0

gives

u (r) = \frac{1}{η} [\frac{G}{2} {b^{2} lo g \frac{r}{a} - \frac{r ^{2} - a ^{2}}{2}} - σ_{y} (r - a)], a \leq r \leq r_{y} .

The unyielded layer translates without shearing at the plug speed

u_{p} = u (r_{y}) = \frac{1}{η} [\frac{G}{2} {b^{2} lo g \frac{r _{y}}{a} - \frac{r _{y}^{2} - a ^{2}}{2}} - σ_{y} (r_{y} - a)] .

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