A Numerical Method for Modeling Time-Dependent Viscoelastic Fluid Flow

A numerical method for modeling time-dependent viscoelastic uid ow, Abstract A numerical method to solve a system of partial diierential equations (PDE), modeling time-dependent viscoelastic uid ow, is developed. The Upper Con-vected Maxwell (UCM) model, and the related Oldroyd-B and Giesekus models are used as constitutive equations, and a linear equation of state allows the uid to be slightly (artiicially) compressible. We show that the periodic Cauchy problem for the system of PDE is well-posed locally in time for all three constitutive equations. The well-posedness properties are independent of the compressibility parameter in the equation of state. We therefore get uniqueness and existence of a solution, locally in time, to the equations for an incompressible viscoelastic uid of UCM type. Also, the solution to the equations for compressible model is close to the incompressible solution. Using a model in which the uid is slightly compressible, we can update the pressure explicitly along with the other variables and no Poisson equation needs to be solved for the pressure. The bounded derivative principle, which is shown to be valid, makes it possible to choose the initial data such that the fast varying sound waves are suppressed. The numerical solver is based on the method of lines. Second-order centered diierences are used to discretize the space variables on overlapping structured grids. The resulting ordinary diierential equation for the time variable is integrated with a second-order Adams predictor-corrector method. The solver is used to study the elastic eeects on a paper coating ow. A vis-coelastic uid is applied to a paper surface with a roll and the excess uid is scraped oo with a steel blade. We investigate the ow in the vicinity of the steel blade. The investigation suggests that there are strong elastic eeects on the ow close to the blade. A weak instability in the pressure is observed along the interpolation boundaries in some cases. The instability appears as a slowly growing, highly oscillatory, mode. To gain understanding on this phenomenon, the simple wave equation is studied as a model problem. We propose that an artiicial dissipation term of fourth order should be introduced to damp the highest frequencies in the numerical solution. This removes the highly oscillatory mode eeciently without aaecting the second-order accuracy of the method. Preface This thesis concerns the numerical solution of a system of partial diierential equations describing time-dependent viscoelastic uid ow. It is based on …