= Solution
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
$$
\frac1r\frac{\partial(ru)}{\partial r}+\frac{\partial w}{\partial z}=0,
$$
$$
\rho\left(u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}\right)
=-\frac{\partial p}{\partial r}
+\mu\left(\frac{\partial^2u}{\partial r^2}+\frac1r\frac{\partial u}{\partial r}-\frac{u}{r^2}+\frac{\partial^2u}{\partial z^2}\right),
$$
$$
\rho\left(u\frac{\partial w}{\partial r}+w\frac{\partial w}{\partial z}\right)
=-\frac{\partial p}{\partial z}
+\mu\left(\frac{\partial^2w}{\partial r^2}+\frac1r\frac{\partial w}{\partial r}+\frac{\partial^2w}{\partial z^2}\right).
$$
The vertical scale is $w\sim V$. <Incompressibility> over radial scale $R$ and gap scale $h$ gives $u\sim VR/h=V/\varepsilon$. In the radial equation, inertia is smaller than transverse viscous stress by
$$
\frac{\rho Vh}{\mu}=\operatorname{Re}\ll1,
$$
while radial viscous derivatives are smaller than $u_{zz}$ by $\varepsilon^2$. The pressure scale is $\mu VR^2/h^3$. In the vertical equation, both viscous and inertial terms are smaller than the corresponding cross-gap pressure-gradient scale by at least $O(\varepsilon^2)$. Thus the leading <lubrication theory> equations are
$$
\boxed{\frac1r(ru)_r+w_z=0,
\qquad p_r=\mu u_{zz},
\qquad p_z=0.}
$$
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