We present a model for the creation of non-thermal particles via diffusive shock acceleration in a colliding-wind binary. Our model accounts for the oblique nature of the global shocks bounding the wind-wind collision region and the finite velocity of the scattering centres to the gas. It also includes magnetic field amplification by the cosmic ray induced streaming instability and the dynamical back reaction of the amplified field. We assume that the injection of the ions and electrons is independent of the shock obliquity and that the scattering centres move relative to the fluid at the Alfv{e}n velocity (resulting in steeper non-thermal particle distributions). We find that the Mach number, Alfv{e}nic Mach number, and transverse field strength vary strongly along and between the shocks, resulting in significant and non-linear variations in the particle acceleration efficiency and shock nature (turbulent vs. non-turbulent). We find much reduced compression ratios at the oblique shocks in most of our models compared to our earlier work, though total gas compression ratios that exceed 20 can still be obtained in certain situations. We also investigate the dependence of the non-thermal emission on the stellar separation and determine when emission from secondary electrons becomes important. We finish by applying our model to WR 146, one of the brightest colliding wind binaries in the radio band. We are able to match the observed radio emission and find that roughly 30 per cent of the wind power at the shocks is channelled into non-thermal particles.
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