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An equation for a viscous incompressible fluid on a spheroidal surface whichis 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 acongruence of null geodesics on the null horizon of the perturbed n-NHEKspacetime, which is dual to a viscous incompressible fluid, is not positivesemi-definite, even if initial conditions on the velocity are smooth. Unlessinitial conditions are elaborated, caustics of null congruence will occur onthe horizon in the future. A similar result is obtained for a perturbedSchwarzschild black hole spacetime which is dual to a viscous incompressiblefluid on $S^2$. An initial condition that $\theta$ be positive semi-definite atany point on $S^2$ is a necessary condition for the existence of smoothsolutions to incompressible Navier-Stokes (NS) equation on $S^2$.

Kerr/Fluid Duality and Singularity of Solutions to the Fluid Equation