Mathematical Analysis if Viscoelastic Non-Newtonian Fluids

This work is devoted to the mathematical analysis of the equations modelling the mechanical behaviour of incompressible, homogeneous, viscoelastic non-Newtonian fluids. These equations form nonlinear systems of partial differential equations, typically of mixed elliptic-hyperbolic or, in the non-stationary case, parabolic-hyperbolic type. We study these systems by suitably decoupling the elliptic and hyperbolic part. Subsequently, we are led to consider (elliptic) Stokes or Oseen systems and (hyperbolic) transport equations. The nonlinear problem are investigated by using fixed point theorems. We address the questions of existence, uniqueness, stability and asymptotic behaviour of steady solutions in different two- and three-dimensional flow geometries, i.e. in bounded domains, in exterior domains and in domains with non-compact boundaries such as a half-space, channels and pipes. Further, we discuss the existence of unsteady flows in bounded domains. Although we focus on the Rivlin-Ericksen fluids of second- and third-grade and on the fluids of the Oldroyd type (Oldroyd-B, Maxwell), our results carry over to a large number of rate and differential type models for non-Newtonian fluidsAvailable from Fundacao para a Ciencia e a Tecnologia, Servico de Informacao e Documentacao, Av. D. Carlos I, 126, 1200 Lisboa / FCT - Fundação para o Ciência e a TecnologiaSIGLEPTPortuga