  We study a low-rank iterative solver for the unsteady Navier-Stokes equations
for incompressible flows with a stochastic viscosity. The equations are
discretized using the stochastic Galerkin method, and we consider an
all-at-once formulation where the algebraic systems at all the time steps are
collected and solved simultaneously. The problem is linearized with Picard's
method. To efficiently solve the linear systems at each step, we use low-rank
tensor representations within the Krylov subspace method, which leads to
significant reductions in storage requirements and computational costs.
Combined with effective mean-based preconditioners and the idea of inexact
solve, we show that only a small number of linear iterations are needed at each
Picard step. The proposed algorithm is tested with a model of flow in a
two-dimensional symmetric step domain with different settings to demonstrate
the computational efficiency.
