  Reynolds' lubrication approximation is used extensively to study flows
between moving machine parts, in narrow channels, and in thin films. The
solution of Reynolds' equation may be thought of as the zeroth order term in an
expansion of the solution of the Stokes equations in powers of the aspect ratio
$\epsilon$ of the domain. In this paper, we show how to compute the terms in
this expansion to arbitrary order on a two-dimensional, $x$-periodic domain and
derive rigorous, a-priori error bounds for the difference between the exact
solution and the truncated expansion solution. Unlike previous studies of this
sort, the constants in our error bounds are either independent of the function
$h(x)$ describing the geometry, or depend on $h$ and its derivatives in an
explicit, intuitive way. Specifically, if the expansion is truncated at order
$2k$, the error is $O(\epsilon^{2k+2})$ and $h$ enters into the error bound
only through its first and third inverse moments $\int_0^1 h(x)^{-m} dx$,
$m=1,3$ and via the max norms $\big\|\frac{1}{\ell!} h^{\ell-1} \partial_x^\ell
h\big\|_\infty$, $1\le\ell\le2k+2$. We validate our estimates by comparing with
finite element solutions and present numerical evidence that suggests that even
when $h$ is real analytic and periodic, the expansion solution forms an
asymptotic series rather than a convergent series.
