Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 37A (Fluid Dynamics II)

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.

The steady rectilinear equation is $u^{\prime \prime }(y)=-G$, with no-slip conditions $u(0)=0$ and $u(a)=U$. Two integrations give the Couette-Poiseuille flowThe first term is driven by the moving wall and the second by the specified pressure gradient.

The baffles create an induced pressure gradient even though no external gradient is applied. Away from them the same Couette-Poiseuille flow has flux per unit widthZero flux therefore gives $G=-6U/a^2$ andThe positive pressure gradient drives the return flow near the stationary lower plate, balancing the forward flow near the moving plate. The rectilinear approximation excludes the short end regions next to the baffles.