  Employing a simplified version of the Israel-Stewart formalism for
general-relativistic shear-viscous hydrodynamics, we perform axisymmetric
general-relativistic simulations for a rotating neutron star surrounded by a
massive torus, which can be formed from differentially rotating stars. We show
that with our choice of a shear-viscous hydrodynamics formalism, the
simulations can be stably performed for a long time scale. We also demonstrate
that with a possibly high shear-viscous coefficient, not only viscous angular
momentum transport works but also an outflow could be driven from a hot
envelope around the neutron star for a time scale $\gtrsim 100\,$ms with the
ejecta mass $\gtrsim 10^{-2}M_\odot$ which is comparable to the typical mass
for dynamical ejecta of binary neutron star mergers. This suggests that massive
neutron stars surrounded by a massive torus, which are typical outcomes formed
after the merger of binary neutron stars, could be the dominant source for
providing neutron-rich ejecta, if the effective shear viscosity is sufficiently
high, i.e., if the viscous $\alpha$ parameter is $\gtrsim 10^{-2}$. The present
numerical result indicates the importance of a future high-resolution
magnetohydrodynamics simulation that is the unique approach to clarify the
viscous effect in the merger remnants of binary neutron stars by the
first-principle manner.
