The objective of this paper is to compare the results and performance of a mixed finite element method and of CONNFFESSIT in the calculation of the start-up of the flow of a FENE-P fluid between eccentric rotating cylinders. Until recently, most viscoleastic flow calculations have been based on a macroscopic mathematical description of both the conservation equations and the constitutive equation, solved by means of appropriate numerical discretization techniques. In the CONNFFESSIT approach, the macroscopic constitutive equation is replaced by a stochastic simulation of a large local ensemble of microscopic entities which act as stress calculators. The stress tensor is obtained as an average over this ensemble. The exact correspondence between the microscopic dynamics and the constitutive equations has a rigorous mathematical basis. Furthermore, many interesting microscopic models for polymer dynamics which cannot be translated into a constitutive equation can be dealt with quite naturally within the CONNFFESSIT approach. The reduction of the statistical error bars to an acceptable level in inherently noisy CONNFFESSIT calculations requires, however, the use of large ensembles and leads to long computations.
In the present paper, we have solved the start-up flow of a FENE-P fluid in a journal bearing geometry using i) the FENE-P constitutive equation as implemented numerically in a mixed finite element method, and ii) the CONNFFESSIT approach with non-interacting dumbbells obeying the FENE-P spring law. The two approaches are mathematically and rheologically equivalent. Differences in the simulation results can only be of numerical or statistical nature. Additionally, a steady-state Newtonian calculation was performed for kinematic reference purposes. |
mk@mat.ethz.ch
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