Learned a lot from my lunch with Thomas Hou today, who recently finally vindicated his 10 year quest of showing the (axisymmetric, incompressible) Euler equation blows up. https://arxiv.org/abs/2210.07191
#PDE #NavierStokes #Euler #Math #Physics #MachineLearning #DeepLearning

Navier–Stokes momentum equations:
\[\overbrace{\overbrace{\underbrace{\dfrac{\partial\mathbf{u}}{\partial t}}}^\text{Variation}_\text{Local}+\overbrace{\underbrace{(\mathbf{u}\cdot\nabla)\mathbf{u}}_\text{Convective}}^\text{Divergence}}^\text{Inertia/Acceleration}=\overbrace{\underbrace{-\dfrac{1}{\rho}\nabla p}_\text{Internal}+\underbrace{\mathbf{g}}_\text{External}}^\text{Source terms}+\overbrace{\underbrace{\nu\nabla^2\mathbf{u}}_\text{Viscous}}^\text{Diffusion}\]
#NavierStokes #FluidDynamics

#SoftMatter have just published the results of a project that Renato Assante, Davide Marenduzzo, Alexander Morozov, and I recently worked on together! What did we do and what’s new? Briefly…

#Microswimmer suspensions behave in a similar way to fluids containing kinesin and microtubules. Both systems can be described by the same system of three coupled nonlinear #PDEs.

A #LinearStabilityAnalysis of these equations suggests that variations in concentration across the system don’t significantly affect emergent #phaseBehaviour. How then can we explain #experiments that show visible inhomogeneities in #microtubule–#kinesin mixtures, for instance?

With increasing activity, we move away from the quiescent regime, past the onset of #SpontaneousFlow, and deeper into the active phase, where #nonlinearities become more important. What role do concentration inhomogeneities play here?

We investigated these questions, taking advantage of the #openSource #Dedalus #spectral framework to simulate the full nonlinear time evolution. This led us to predict a #novel regime of #spontaneous #microphaseSeparation into active (nematically ordered) and passive domains.

Active flow arrests macrophase separation in this regime, counteracting domain coarsening due to thermodynamic coupling between active matter concentration and #nematic order. As a result, domains reach a characteristic size that decreases with increasing activity.

This regime is one part of the #PhaseDiagram we mapped out. Along with our other findings, you can read all about it here!

low #ReynoldsNumber #turbulence #ActiveTurbulence #CahnHilliard #ActiveMatter #NavierStokes #BerisEdwards #CondensedMatter #PhaseTransitions #TheoreticalPhysics #BioPhysics #StatisticalPhysics #FluidDynamics #ComputationalPhysics #Simulation #FieldTheory #paperthread #NewPaper #science #research #ActiveGel #activeNematic #analytic #cytoskeleton #hydrodynamics #MPI #theory

#SoftMatter have just published the results of a project that Renato Assante, Davide Marenduzzo, Alexander Morozov, and I recently worked on together! What did we do and what’s new? Briefly…

The #hydrodynamic behaviour of inhomogeneous #activeNematic gels (such as extensile bundles of #cytoskeletal filaments or suspensions of low #ReynoldsNumber swimmers) can be described by the time evolution of three coupled #PDEs.

Standard #ActiveGel #theory concludes, from a #LinearStabilityAnalysis of these equations, that fluctuations in concentration don’t significantly affect emergent #phaseBehaviour. However, this leaves #experimental #observations of visible inhomogeneities in #microtubule–#kinesin mixtures unexplained. As we move away from the passive (quiescent) regime, past the onset of #SpontaneousFlow, and deeper into the active phase, #nonlinearities become more important. What role do concentration inhomogeneities play here?

Alongside #analytic techniques, we used an in-house #MPI-parallel code developed within the #Dedalus #spectral framework to investigate. We predict a #novel regime of #spontaneous #microphaseSeparation into active (nematically ordered) and passive domains. In this regime, active flow arrests macrophase separation, which is itself driven by the thermodynamic coupling between active matter concentration and #nematic order. As a result, domains do not #coarsen past a typical size, which decreases with increasing activity. This regime is one part of the #PhaseDiagram we mapped out.

Along with our other findings, you can read all about it here!

