FINITE-ELEMENT METHOD FOR FLUIDS WITH MEMORY

Numerical methods offer the best hope for the analysis of complex polymer processing flows. The construction of numerical programs with the necessary capabilities involve two essential choices. Firstly, a method must be selected which can successfully handle the velocity and stress boundary conditions encountered in polymer processing. This requirement coupled with the additional capability of handling complex geometries points decisively to the finite element method (FEM). Secondly, a choice must be made of a constitutive equation which is able to represent the behavior of real polymer fluids under a variety of conditions, and which must also conform to the needs of the numerical method. The main choice is between differential models which demand a high degree of differentiation of field variables, and integral models which involve the complexity of memory integrals but retain the same low order of differentiation as the Newtonian fluid. In this paper a Galerkin, finite element formulation is presented for the simplest fluid of the single integral memory type (linear in C and C−1). The memory fluid gives rise to basic difficulties in the assembly of the elements if particle paths are regarded as unknown. In this paper an iterative procedure is used in which the displacement is decomposed into a part determined from previous iterations and an unknown part to be determined. This device avoids the assembly problem. An important feature of the memory integral formulation is that it can be carried out with simple constant strain rate, triangular elements. In contrast elements having at least linear strain rate interpolation are required for any differential model. A less obvious aspect of the FEM for incompressible fluids is the interpolation of the scalar pressure. Experience with Newtonian fluids dictates consistency rules which do not hold for fluids with normal stresses. The problem is accentuated with nonlinear interpolation, and here again the memory fluid lends itself most readily to the resolution of the problem. © 1979, All rights reserved.