  The Petrov type I condition for the solutions of vacuum Einstein equations in
both of the non-relativistic and relativistic hydrodynamic expansions is
checked. We show that it holds up to the third order of the non-relativistic
hydrodynamic expansion parameter, but it is violated at the fourth order even
if we choose a general frame. On the other hand, it is found that the condition
holds at least up to the second order of the derivative expansion parameter.
Turn the logic around, through imposing the Petrov type I condition and
Hamiltonian constraint on a finite cutoff surface, we show that the stress
tensor of the relativistic fluid can be recovered with correct first order and
second order transport coefficients dual to the solutions of vacuum Einstein
equations.
