= Solution
Linearize about a relaxed state $\mathsf T=0$ with constant <mass density> $\rho_0$. In the unforced <momentum conservation> equation $\rho_0D_t\mathbf u=-\nabla p+\nabla\cdot\mathsf T$, the $x$ component of the <pressure gradient> is zero for a freely evolving <unidirectional flow> independent of $x$. A prescribed external <pressure gradient> would add a forcing term.
For $\mathbf u=u(y,t)\widehat{\mathbf x}$, <incompressibility> is automatic and the only nonzero entries of the <rate-of-strain tensor> are $E_{xy}=E_{yx}=\frac12\partial_yu$. Products of <stress> and <velocity gradient> are quadratic in the small amplitude. The <constitutive equation> consequently becomes $\mathsf T+\tau\partial_t\mathsf T=2\mu\mathsf E$ to first order. With $T=T_{xy}=T_{yx}$, its shear component and the $x$ component of <momentum conservation> are
$$
\boxed{\rho_0\partial_tu=\partial_yT,\qquad T+\tau\partial_tT=\mu\partial_yu.}
$$
The diagonal <stress> components have only homogeneous relaxation at this order and vanish if initially relaxed. In particular, $\alpha$ does not enter this linear response.
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