Chaotic advection in creeping flow of viscoelastic fluids between slowly modulated eccentric cylinders

Recent experiments show that very low levels of elasticity can either enhance or diminish the area over which chaotic advection occurs in creeping flows [T. C. Niederkorn and J. M. Ottino, J. Fluid Mech. 256, 243 (1993)]. No mechanistic explanation of this phenomenon is currently available. This has motivated us to consider the problem of two-dimensional flow between counter-rotating eccentric cylinders where the angular velocities are subject to slow, continuous modulation. Regular perturbation theory for low levels of elasticity is used to semi-analytically determine the viscoelastic correction to the Newtonian flow field based on the Oldroyd-B constitutive model. The geometric theory of Kaper and Wiggins [J. Fluid Mech. 253, 211 (1993)] is then applied to make predictions about how elasticity affects chaotic advection in quasi-steady flows. It is found that elasticity can act to either increase or decrease the area over which chaotic advection occurs, depending on the boundary motion. This is accomplished through three distinct mechanisms: (1) area changes of the maximum area over which chaotic advection can occur, the potential mixing zone (PMZ); (2) area changes of the region in the PMZ where fluid particles execute non-chaotic trajectories below a critical modulation frequency; (3) area changes of the region between the extrema of the Newtonian stagnation streamlines which does not belong to the PMZ. The mechanism responsible for these area changes is a modified pressure gradient in the angular direction, which in turn appears to be due to first normal stress differences caused by shearing Numerical calculations of fluid particle trajectories confirm the predictions of the geometric theory. For the boundary motions considered here, the calculations yield two additional results about the effect of low levels, of elasticity on chaotic advection. First, the critical modulation frequency is decreased Second, the rate of chaotic mixing, as measured by the largest Liapunov exponent, is increased for modulation frequencies greater than the critical Newtonian value.