#CahnHilliard #ActiveMatter #NavierStokes #BerisEdwards #CondensedMatter #PhaseTransitions #TheoreticalPhysics #BioPhysics #StatisticalPhysics #FluidDynamics #ComputationalPhysics #Simulation #FieldTheory #paperthread #NewPaper #science #research

"For more than 250 years, mathematicians have wondered if the Euler equations might sometimes fail to describe a fluid’s flow. A new computer-assisted proof marks a major breakthrough in that quest." #water #engineering #NavierStokes https://www.quantamagazine.org/computer-helps-prove-long-sought-fluid-equation-singularity-20221116/

> One of the central problems in fluid dynamics is to figure out if the [Euler] equations ever fail, outputting nonsensical values that render them unable to predict a fluid’s future states.
> Mathematicians have long suspected that there exist initial conditions that cause the equations to break down. But they haven’t been able to prove it.
> In a preprint posted online last month, a pair of mathematicians has shown that a particular version of the Euler equations does indeed sometimes fail.
...
> Perhaps in some situations, the equations will proceed as expected, producing precise values for the state of the fluid at any given moment, only for one of those values to suddenly skyrocket to infinity. At that point, the Euler equations are said to give rise to a “singularity” — or, more dramatically, to “blow up.”
> Once they hit that singularity, the equations will no longer be able to compute the fluid’s flow.
...

Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations
https://www.quantamagazine.org/computer-helps-prove-long-sought-fluid-equation-singularity-20221116/

Very important and interesting work, if you're a mathematician, a physicist or an engineer that has ever 'crashed' with Euler/Navier-Stokes equations.

#math #physics #fluids #euler #navierstokes #equations #singularities

Als ik dit artikel goed lees lijkt het ook op XY: los de problemen met vloeistofdynamica op met de #navierstokes . Het sleutelen aan de vergelijking hint al naar het idee om eens te kijken hoe we vloeistofdynamica oplossen misschien wel met iets anders dan deze formule.

"Het Amerikaanse Clay Mathematics Institute looft een miljoen dollar uit voor iemand die zo’n probleem zoals het Navier-Stokesprobleem oplost."

Als al zoveel briljante hoofden er naar gekeken hebben is een miljoen natuurlijk veel te weinig, wat zijn dat voor krenten bij dat Clay institute? Kapitalisme heeft zijn keerzijde maar te lang op een wiskundig instituut rondlopen geeft je blijkbaar niet het inzicht wat iets nou eigenlijk waard is.

https://www.nrc.nl/nieuws/2022/12/09/alles-stroomt-maar-de-wiskunde-daarachter-is-nog-goeddeels-onbegrepen-a4150762?utm_source=SIM&utm_medium=email&utm_campaign=nrcwetensch&utm_content=&utm_term=20221210

#NRC #navierstokes

Hoe staat het met de Navier-Stokesvergelijkingen? Ze zijn 200 jaar oud, bruikbaar in de praktijk dankzij computerberekeningen, maar veel van de achterliggende #wiskunde is nog altijd onbegrepen. Over de huidige stand van zaken schreef ik voor #NRC.
#NavierStokes
#millenniumproblem
#milliondollar
https://www.nrc.nl/nieuws/2022/12/09/alles-stroomt-maar-de-wiskunde-daarachter-is-nog-goeddeels-onbegrepen-a4150762

Hello! As I’m new both to #Mastodon and to the #QOTO instance, I’d like to make my #introductions here.

I’m Dom, a postgraduate researcher studying #CondensedMatter, in particular #SoftMatter, #ActiveMatter, #MolecularBiology, #CellBiology, #PhaseTransitions, and #ComplexSystems using the tools of #TheoreticalPhysics, #BioPhysics, #StatisticalPhysics, #FluidDynamics, #ComputationalPhysics, and #Simulation.

I’m working on a few projects at the moment. One addresses #universality in active #turbulence with #spectral techniques. Another predicts a new microphase separation regime in inhomogeneous active #nematics by simulating the coupled #CahnHilliard, #NavierStokes, and #BerisEdwards equations. In another project I’m developing a #FieldTheory of the #orderphobic effect, a nonspecific interaction between #transmembrane bodies anchored in a #lipid #bilayer.

I’m also interested in…
#cycling
#ArchLinux
#Linux
#LibreSoftware
learning #Korean
#yoga
#calisthenics
#singing
#autonomy
#Cooperatives
#DeliberativeDemocracy
#CitizensAssemblies
#ParticipatoryDemocracy
#DemocraticInnovation
#OpenDemocracy
#LibertarianSocialism
#RightToRepair
#democracy
#humanism
#rationalism
#OpenScience
#science
#CommunityEmpowerment
#community
#commons
#research
#wikis
#governance

Quanta article about how a team of mathematicians recently managed to coax the Navier-Stokes equations into producing some non-unique solutions for certain initial conditions, exposing a weakness in the equations' modeling of physical reality

https://www.quantamagazine.org/mathematicians-coax-fluid-equations-into-nonphysical-solutions-20220502/

#FluidDynamics #Physics #Mathematics #NavierStokes

#ScienceÉtonnante #61 : « Comment les #avions volent-ils ? »

« Par quel moyen les avions arrivent-ils à se maintenir en l'air ? Un sujet pas si simple, et même controversé à en croire certains »
#NavierStokes #Bernouilli #Newton #KuttaJukowski #Science
https://sciencetonnante.wordpress.com/2019/09/03/comment-les-avions-volent-ils/