We simulate numerically the surface flow of a gas-supplying companion star in a semi-detached binary system. Calculations are carried out for a region including only the mass-losing star, thus not the mass accreting star. The equation of state is that of an ideal gas characterized by a specific heat ratio gamma, and the case with gamma=5/3 is mainly studied. A system of eddies appears on the surface of the companion star: an eddy in the low pressure region near the L1 point, one around the high pressure at the north pole, and one or two eddies around the low pressure at the opposite side of the L1 point. Gas elements starting near the pole region rotate clockwise around the north pole (here the binary system rotates counter-clockwise as seen from the north pole). Because of viscosity, the gas drifts to the equatorial region, switches to the counter-clockwise eddy near the L1 point and flows through the L1 point to finally form the L1 stream. The flow field in the L1 region and the structure of the L1 stream are also considered.
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