北京大学硕士论文.pdf

К???????????? ????????????? ?? ??? ?? 1 0 1 0 1 0 4 0 ? ?? ??????? ?? ????? ?? ????????? ?? ???????????К????????? 2004?5????????????Ч?????Ч?????????Ч??????????Ч???????????????????????????????????θ???????????????????? ??????????????????????????ē?????????????????????Maier-Saupe??????????????ē?????????????????????ē?????????????????????????????????????????????–????????????????Galerkin?????????–??????????????????ē???????????????????????????????????????????????????????????????????????Я???????????????????????????????????????????????O(h 2+δt+ 1 √M ). Ч?????????????????????– spurt?????????????????spurt??????????spurt????????э?????????ē?????????????????????ē?????1 ??????Hookean???FENE???FENE??????э?????????????????????spurt??????ē????2?FENE????????????????Fokker Planck?????????????????????????????????????????ē??3?FENE??????????????1???????2?FENE???3?FENE????э????????Ч????ē ???: Doi?????????????????????????????????spurt???????э?????????????FENE??ē i ii Abstract In Part One, We consider the stochastic model of concentrated Liquid Crystal Polymers(LCPs) in a shear ?ow. The dynamic equation for liquid crystal poly- mers is described by a nonlinear stochastic di?erential equation with Maier-Saupe interaction potential. The stress tensor is obtained from an ensemble average of microscopic polymer con?gurations. We present the local existence and uniqueness theorem for the solution of the coupled ?uid-polymer system. We also analyze the error of a fully ?nite di?erence-Monte Carlo hybrid numerical scheme by investigat- ing the asymptotic behavior of weakly interacting processes. It is proved that the rate of convergence of the full discretized scheme isO(h 2+δt+ 1 √M ). The second part of the thesis is concerned with spurt phenomena in viscoelastic ?uids. Both the experimental spurt e?ect in the slit-die ?ow and its possible expla- nations from rheology theory are summarized, with emphasis on the nonmonotone stress-strain relationship determined by the material property of viscoela