I. INTRODUCTION The problem of coarse graining in terms of bridging time and length scales between microscopic and macroscopic levels of description is a crucial issue in the physics of com- plex fluids, like polymer melts, colloids, liquid crystals and emulsions, out of equilibrium. The wide span of length and time scales is particularly evident in these systems, due to their internal structure leading to additional mesoscopic levels of description, intermediate between microscopic and macroscopic [1]. Many of the practical applications of these flu- ids crucially depend on the evolution of their complex, multiphase morphologies developed through equilibrium self-assembling or during non-equilibrium processing of these systems. Some examples of such applications are food processing, membrane technology, encapsula- tion systems, drug delivery, coating, production of paint and cosmetics etc [1–4]. Therefore, the goal of connecting different levels of description of complex fluids is of great importance. In the present work, this problem is approached by considering the phase separation of binary fluid mixtures subjected to a shear flow. When a binary (AB) fluid mixture is quenched from a high temperature homogeneous phase to an unstable region below the coexistence curve, it becomes unstable with respect to long-wavelength fluctuations in the composition and starts to phase separate [5]. The interface between the A- and B-rich domains is initially very diffuse, but sharpens with time. In the later stages of phase separation, local equilibrium is achieved within each domain and the effects of interfacial energy become important. The domain pattern further coarsens in time, while the dynamics is governed by the minimization of the excess interfacial energy of the system. The kinetics of this nonequilibrium phenomena is an area of extensive research [5]. Especially the effects of a shear flow on the deformation and kinetics of domain growth, and the corresponding rheological behavior, are challenging and technologically important topics [6]. The complexity of the problem lies in the presence of a complex interface and its motions due to coagulation, rupture and deformation of domains, which significantly influences the macroscopic properties of a mixture. Although numerous experimental [7– 11], numerical [12–24] and analytical [14–17, 24–28] studies have been done in order to understand the flow effects on the morphology and the dynamics in this system, many questions still remain open, like the possibility of achieving a nonequilibrium steady state [20–24]. 2