From Reynolds's stretching and folding to mixing studies using horseshoe maps

Osborne Reynolds's seminal idea of stretching and folding being the basis of fluid mixing has a direct bearing on the interpretation of mixing processes involving dynamical systems tools, in particular, horseshoe maps. Horseshoes offer the only direct, mathematically rigorous, experimental verification of chaos in a flow. In this work these ideas are formalized and developed, with the goal of exploiting the concepts in experimental mixing studies, particularly in the case of alternating doubly symmetric flows. Methods to represent and to identify horseshoes are developed. Application examples to three different flows - focusing primarily on errors arising from imperfect placement and reconstruction - are presented.