I'm attempting to determine an algebraic equation for $\gamma'$ over $[0,1]$ from Equation (1) below; however, I'm having difficulty doing so. This is in relation to a shear-thinning generalised Newtonian fluid problem (or Carreau's Law, not that this is important).
Specifically, given
$$\frac{dp}{dx}R = \frac{d}{dy}\Big(\mu(\gamma')\frac{d}{dy}u(y)\Big) \tag{1}$$
and
$$\mu(\gamma')=\frac{\mu_\infty}{\mu_0}+\Big(1-\frac{\mu_\infty}{\mu_0}\Big)\Big(1+(\lambda \gamma')^2\Big)^\frac{(n-1)}{2} \tag{2}$$
with $\frac{dp}{dx} R = -1$, $n=0.5$, $\mu_\infty/\mu_0=0.01$ and $\lambda=1$. Furthermore, $\mu(\gamma')$ and $u(y)$ are known to be monotone decreasing functions, $\gamma'=|\frac{du}{dy}|$, $\frac{d}{dy}u(y)=0$ at $y=0$ and $u(y)=0$ at $y=1$.
Using all the information (minus the last two points about $y=0$ and $y=1$), I managed to arrive at an equation
$$100y=(1 + 99(1+\gamma'^2)^{-1/4})\gamma' \tag{3}$$
but I can't see how to proceed any further to get $\gamma'$. Any tips would be appreciated.
This non-linear equation (Equation 3) is a correct algebraic equation for $\gamma'$, however it cannot be rearranged for $\gamma'$.
To solve for $\gamma'$ one must use a numerical approach.