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For a constant-density incompressible fluid, the Navier-Stokes equation and incompressibility constraint areScale length by $L_{\ast }$, velocity by $U_{\ast }$, time by $L_{\ast }/U_{\ast }$ and pressure by $\rho U_{\ast }^2$. In dimensionless variables they becomeA rectilinear flow has velocity $\mathbf{u}=U(y,z)\hat{x}$, 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.

A rectilinear flow has velocity everywhere parallel to a fixed direction and independent of the coordinate in that direction, for example $U(y,z)\hat{x}$. Its convective acceleration vanishes. A steady incompressible viscous rectilinear flow obeys the linear transverse Poisson equation $\nu \Delta U={\rho }^{-1}dp/dx$, rather than an inertia-viscosity balance. This explains Reynolds-number independence of the spatial equation with fixed scaled forcing and boundary data.