Mixing of a viscoelastic fluid in a time-periodic flow

Abstract

We present an experimental and computational investigation of mixing of a viscoelastic fluid in two-dimensional time-periodic flows generated in an eccentric cylindrical geometry. The objective of the study is to investigate the impact of fluid elasticity on the morphological structures produced by the advection of passive tracers in chaotic flows. The relevant dimensionless numbers that quantify the rheological differences with respect to the Newtonian fluid are the Deborah number (De), defined as the ratio of the fluid timescale to the flow timescale, and the Weissenberg number (We), defined as the product of the fluid timescale and the mean shear rate. The effects of elasticity are investigated in the limit of slow flows, De ≈ 0 and We < 0.1. The experimental window of We is limited to Newtonian behaviour on the low end and the transition to three-dimensional flow on the high end; experiments show that this window is small, 0.02 < We < 0.1. Typical values of the Reynolds number and the Strouhal number are O(0.001) and O(0.1), respectively.Results from experiments with a constant-viscosity elastic fluid and computations using the upper-convected Maxwell constitutive equation are presented. Even though the streamlines for the elastic flow are nearly indistinguishable from the Newtonian flow, small deviations in the velocity field produce large effects on chaotically advected patterns. Elasticity affects both the asymptotic coverage of a dyed passive tracer and the rate at which the tracer is stretched. In all cases the tracer undergoes exponential stretching, but on a longer timescale as the elasticity increases. According to flow conditions, elasticity might increase or decrease the degree of regularity; however, island symmetry does not seem to be affected. Similar phenomena are observed in both the experiments and computations; therefore, an analysis of the chaotic dynamics of the periodic flow using numerical techniques is possible.