Fluid/gravity duality with Petrov-like boundary condition in a spacetime   with a cosmological constant
  Recently it has been shown that imposing Petrov type I condition on the boundary may reduce the Einstein's equation to the Navier-Stokes equation in the non-relativistic and near-horizon limit. In this paper we extend this framework to a spacetime with a cosmological constant. By explicit construction we show that the Navier-Stokes equation can be derived from both black brane background and spatially curved spacetime. We also conjecture that imposing Petrov type I condition on the boundary should be equivalent to the conventional method using the hydrodynamical expansion of the metric in the near horizon limit. 
Tai-Zhuo Huang, Yi Ling, Wen-Jian Pan, Yu Tian and Xiao-Ning Wu
hep-th gr-qc