The simplest deviation of a fluid from non$-$Newtonian behavior is to assume that Newton's law of viscosity still holds for the deviatoric stress, i$.$e$.,$

$=Itr(e)+2e$

but where the shear viscosity is$,$ in general, a function of the rate$-$of$-$strain tensor, $e=\frac{1}{2}[u+(u{)}^T],$ where $$ is the gradient.

$=(e)$

Such a fluid is called a generalized Newtonian fluid.

$($a$)$ Consider the case of steady planar shear flow $($Couette flow$)$ between two parallel plates separated by distance $d$ with one plate moving at velocity $U$ relative to the other. Show that the steady state velocity profile is linear for a power law fluid, i$.$e$.,$

$=2m||e|{|}^{n-1}$

regardless of the particular value of $n.$ Here, $m,n$ are constants. The magnitude of a tensor is the square root of the sum of its component products,

$||e||=\sqrt[\phantom{2}]{e_{11}{e}_{11}+e_{12}{e}_{12}+...+e_{23}{e}_{23}+e_{33}{e}_{33}}$

$($b$)$ Repeat part $($a$)$ for the case of any generalized Newtonian fluid. That is$,$ prove that the velocity profile is linear as long as the viscosity only depends on the magnitude of $e.$