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In NS equations, non-linear term dominate viscosity term when order of velocities is higher than wavenumber (3). For lower velocity, viscosity term dominate (4). Non-linear terms are coupled (across different length scales), hence onset of turbulence causes distribution of velocities across all frequency domains. It can be understood on performing Fourier transform of NS andContinueContinue reading “Turbulence: part 1”

To prove (or disprove) that the Navier-Stokes equations blow up at finite time, lot of efforts from geometers and topologists are recently garnering attention, after Tao introduced the fluid computer program in 2014. From a physicist perspective, you see a jet of liquid pinching off at a certain time and position, and Eggers especially hasContinueContinue reading “Turing completeness of Euler equations”

https://media.giphy.com/media/c1uPY08OL1hYP9GGx3/giphy.gif As I punt down the Cam river, I see my friend paddling leisurely. Much of this action can be deemed trivial and of not-so-importance use, until: you zoom at the moment, when the paddle touched the surface of water; let this moment play for a while, you see two small whirlpools (and bubbles underneath)ContinueContinue reading “Punting and finite-time singularity problem”

Batchelor (1967) discusses that numerous experimental evidences have pointed to the assumption which every fluid dynamicist make in day-to-day life: assuming fluid flow as smooth and continuous. The length scale for such assumption should be large enough to accomodate molecular fluctuations of density and other properties and small enough to not any effect of spatialContinueContinue reading “Missing mathematical basis of continuum hypothesis (fluid mechanics)”

Moffatt (2021) [1] discusses the two main and complementary approaches to solve the regularity problem. The view by pure mathematicians on the turbulence models is best summarized by Lemarie-Rieusset (2016) as follows: View of large scales and small scale fluid parcels by Lemarie-Rieusset (2018): [1] Moffatt, H. K. (2021). Some topological aspects of fluid dynamics. JournalContinueContinue reading “Turbulence VS Functional analysis”

Euler was the first mathematician to derive the equations, for an inviscid fluid, conserving mass and momentum. First person to express Newton’s law (for a point mass) for the case of a continuous medium, he could do so by (in the words of Truesdell):looking within the interior moving fluid, where neither eye nor experiment mayContinueContinue reading “Landmark ideas”