  An equation for a viscous incompressible fluid on a spheroidal surface which
is dual to the perturbation around the near-near horizon extreme Kerr (n-NHEK)
black hole is derived. It is also shown that an expansion scalar $\theta$ of a
congruence of null geodesics on the null horizon of the perturbed n-NHEK
spacetime, which is dual to a viscous incompressible fluid, is not positive
semi-definite, even if initial conditions on the velocity are smooth. Unless
initial conditions are elaborated, caustics of null congruence will occur on
the horizon in the future. A similar result is obtained for a perturbed
Schwarzschild black hole spacetime which is dual to a viscous incompressible
fluid on $S^2$. An initial condition that $\theta$ be positive semi-definite at
any point on $S^2$ is a necessary condition for the existence of smooth
solutions to incompressible Navier-Stokes (NS) equation on $S^2$.
