  Clay Mathematical Institute, in the year 2000, formulated a list of seven
unsolved mathematical problems which might influence and direct the course of
the research in the 21st century. Navier-Stokes regularity problem is one of
the seven problems and has not been solved till date. These equations govern
the motion of the viscous fluids, such as liquids, gases, gels, polymers. The
present article is a spotlight on the NSE problem: a review of the past and
current efforts to solve it, and our changing perspectives and understanding of
these equations. The major debate is to either prove that the solutions to
Navier-Stokes equations (NSE) are smooth or to prove that the solutions reach
singularity at a finite-time (blowup); given some initial conditions. While in
the past, the focus was on finding bounds for the smooth solutions, recently,
there has been growing interest in finding solutions that blowup. The current
article places the Turing-completeness of Euler equations (a subclass of NSE)
into the centre-stage, by discussing the recent developments that are pointing
towards that the idea that the fluid equations should be viewed as a 'computer
program'. Given this is the case, the article then argues to see 'continuum
fluid' from a complexity notion, and thus, as a Turing machine. This notion
connects the discipline of fluid mechanics to disciplines like, 'neuromorphic
computing, reservoir computing' and potentially leading to emerging new
discipline, "soft matter complexity, computing, and learning", each of which
has been discussed in brief.
