  Concerned with the Stokes systems with rapidly oscillating periodic
coefficients, we mainly extend the recent works in \cite{SGZWS,G} to those in
term of Lipschitz domains.
  The arguments employed here are quite different from theirs, and the basic
idea comes from \cite{QX2}, originally motivated by \cite{SZW2,SZW12,TS}. We
obtain an almost-sharp $O(\varepsilon\ln(r_0/\varepsilon))$ convergence rate in
$L^2$ space, and a sharp $O(\varepsilon)$ error estimate in
$L^{\frac{2d}{d-1}}$ space by a little stronger assumption.
  Under the dimensional condition $d=2$, we also establish the optimal
$O(\varepsilon)$ convergence rate on pressure terms in $L^2$ space. Then
utilizing the convergence rates we can derive the $W^{1,p}$ estimates uniformly
down to microscopic scale $\varepsilon$ without any smoothness assumption on
the coefficients, where $|\frac{1}{p}-\frac{1}{2}|<\frac{1}{2d}+\epsilon$ and
$\epsilon$ is a positive constant independent of $\varepsilon$. Combining the
local estimates, based upon $\text{VMO}$ coefficients, consequently leads to
the uniform $W^{1,p}$ estimates. Here the proofs do not rely on the well known
compactness methods.
