We present a general model for the dynamics of phase separation in polymer-solvent mixtures under shear flow, which unifies previous phenomenological theories. For the dilute case, the model can be derived in the absence of hydrodynamic interactions via the appropriate Smoluchowski equation. Linear analysis shows that the shear flow does not change the equilibrium phase boundary. We then generalize the model to the semidilute case, and find that the phase separation temperature is indeed shifted by the shear flow. The results indicate that a nonlinear concentration dependence of the modulus is necessary for a shift in the phase separation temperature. ͓S1063-651X͑97͒50306-0͔PACS number͑s͒: 61.25.Hq, 47.10.ϩg, 62.10.ϩs The dynamics of phase separation in polymer mixtures under shear flow has generated a great deal of interest over the last two decades ͓1-8͔. A number of groups have reported that the phase separation of polymer-solvent mixtures can be dramatically changed by macroscopic flow fields. Indeed, a greatly enhanced turbidity is observed in flowing polymer solutions at temperatures much higher than the equilibrium critical temperature ͓1-3͔. However, the mechanism of the observed phenomenon is not completely clear as current theoretical models provide conflicting explanations. Helfand and Fredrickson ͑HF͒ argued that the observed phenomenon is not a real phase separation, but is only a result of large-scale fluctuations in the monomer concentration induced by the shear flow ͓4͔. On the other hand, Onuki suggested that the equilibrium phase boundary is shifted to higher temperatures by the shear flow ͓6͔. Both groups are correct, and we believe the divergence in views may be attributable to the fact that the two groups used different models ͓7͔. A key to resolving this issue is to develop a more fundamental theory that will account for the complexity of the system, and this is what we report here.In the dynamics of phase separation, the basic stochastic variable is the monomer concentration (r,t), which describes the coarse-grained configuration of the system. However, in the presence of shear flow, (r,t) must couple with the fluid velocity v(r,t), and the stress tensor (r,t) of the deformed polymer chains. Since in general, the strain tensor w and are not independent variables, the stress tensor is chosen as the independent variable. The state of the system can then be described by a set of collective variables ͕,v,͖. The dynamics of the system is determined by a set of coupled Langevin equations governing the time evolution of the state ͕,v,͖. The time evolution of v is to be described by the Navier-Stokes equation. One must construct the Langevin equations for and .In principle, the Langevin equations for and can be projected out of the equation describing the evolution of the full probability distribution ͓9͔. While seminal work has already been carried out along this line ͓4,7͔, and some general features of the equation for were derived ͓4,7͔, one knows little about the equation for the stress tens...