ABSTRACT We study the effects of Kelvin–Helmholtz Instability (KHI) on the cold streams that feed massive haloes at high redshift, generalizing our earlier results to include the effects of radiative cooling and heating from a UV background, using analytic models and high resolution idealized simulations. We currently do not consider self-shielding, thermal conduction, or gravity. A key parameter in determining the fate of the streams is the ratio of the cooling time in the turbulent mixing layer which forms between the stream and the background following the onset of the instability, $t_{\rm cool,\, mix}$, to the time in which the mixing layer expands to the width of the stream in the non-radiative case, tshear. This can be converted into a critical stream radius, Rs, crit, such that $R_{\rm s}/R_{\rm s,crit}=t_{\rm shear}/t_{\rm cool,\, mix}$. If Rs < Rs, crit, the non-linear evolution proceeds similarly to the non-radiative case studied by Mandelker et al. If Rs > Rs,crit, which we find to almost always be the case for astrophysical cold streams, the stream is not disrupted by KHI. Rather, background mass cools and condenses on to the stream, and can increase the mass of cold gas by a factor of ∼3 within 10 stream sound crossing times. The mass entrainment induces thermal energy losses from the background and kinetic energy losses from the stream, which we model analytically. Roughly half of the dissipated energy is radiated away from gas with $T \lt 5\times 10^4\, {\rm K}$, suggesting much of it will be emitted in Ly α.