 Solution

Let $\mathrm{Re}=\rho UL/\mu$ and use the given anisotropic scales. Continuity makes the transverse velocity scale $\delta U$ consistent with the gap height $\delta L$. The streamwise viscous stress divergence has scale $\mu U/({\delta }^2{L}^2)$, while the streamwise pressure gradient has scale $\Pi /L$. Their balance selectsThe streamwise equation then has the dimensionless formThe normal equation becomesWhen $\delta \ll 1$ and ${\delta }^2\mathrm{Re}\ll 1$, neglecting the indicated smaller terms givesThe bottom conditions express no slip on a plate moving at speed $U$ and no penetration. The top conditions express no slip and no penetration on the stationary wall. The leading pressure is a function of the streamwise coordinate alone.