Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 37A / Solution

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.