An Modified Nonlinear Galerkin Method for the Viscoelastic Fluid Motion Equations

In this article we rst provide a priori estimates of the solution for the nonsta tionary two dimensional viscoelastic uid motion equations with periodic bound ary condition We then present an modi ed nonlinear Galerkin method for solving such equations By comparing the convergence rates of the proposed method with the standard Galerkin method we conclude that the modi ed nonlinear Galerkin method is better than the standard Galerkin method be cause the former can save a large amount of computational work and maintain the convergence rate of the latter Introduction In this paper we consider Oldroyd s mathematical model of two dimensional viscoelas tic uid motion Such model see can be de ned by the rheological relation k k t t k x x where is the stress tensor and is the strain tensor and k k are positive constants In fact i j is the matrix with components ij ui xj uj xi with u u u being the rate of the uid motion If k k in we shall obtain Newton s model of uncompressible viscous uid Relation and the motion equation in Cauchy form leads us to the initial boundary value problem u t u r u u Z t expf t s g uds rp F divu t x u t x u x u x x p Z p x t dx Mathematics Subject Classi cation L M D A