Proper Orthogonal Decomposition (POD) of the flow dynamics for a viscoelastic fluid in a four-roll mill geometry at the Stokes limit

Abstract

Numerical simulations of viscoelastic fluids in the Stokes limit with a four-roll mill background force were performed at a range of Weissenberg number (non-dimensional relaxation time). For small Weissenberg number the flow is steady and symmetric but upon increasing the Weissenberg number (corresponding to increased elasticity or flow memory time), the flow becomes unstable leading to a variety of temporal evolutions to different periodic and aperiodic solutions. These dynamics were analyzed using a Proper Orthogonal Decomposition (POD) that extracted elastic modes in terms of their contribution to the energy of the system. The temporal behavior of the system, captured by the decomposition, indicates that the motion of the stagnation points drives the different flow transitions. In particular, a transition to an asymmetric state occurs when the extensional stagnation points lose their pinning to the background forcing. A further transition to higher frequency modal dynamics occurs when the stagnation points that were initially tied by the forcing to the centers of the rolls, begin to move. The relative frequencies of the motion of these stagnation points is a critical factor in determining the complexity of the flow, measured by the number of modes needed to capture most of the energy in the system. Even when the flows are more complex a small number of modes is sufficient to capture the time evolution of these flows, demonstrating the usefulness of the POD applied to viscoelastic fluids at zero Reynolds number.