This is from a previous exam paper that I’m going through for my upcoming exam regarding the modelling of PDE’s.
Parts a) and b) are fine but I’ve been stuck on part c) for a while. I’ve tried googling it and all I’ve come across is something called the power law. I just can’t seem to figure this out, even with the hint. Could someone please start the problem off for me or give me some hints (other than the hint given)?
The velocity is zero at the walls where $y=0,h$ due to the no-slip condition. By symmetry, the maximum velocity must be attained at the centerline $y = h/2$. Hence, the velocity gradient is positive for $0 \leqslant y \leqslant h/2$ and $\left|\frac{du}{dy} \right| = \frac{du}{dy}$ in this region.
Let $q(y) = \frac{du}{dy}$. The differential equation applicable for $0 \leqslant y \leqslant h/2$ is
$$\mu_0 \left|\frac{du}{dy} \right|^{n-1} \frac{d^2 u}{dy^2} = \mu_0q^{n-1} \frac{dq}{dy} = G,$$
where $G = \frac{dp}{dx} < 0$ is the pressure gradient which is constant for fully-developed flow.
This implies
$$q^{n-1} \frac{dq}{dy} =\frac{1}{n}\frac{dq^n}{dy} = \frac{G}{\mu_0}$$
Integrating we get
$$\left(\frac{du}{dy}\right)^n = q^n = \frac{nG}{\mu_0}y + C.$$
We can evaluate the constant $C$ by applying the condition $\frac{du}{dy} = 0$ at $y = h/2$ where the velocity has a maximum to find
$$\frac{du}{dy} = \left[\frac{nG}{\mu_0}\left( y - \frac{h}{2}\right)\right]^{1/n}.$$
I'm sure you can complete the problem from this point.