Heat-transfer enhancement due to slender recirculation and chaotic transport between counter-rotating eccentric cylinders

Abstract

Using Stokes flow between eccentric, counter-rotating cylinders as a prototype for bounded, nearly parallel lubrication flow, we investigate the effect of a slender recirculation region within the flow field on cross-stream heat or mass transport in the important limit of high Péclet number Pe where the enhancement over pure conduction heat transfer without recirculation is most pronounced. The steady enhancement is estimated with a matched asymptotic expansion to resolve the diffusive boundary layers at the separatrices which bound the recirculation region. The enhancement over pure conduction is shown to vary as ε½ at infinite Pe, where ε½ is the characteristic width of the recirculation region. The enhancement decays from this asymptote as Pe−½. If one perturbs the steady flow by a time-periodic forcing, fast relative to the convective and diffusive times, the separatrices undergo a homoclinic entanglement which allows fluid elements to cross the separatrices. We establish the existence of this homoclinic entanglement and show that the resulting chaotic particle transport further enhances the cross-stream flux. We estimate the penetration of the fluid elements across the separatrices and their effective diffusivity due to this chaotic transport by a Melnikov analysis for small-amplitude forcing. These and the steady results then provide quantitative estimates of the timeaveraged transport enhancement and allow optimization with respect to system parameters. An optimum forcing frequency which induces maximum heat-transfer enhancement is predicted and numerically verified. The predicted optimum frequency remains valid at strong forcing and large Pe where chaotic transport is as important as the recirculation mechanism. Since most heat and mass transport devices operate at high Pe, our analysis suggests that chaotic enhancement can improve their performance and that a small amplitude theory can be used to optimize its application.