= Solution
Write $\mu$ for <dynamic viscosity> and $\nu=\mu/\rho$ for <kinematic viscosity>, consistently with the introductory <Navier-Stokes equation>. The phrase “dynamic viscosity $\nu$” in this request mixes these conventions. A fully developed <plane Poiseuille flow>, $\mathbf u=(u(z),0,0)$, has zero <convective acceleration>. The <Navier-Stokes equation> gives $\mu u''=-f$ and $p_z=-\rho g$. Applying the <no-slip boundary condition> at both stationary plates gives
$$
\boxed{u(z)=\frac{f}{2\mu}z(b-z)=\frac{f}{2\rho\nu}z(b-z).}
$$
The material coefficients must be effectively constant and positive. The profile is an exact parallel-flow solution for every $\mu>0$; experimental validity additionally requires that the realized flow remain in the <laminar flow> regime. A sufficient conservative condition is
$$
\boxed{\mathrm{Re}=\frac{\bar u b}{\nu}=\frac{fb^3}{12\rho\nu^2}\ll1,
\qquad \bar u=\frac{fb^2}{12\mu}.}
$$
Equivalently $\nu\gg\sqrt{fb^3/(12\rho)}$, or $\mu\gg\sqrt{\rho fb^3/12}$. This sufficient bound is stronger than necessary; there is no need to discard <convective acceleration> by an approximation here because it vanishes identically for the assumed profile.
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