A disc of radius $R$ and weight $W$ hovers at a height $h$ on a cushion of air above a horizontal air table - a fine porous plate through which air of density $\rho$ and dynamic viscosity $\mu$ is pumped upward at constant speed $V$. You may assume that the air flow is axisymmetric with no flow in the azimuthal direction, and that the effect of gravity on the air may be ignored.

(a) Write down the relevant components of the Navier-Stokes equations. By estimating the size of the individual terms, simplify these equations when $\varepsilon:=h / R \ll 1$ and $R e:=\rho V h / \mu \ll 1$.

(b) Explain briefly why it is reasonable to expect that the vertical velocity of the air below the disc is a function of distance above the air table alone, and thus find the steady pressure distribution below the disc. Hence show that

$$W=\frac{3 \pi \mu V R}{2 \varepsilon^{3}}$$