  We consider the Helmholtz-Weyl decomposition on a time dependent bounded
domain $\Omega(t)$ in $\mathbb{R}^3$.
  Especially, we investigate the domain dependence of each component in the
decomposition, namely, the harmonic vector fields (i.e., $\mathrm{div}$ and
$\mathrm{rot}$ free vectors), vector potentials, and scalar potentials equipped
with suitable boundary conditions, when $\Omega(t)$ moves along to
$t\in\mathbb{R}$.
  As an application, we construct a time periodic solution of the
incompressible Navier-Stokes equations for some large boundary data with
non-zero fluxes.
