Matter makes a transition from non‐relativistic to relativistic regime, as it falls onto a black hole. We employ a relativistic equation of state, abbreviated as RC, to study multi‐species fluid flow around black holes. We show that pair‐plasma fluid around a black hole is thermally not relativistic. In order to make it relativistic, a finite baryon loading is necessary. As a consequence of this, pair‐plasma flow do not suffer centrifugal pressure driven shock in accretion. However, fluid with finite baryon content may undergo shock transition.