Recollements from exact model structures and heart constructions in triangulated categories

We consider a complete hereditary cotorsion pair (A,B) in a Grothendieck category G such that A contains a generator of finite projective dimension. The derived category D(B) of the exact category B is defined as the quotient category of the category Ch(B), of unbounded cochain complexes with terms in B, modulo the subcategory tilde(B) consisting of acyclic complexes with terms and cycles in B. We prove that there are two recollements anologous to the classical one, with middle term being respectively D(B) and K(B). We study also some cases where there is a recollement involving both K(B) and D(B). Simmetrically, we prove analogous results for the exact category A. We also introduce the notion of Nakaoka contexts in additive categories as couples of torsion pairs t_1=(T_1,F_1) and t_2=(T_2,F_2) such that T_2 is included in T_1. We give a set of axioms for a Nakaoka context in order to ensure that the heart, i.e. the intersection between T_1 and F_2, is Abelian. Then, we inspect the properties of Nakaoka contexts in Abelian and triangulated categories. In particular, given a t-structure t_1 in a triangulated category, we are able to find a bijection between the Nakaoka contexts (t_1,t_2) with Abelian heart and the cohereditary torsion pairs in the heart of t_1.