Finite element simulation of viscoelastic flow

Abstract A finite element simulation has been carried out for a viscoelastic fluid of the single integral, memory type. In the current work, the memory kernel is chosen to be a single exponential in the time lapse (Maxwell model). However, the formulation is such that it can easily be generalized to more realistic models such as the BKZ theory. From the point of view of numerical analysis, differential models are appealing because they avoid the complexities of memory integrals. However, in these models the viscoelastic effect always enters through terms having the highest-order derivatives. The disadvantage of this situation for numerical analysis appears to be borne out in the experiences reported recently by several workers. In a memory integral formulation, the demand on differentiability of the velocity field is no greater than for the Newtonian fluid. The basic idea in the formulation is the approximation of the memory integral by a Laguerre numerical quadrature formula. The kinematical problem is the computation of the displacement vector from every node to the Laguerre points upstream along particle paths. Since this operation requires the velocity field to be known, the method is restricted to the calculation of non-linear effects as body forces. Thus, the equations being solved in any iteration are those of linear viscous flow with an arbitrary body force. In spite of this limitation, the method converges at dimensionless relaxation times greater than the largest values attained with formulations based on differential models. In the present paper, the method is illustrated with the die entry flow in which the fluid is forced through a four-to-one axisymmetric contraction.