The axial velocity $u(r)$ decreases toward the wall, so $\dot\gamma=-du/dr\geq0$. For a <power-law fluid>, the stress magnitude is
$$
K\dot\gamma^n=\frac{Gr}{2}.
$$
Hence
$$
\boxed{\dot\gamma(r)=
\left(\frac{Gr}{2K}\right)^{1/n}}.
$$
Integrating inward from the <no-slip boundary condition> $u(R)=0$ gives
$$
\boxed{u(r)=\frac n{n+1}
\left(\frac G{2K}\right)^{1/n}
\left(R^{1+1/n}-r^{1+1/n}\right)}.
$$
For $n=1$ this reduces to the parabolic Hagen--Poiseuille profile.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-352/1/d/solution