Three-dimensional viscoelastic instabilities in a four-roll mill geometry at the Stokes limit

Abstract

Three-dimensional numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force (extended to the third dimension) were performed. 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 a small Weissenberg number, all the solutions are constant in the third dimension, displaying strictly two-dimensional temporal evolutions. However, for a sufficiently large Weissenberg number, three-dimensional instabilities were observed, creating complex temporal behaviors. For certain Weissenberg values and models, 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 for 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 a sufficiently large Weissenberg number, and possible mechanisms for this instability are discussed.