= Solution
For a constant-density incompressible fluid, the <Navier-Stokes equation> and incompressibility constraint are
$$
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u
=-\rho^{-1}\nabla p+\nu\nabla^2\mathbf u,\qquad\nabla\cdot\mathbf u=0.
$$
Scale length by $L_*$, velocity by $U_*$, time by $L_*/U_*$ and pressure by $\rho U_*^2$. In dimensionless variables they become
$$
\partial_{t'}\mathbf u'+(\mathbf u'\cdot\nabla')\mathbf u'
=-\nabla'p'+\operatorname{Re}^{-1}\nabla'^2\mathbf u',\quad
\operatorname{Re}=U_*L_*/\nu.
$$
A <rectilinear flow> has velocity $\mathbf u=U(y,z)\widehat x$, parallel everywhere to one fixed direction and independent of that direction. Its convective acceleration is identically zero. A steady such flow therefore satisfies the linear equation $\nu\Delta_\perp U=\rho^{-1}dp/dx$, with no inertial term and no Reynolds-dependent change of equation or spatial shape for fixed scaled boundary/forcing data. Relative changes in imposed pressure forcing can of course change the profile; Reynolds independence is not independence of the boundary data.
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