  We study the time-dependent Navier-Stokes equations in the context of
stochastic finite element discretizations. Specifically, we assume that the
viscosity is a random field given in the form of a generalized polynomial chaos
expansion, and we use the stochastic Galerkin method to extend the methodology
from [D. A. Kay et al., \textit{SIAM J. Sci. Comput.} 32(1), pp. 111--128,
2010] into this framework. For the resulting stochastic problem, we explore the
properties of the resulting stochastic solutions, and we also compare the
results with that of Monte Carlo and stochastic collocation. Since the
time-stepping scheme is fully implicit, we also propose strategies for
efficient solution of the stochastic Galerkin linear systems using a
preconditioned Krylov subspace method. The effectiveness of the stochastic
Galerkin method is illustrated by numerical experiments.
