A topological approach to three dimensional laminar mixing

In recent years, topological concepts have yielded valuable insights into the long standing problem of laminar fluid mixing. Topologically complex stirring protocols are typically far superior to topologically simple protocols, guaranteeing chaotic advection of fluid particles and the associated exponential dilation of material elements. Furthermore, topological approaches to mixer design are typically intuitive and insensitive to precise geometry or fluid properties. However, results to date have been limited to two dimensional flows (for example, batch stirrers in food or polymer manufacturing) and quasi three dimensional protocols (for example, continuous flow micromixers). Motivated by a simple stretching and folding argument that works well in two dimensions, we propose a topological approach to fully three dimensional fluid mixing. A transition matrix is derived to describe the mapping induced by a three dimensional `braid' on area elements, and the associated Perron--Frobenius eigenvalue provides a prediction of the large time asymptotic area growth rate. We show that these theoretical predictions agree well with numerical data obtained from simulations in a prototype three dimensional mixing device. References M. Bestvina and M. Handel. Train tracks for surface homeomorphisms, Topology 34, 1995, 109--140. doi:10.1016/0040-9383(94)E0009-9 P. L. Boyland, H. Aref and M. A. Stremler. Topological fluid mechanics of stirring, J. Fluid Mech., 403, 2000, 277--304. doi:10.1017/S0022112099007107 J. Chen and M. A. Stremler. Topological chaos and mixing in three dimensional channel flow. {Physics of Fluids}, 21, 2009, 021701. doi:10.1063/1.3076247 M. D. Finn and J.--L. Thiffeault. Topological entropy of braids on the torus. SIAM J. Appl. Dyn. Sys., 6, 2007, 79--98. doi:10.1137/060659636 O. S. Galaktionov, P. D. Anderson, G. M. W. Peters and F. N. van de Vosse. An adaptive front tracking technique for three dimensional transient flows. Int. J. Numer. Meth. Fluids, 32, 2000, 201--217. E. Gouillart, N. Kuncio, O. Dauchot, B. Dubrulle, S. Roux and J.--L. Thiffeault. Walls inhibit chaotic mixing. Phys. Rev. Lett. 99, 2007, 114501. T. G. Kang, M. K. Singh, T. H. Kwon and P. D. Anderson. Chaotic mixing using periodic and aperiodic sequences of mixing protocols in a micromixer. Microfluid Nanofluid, 4, 2008, 589--99. doi:10.1007/s10404-007-0004-x J.--O. Moussafir. On computing the entropy of braids. Func. Anal. Other Math., 1, 2006, 37--46. doi:10.1007/s11853-007-0206-z H. A. Stone and S. Kim. Microfluidics: basic issues, applications and challenges. American Institute of Chemical Engineers Journal, 47, 2001, 1250--54. doi:10.1002/aic.690470602 M. A. Stremler and J. Chen. Generating topological chaos in lid driven cavity flow, Physics of Fluids, 19, 2007, 103602. doi:10.1063/102772881 M. A. Stremler, F. R. Haselton and H. Aref. Designing for chaos: applications of chaotic advection at the microscale. Phil. Trans. R. Soc. Lond. A, 362, 2004, 1019--36. J.-L. Thiffeault and M. D. Finn. Topology, braids and mixing in fluids. Phil. Trans. R. Soc. Lond. A, 364, 2006, 3251--66. doi:10.1098/rsta.2006.1899 C. R. Thomas. Problems of shear in biotechnology. Critical Reports on Applied Chemistry, 29, 1990, 23--93. S. O. Unverdi and G. Tryggvason. A front tracking method for viscous, incompressible, multi-fluid fluids. J. Comp. Phys., 100, 1992, 25--37. G. M. Whitesides and A. D. Stroock. Flexible methods for microfluidics. Physics Today, 54, 2001, 42--48