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

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- Solution d

Solution

The axial velocity

u (r)

decreases toward the wall, so

\overset{γ}{˙} = - d u / d r \geq 0

. For a power-law fluid, the stress magnitude is

K \overset{γ}{˙}^{n} = \frac{G r}{2} .

Hence

\overset{γ}{˙} (r) = (\frac{G r}{2 K})^{1/ n} .

Integrating inward from the no-slip boundary condition

u (R) = 0

gives

u (r) = \frac{n}{n + 1} (\frac{G}{2 K})^{1/ n} (R^{1 + 1/ n} - r^{1 + 1/ n}) .

For

n = 1

this reduces to the parabolic Hagen--Poiseuille profile.

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