A complexity perspective on fluid mechanics (2212.00153v2)

Abstract: This article attempts to use the ideas from the field of complexity sciences to revisit the classical field of fluid mechanics. For almost a century, the mathematical self-consistency of Navier-Stokes equations has remained elusive to the community of functional analysts, who posed the Navier-Stokes problem as one of the seven millennium problems in the dawn of 21st century. This article attempts to trace back the historical developments of fluid mechanics as a discipline and explain the consequences of not rationalising one of the commonly agreed upon tenets - continuum hypothesis - in the community. The article argues that 'fluids' can be treated as 'emergent' in nature, in that the atoms and molecules in the nanometre length scale can likely be correlated with the continuum physics at the microscale. If this is the case, then one might start trying to find a theoretical framework that models the emergence of fluids from atoms, effectively solving the multi-scale problem using a single abstract framework. Cantor set with layers $N$ (N can have up to two orders of magnitude) is presented as a potential contender (analytical framework) for connecting the energy in molecular level $C_{1}$ at length scale $l_{cut}$ to the energy at continuum level $C_N$ with length scale $L$. Apart from fluid mechanics, Cantor set is shown to represent the conceptual understanding of VLSI hardware design ($N=5$). Apart from Cantor set, an experimental technique of architecting metafluids is also shown to solve emergence experimentally (i.e. connect physics at specific lower scales to higher scales).