Chaotic mixing and heat transfer between confocal ellipses: Experimental and numerical results

Abstract

The annular region between two concentric, confocal ellipses is a new geometry which is particularly effective for either mixing of viscous fluids or heat transfer enhancement in the important limit of a high Péclet number. This geometry is in many respects similar to the annular space between two eccentric, rotating cylinders although it possesses two (instead of one) axes of symmetry. The recently obtained analytical solution of the Stokes flow equations in this geometry shows that at steady state and for counter-rotation of the inner and outer ellipses, two opposite saddle points (connected by two different streamlines) appear in the region of minimum gap. This flow characteristic is also exhibited by the eccentric cylinder system for some cases of co-rotation. The Poincaré sections obtained when the inner and outer ellipses are displaced using a discontinuous velocity protocol show that a more effective long term mixing is obtained for the counter-rotating case, this is confirmed by the experimental data we have obtained. The opposite conclusions (more effective mixing for co-rotation) have been given in the eccentric cylinder geometry. Photographs of the fluorescent dye after 5 periods are compared with remarkable success to numerical blob deformation experiments. Experimental results also confirm previous results based on an analysis of Poincaré sections. In particular, better mixing is obtained when the inner ellipse displacement per period increases. Finally, this geometry is shown to be particularly effective as a heat exchanger. For steady, counter-rotation of the two boundaries, the recirculation zones can lead to a heat transfer rate increase of 80% over that of pure conduction at high Péclet numbers, and, by an appropriate sinusoidal modulation of the angular velocity of one boundary, the heat transfer rate can be more than double that of pure conduction. Since an analysis of the experimental data also suggests that the mixing rate for a sinusoidal modulation of the angular velocity of the boundaries is better than for a discontinuous velocity protocol, we propose that the average Nusselt number per period could be one of the several useful tools in the global optimization of the mixing protocol.