  A minimum uncertainty state for position and momentum of a fluid element is
obtained. We consider a general fluid described by the Navier-Stokes-Korteweg
(NSK) equation, which reproduces the behaviors of a standard viscous fluid, a
fluid with the capillary action and a quantum fluid, with the proper choice of
parameters. When the parameters of the NSK equation is adjusted to reproduce
Madelung's hydrodynamic representation of the Schreodinger equation, the
uncertainty relation of a fluid element reproduces the Kennard and the
Robertson-Schreodinger inequalities in quantum mechanics. The derived minimum
uncertainty state is the generalization of the coherent state and its
uncertainty is given by a function of the shear viscosity. The viscous
uncertainty can be smaller than the inviscid minimum value when the shear
viscosity is smaller than a critical value which is similar in magnitude to the
Kovtun-Son-Starinets (KSS) bound. This uncertainty reflects the information of
the fluctuating microscopic degrees of freedom in the fluid and will modify the
standard hydrodynamic scenario, for example, in heavy-ion collisions.
