= Solution
Let the flow direction be $x$, the gradient direction $y$, and the vorticity direction $z$, with <simple shear flow> $u=(s y,0,0)$ and signed <shear rate> $s=\dot\gamma$. Use the gradient convention established in part (a):
$$
\omega=\begin{pmatrix}0&-s&0\\s&0&0\\0&0&0\end{pmatrix},\qquad
\dot\gamma=\begin{pmatrix}0&s&0\\s&0&0\\0&0&0\end{pmatrix}.
$$
At steady homogeneous shear the ordinary time derivative and advection vanish. Put $X=\sigma_{xx}$, $Y=\sigma_{yy}$ and $S=\sigma_{xy}$. The <corotational Maxwell fluid> law then gives
$$
X-\lambda s S=0,\qquad Y+\lambda s S=0,\qquad
S+\frac{\lambda s}{2}(X-Y)=\mu s.
$$
The $zz$ equation gives $\sigma_{zz}=0$, and the unforced $xz,yz$ equations force those components to zero. Solving the three displayed equations gives the <steady shear of a corotational Maxwell fluid>:
$$
S=\frac{\mu s}{1+\lambda^2s^2},\qquad
X=\frac{\mu\lambda s^2}{1+\lambda^2s^2},\qquad Y=-X.
$$
The <deviatoric stress> is indeed traceless. Its isotropic pressure contribution is irrelevant to either shear viscosity or a <normal-stress difference>.
Consequently the steady <shear viscosity> is
$$
\boxed{\eta_{\rm sh}(s)=\frac{S}{s}=\frac{\mu}{1+\lambda^2s^2},\qquad \eta_{\rm sh}(0)=\mu.}
$$
For $s>0$, $d\eta_{\rm sh}/ds=-2\mu\lambda^2s/(1+\lambda^2s^2)^2<0$ when $\mu,\lambda>0$; it decreases with $|s|$ and tends to zero at large rate. This proves <shear thinning> directly, without confusing a decreasing viscosity with a necessarily decreasing shear stress.
With the standard definitions $N_1=\sigma_{xx}-\sigma_{yy}$ and $N_2=\sigma_{yy}-\sigma_{zz}$,
$$
\boxed{N_1=\frac{2\mu\lambda s^2}{1+\lambda^2s^2}>0,\qquad
N_2=-\frac{\mu\lambda s^2}{1+\lambda^2s^2}=-\frac12N_1<0.}
$$
At zero shear both vanish; at small shear they are quadratic, consistently with part (b). The signs and the fact $|N_2|<N_1$ agree qualitatively with many polymeric-fluid experiments. However the fixed ratio $|N_2|/N_1=1/2$ often overestimates the measured second difference and cannot capture variation of that ratio with material and shear rate. For example, https://onlinelibrary.wiley.com/doi/full/10.1002/app.52094[measurements on polyethylene] show a smaller, rate-dependent relative second normal-stress difference. Thus \b[the qualitative normal-stress signs are reasonable, but the predicted relative magnitude is not generally quantitatively satisfactory]; these signs are not a universal statement about every non-Newtonian fluid.
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