  We investigate the effect of antikaon condensed matter on bulk viscosity in
rotating neutron stars. We use relativistic field theoretical models to
construct the equation of state of neutron stars with the condensate, where the
phase transition from nucleonic to $K^-$ condensed phase is assumed to be of
first order. We calculate the coefficient of bulk viscosity due to the
non-leptonic weak interaction n --> p + K^-. The influence of antikaon bulk
viscosity on the gravitational radiation reaction driven instability in the
r-modes is investigated. We compare our results with the previously studied
non-leptonic weak interaction $n + p --> p + \Lambda$ involving hyperons on the
damping of the r-mode oscillations.
  We find that the bulk viscosity coefficient due to the non-leptonic weak
process involving the condensate is suppressed by several orders of magnitude
in comparison with the non-superfluid hyperon bulk viscosity coefficient.
Consequently, the antikaon bulk viscosity may not be able to damp the r-mode
instability, while hyperon bulk viscosity can effectively suppress r-mode
oscillations at low temperatures. Hence neutron stars containing $K^-$
condensate in their core could be possible sources of gravitational waves.
