Left and middle: sketch of the thermodynamic path for cases (1) and (2) (see text). Σ₁ (red triangle) indicates the density and temperature of the pre-shock gas. The gas is shocked reaching the point Σ_ps. Subsequently, the gas cools adiabatically due to the expansion of the stream until the halo pressure is reached at Σ_h. Two phases separate. For case (1), phase separation occurs at Σ_h, while for case (2), it occurs at Σ_ps. In both cases, the hot component mixes with the surrounding halo gas, reaching Σ_H. For case (1), the warm component cools radiatively and isobarically to the point Σ_w. For case (2), the same point is reached but the gas takes a different thermodynamic path. Dashed black lines represent adiabats and isobars labeled A and I, respectively. The two blue-dashed curves in the left panel indicate contours of constant cooling time, t_cool ~ 0.2t_dyn,halo (upper curve) and t_cool ~ 0.12t_dyn,halo (lower curve). These curves indicate that the cooling time during the expansion remains approximately constant. (right) Analysis of the isobaric differential cooling “instability” (Eq. (8)) of low-, 10⁻³ solar (solid line) and solar-metallicity (dashed line) gas as a function of temperature. When 2 − d lnΛ∕ dlnT > 0 the gas can become heterogeneous through differential cooling if δ is sufficiently large. The post-shock temperature of our illustration lies within the region where the stream can become unstable.