Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll
mill background force (extended to the third dimension). Both the Oldroyd-B model and FENE-P
model of viscoelastic fluids were used. Different temporal behaviors were observed depending on
the Weissenberg number (non-dimensional relaxation time), model, and initial conditions. Temporal
dynamics evolve on long time scales and simulations were accelerated by using a Graphics
Processing Unit (GPU). Previously, parameter explorations and long-time simulations in 3D were
prohibitively expensive. For small Weissenberg number, all the solutions are constant in the third
dimension, displaying strictly two-dimensional temporal evolutions. However, for sufficiently large
Weissenberg number, three-dimensional instabilities were observed, creating complex temporal behaviors.
In some of the cases, the instability that first emerges is two-dimensional (in the x; y
plane), and then the solution develops an instability in the z-direction whereas in others the z
instability comes first. Using a linear perturbation from a steady two-dimensional background
solution, extended to three dimensions as constant in the third dimension, it is demonstrated that
there is a linear instability for sufficiently large Weissenberg number, and possible mechanisms for
this instability are discussed.
