  In this note, we have compared two different perturbation techniques that are
used to generate dynamical black-brane solutions to Einstein equation in
presence of negative cosmological constant. One is the `derivative expansion',
where the gravity solutions are in one-to-one correspondence with the solutions
of relativistic Navier-Stokes equation. The second is the expansion in terms of
inverse power of space-time dimensions and here the gravity solutions are dual
to a co-dimension one dynamical membrane, embedded in AdS space and coupled to
a velocity field. We have shown that in large number of space-time dimensions,
there exists an overlap regime between these two perturbation techniques and we
matched the two gravity solutions along with their dual systems upto the first
non-trivial order in the expansion parameter on both sides. In the process, we
established a one-to-one map between dynamical black-brane geometry and the AdS
space, which exists even when the number of dimensions is finite.
