Fene In Complex Fluid Dynamics Model Of Qualitative Research

The present Ph.D. thesis is concerned with some mathematical problems on hydro-dynamic system for complex fluids. Complex fluids are non-Newtonian fluids, they can be viewed as the intermediate states between the fluids and the solids. An important flu-id of the complex fluid is the polymer fluid, which has aroused extensive attention of physicists, chemists and rheologists. The author considers the finite extensible nonlinear elasticity (FENE) systems for polymeric fluids.This thesis has four parts.In Chapter1, we introduce the background of complex fluids and the FENE model. This system couples the incompressible Navier-Stokes equation for the fluid velocity with the Fokker-Planck equation describing the evolution of the polymer density. Moreover, this chapter also includes some recent results and our main works.In Chapter2, since the FENE model is of singularity at the boundary, in order to discuss the behavior of solutions near the boundary to the FENE model and the exact formulation of the well-posedness of boundary value problems, an initial boundary value problem for some modeling degenerate parabolic equation is introduced and its existence is proved by the method of regularization, and the higher order regularity of solutions is studied in Sobolev spaces without any boundary condition.In Chapter3, we focus our attention on the microscopic description in the FENE model. We prove the existence of global smooth solutions in some weighted Sobolev spaces to the microscopic FENE model if the non-dimensional parameter b>2. The regularity obtained here is intrinsic and the best when b is not even.In Chapter4, we prove the local existence of smooth solutions for the FENE dumb-bell model of polymeric flows in some weighted spaces by the fixed-point theorem if the non-dimensional parameter b>2.