Stars lose mass in the form of supersonic winds. In a binary star system, these winds collide to produce shock waves. Such stellar wind collisions are observed in many binary star systems where, due to the orbital motion, a trailing spiral structure is produced. We present a solution method in the co-rotating frame of the stars, which allows us to consider steady-state solutions. This requires the inclusion of Coriolis and centrifugal forces. We assume efficient post-shock cooling, which allows us to consider a geometrically thin shell. A set of four ordinary differential equations (ODEs) describe the conservation of mass and momentum within the shell. It was necessary to develop Taylor series expansions to find self-consistent starting values that allow for integration of the equations out of the initial singularity. Numerical integration of the equations yields the shell shape for systems where the wind speed greatly exceeds the orbital speed. The solution generalizes the analytic solution of Cantó, Raga & Wilkin (1996) to include orbital motion.