  In this paper we have examined numerically exact configurations of close
binary systems composed of incompressible fluids with internal flows. Component
stars of binary systems are assumed to be circularly orbiting each other but
rotating nonsynchronously with the orbital motion, i.e. stars in binary systems
have steady motions seen from a rotational frame of reference. We have computed
several equilibrium sequences by taking fully into account the tidal effect of
Newtonian gravity without approximation. We consider two binary systems
consisting of either 1) a point mass and a fluid star or 2) a fluid star and a
fluid star. Each of them corresponds to generalization of the Roche--Riemann or
the Darwin--Riemann problem, respectively. Our code can be applied to various
types of incompressible binary systems with various mass ratios and spin as
long as the vorticity is constant. We compare these equilibrium sequences of
binaries with approximate solutions which were studied extensively by Lai,
Rasio and Shapiro (LRS) as models for neutron star--neutron star (NS--NS)
binary systems or black hole--neutron star (BH--NS) binary systems. Our results
coincide qualitatively with those of LRS but are different from theirs for
configurations with small separations. For these binary systems, our sequences
show that dynamical or secular instability sets in as the separation decreases.
The quantitative errors of the ellipsoidal approximation amount to 2\sim 10%
for configurations near the instability point. Compared to the results of LRS,
the separation of the stars at the point where the instability sets in is
larger and correspondingly the orbital frequency at the critical state is
smaller for our models.
