A numerical hydrodynamic scheme, applicable to gas flow in binary stars, is presented which incorporates the effects of pressure and gravitational forces, radiative cooling, and shock formation. The method is used to study gas flow originating from Roche lobe overflow through the inner Lagrangian point, L$sub 1$, in a binary star modeled to resemble the dwarf nova Z Cam. Three mass transfer rates, 3times (10$sup -9$, 10$sup -8$, 10$sup -7$) M/sub sun/ year $sup - 1$, are followed for a total time of 1.50 binary orbital periods. The calculations indicate that rapid energy dissipation, via radiative cooling in a shock front formed between the stream from L$sub 1$ and the returning gas which has circuited the star, quickly transforms the flow into a stream-ring configuration. An optically thick, confined shock forms at the intersection of the stream and ring. At a transfer rate of 3times10$sup -9$ M/sub sun/ year$sup - 1$ the shock's location, temperature (23,000 K at maximum), and energy release (8times10$sup 32$ ergs s$sup -1$) are in substantial agreement with observations of Z Cam, and confirm the major elements of the ''hot spot'' model proposed to more » account for observations of cataclysmic variable stars. (AIP) « less