  We have investigated an influence of the r-mode instability on
hypercritically accreting ($\dot{M}\sim 1M_\odot {y}^{-1}$) neutron stars in
close binary systems during their common envelope phases based on the scenario
proposed by Bethe et al. \shortcite{bethe-brown-lee}. On the one hand neutron
stars are heated by the accreted matter at the stellar surface, but on the
other hand they are also cooled down by the neutrino radiation. At the same
time, the accreted matter transports its angular momentum and mass to the star.
We have studied the evolution of the stellar mass, temperature and rotational
frequency.
  The gravitational-wave-driven instability of the r-mode oscillation strongly
suppresses spinning-up of the star, whose final rotational frequency is well
below the mass-shedding limit, typically as small as 10% of that of the
mass-shedding state. On a very short time scale the rotational frequency tends
to approach a certain constant value and saturates there as far as the amount
of the accreted mass does not exceed a certain limit to collapse to a black
hole. This implies that the similar mechanism of gravitational radiation as the
so-called Wagoner star may work in this process. The star is spun up by
accretion until the angular momentum loss by gravitational radiation balances
the accretion torque. The time-integrated dimensionless strain of the radiated
gravitational wave may be large enough to be detectable by the gravitational
wave detectors such as LIGO II.
