Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 38A (Fluid Dynamics)

Let $u(r,z)$ and $w(r,z)$ be the radial and vertical velocity components. For steady incompressible axisymmetric flow without swirl, mass conservation and the relevant cylindrical components of the Navier-Stokes equation are

The vertical scale is $w\sim V$. Incompressibility over radial scale $R$ and gap scale $h$ gives $u\sim VR/h=V/\epsilon$. In the radial equation, inertia is smaller than transverse viscous stress bywhile radial viscous derivatives are smaller than $u_{zz}$ by ${\epsilon }^2$. The pressure scale is $\mu V{R}^2/h^3$. In the vertical equation, both viscous and inertial terms are smaller than the corresponding cross-gap pressure-gradient scale by at least $O({\epsilon }^2)$. Thus the leading lubrication theory equations are

Away from the rim's $O(h)$ edge region, the geometry and uniform injection have no radial scale other than the factor required by continuity. It is therefore consistent to take $w=w(z)$. Regularity at $r=0$ then lets the continuity equation integrate toThe no-slip boundary condition and prescribed normal velocities areSince $p$ is independent of $z$, the radial equation implies that $w^{\prime \prime \prime }$ is constant. The four boundary conditions giveand henceUsing $w^{\prime \prime \prime }=12V/h^3$ in $p_r=\mu u_{zz}$ givesTaking the pressure at the rim to be atmospheric, $p(R)=p_0$, produces the porous-plate lubrication cushion pressureThe upward pressure force balances the disc's weight:Since $\epsilon =h/R$,