  Over the past few decades, a host of theoretical evidence have surfaced that
suggest a connection between theories of gravity and Navier-Stokes (NS)
equation of fluid dynamics. It emerges out that gravity theory can be treated
as some kind of fluid on a particular surface. Motivated by the work carried
out by Bredberg et al (JHEP 1207, 146 (2012)) \cite{Bredberg:2011jq}, our paper
focuses on including certain modes to the metric which are consistent with the
so called hydrodynamic scaling and discuss the consequences, one of which
appear in the form of Damour Navier Stokes (DNS) equation with the
incompressibility condition. We also present an alternative route to the
results by considering the metric as a perturbative expansion in the
hydrodynamic scaling parameter $\epsilon$ and with a specific gauge choice,
thus modifying the metric. It is observed that the inclusion of certain modes
in the metric corresponds to the solution of Einstein's equations in presence
of a particular type of matter in the spacetime. This analysis reveals that
gravity has both the NS and DNS description not only on a null surface, but
also on a timelike surface. So far we are aware of, this analysis is the first
attempt to illuminate the possibility of presenting the gravity dual of DNS
equation on a timelike surface. In addition, an equivalence between the
hydrodynamic expansion and the near-horizon expansion has also been studied in
the present context.
