Podcasts about navier stokes

Terence Tao is widely considered to be one of the greatest mathematicians in history. He won the Fields Medal and the Breakthrough Prize in Mathematics, and has contributed to a wide range of fields from fluid dynamics with Navier-Stokes equations to mathematical physics & quantum mechanics, prime numbers & analytics number theory, harmonic analysis, compressed sensing, random matrix theory, combinatorics, and progress on many of the hardest problems in the history of mathematics. Thank you for listening ❤ Check out our sponsors: https://lexfridman.com/sponsors/ep472-sc See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc. Transcript: https://lexfridman.com/terence-tao-transcript CONTACT LEX: Feedback - give feedback to Lex: https://lexfridman.com/survey AMA - submit questions, videos or call-in: https://lexfridman.com/ama Hiring - join our team: https://lexfridman.com/hiring Other - other ways to get in touch: https://lexfridman.com/contact EPISODE LINKS: Terence's Blog: https://terrytao.wordpress.com/ Terence's YouTube: https://www.youtube.com/@TerenceTao27 Terence's Books: https://amzn.to/43H9Aiq SPONSORS: To support this podcast, check out our sponsors & get discounts: Notion: Note-taking and team collaboration. Go to https://notion.com/lex Shopify: Sell stuff online. Go to https://shopify.com/lex NetSuite: Business management software. Go to http://netsuite.com/lex LMNT: Zero-sugar electrolyte drink mix. Go to https://drinkLMNT.com/lex AG1: All-in-one daily nutrition drink. Go to https://drinkag1.com/lex OUTLINE: (00:00) - Introduction (00:36) - Sponsors, Comments, and Reflections (09:49) - First hard problem (15:16) - Navier–Stokes singularity (35:25) - Game of life (42:00) - Infinity (47:07) - Math vs Physics (53:26) - Nature of reality (1:16:08) - Theory of everything (1:22:09) - General relativity (1:25:37) - Solving difficult problems (1:29:00) - AI-assisted theorem proving (1:41:50) - Lean programming language (1:51:50) - DeepMind's AlphaProof (1:56:45) - Human mathematicians vs AI (2:06:37) - AI winning the Fields Medal (2:13:47) - Grigori Perelman (2:26:29) - Twin Prime Conjecture (2:43:04) - Collatz conjecture (2:49:50) - P = NP (2:52:43) - Fields Medal (3:00:18) - Andrew Wiles and Fermat's Last Theorem (3:04:15) - Productivity (3:06:54) - Advice for young people (3:15:17) - The greatest mathematician of all time PODCAST LINKS: - Podcast Website: https://lexfridman.com/podcast - Apple Podcasts: https://apple.co/2lwqZIr - Spotify: https://spoti.fi/2nEwCF8 - RSS: https://lexfridman.com/feed/podcast/ - Podcast Playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 - Clips Channel: https://www.youtube.com/lexclips

Richard Feynman once dubbed turbulence “the last unsolved problem of classical physics.” Beyond the Navier–Stokes equations, no comprehensive statistical framework exists to predict how fluids spin, eddy, and cascade energy—whether in galactic jets, ocean currents, or the swirl of your morning coffee. But that might all be changing. In this episode, I sit down with theoretical physicist Sasha Migdal, who emerged from retirement with a bold, loop‑equation approach to turbulence. His formalism recasts the Navier–Stokes equations as an infinite number of linear, readily solvable equations that track circulation around closed loops in the fluid—offering, for the first time, a tractable statistical framework for chaotic flow. We chart Sasha's remarkable journey: from a promising career at the Landau Institute and Russian Space Institutes, where he resisted KGB pressure to compromise his work, to his defection to the United States in the late 1980s. We discuss his time outside of physics, and then delve into his return, and his novel approach to solving Navier–Stokes.  Since we recorded, his theory has advanced rapidly. A rigorous proof now shows that it is the solution  to the Navier–Stokes (NS) loop equation. The theory has also been extended to magnetohydrodynamics and turbulent mixing. Recent large-scale direct numerical simulations (DNS) match the theoretical predictions with even greater accuracy.  ►Watch On YouTube: https://youtu.be/BNORi7mxxzg ►Find out more about Sasha's work here: https://alexandermigdal.com/ https://sashamigdal.github.io/TurbulenceDuality/index.html ►For a more technical overview of Sasha's work you can check out his recent talk at the IAS: https://www.youtube.com/watch?v=mgvGA6q7oPY  

Episode: 3314 The Navier Stokes Equations in Movies, Video Games and Hurricane Forecasts.  Today, the Navier Stokes equations.

Outline00:00 - Intro01:07 - Early steps02:47 - Why control?05:20 - The move to the US07:40 - The first journal paper13:30 - What is backstepping?17:08 - Grad school25:10 - Stochastic stabilization29:53 - The interest in PDEs43:24 - Navier-Stokes equations52:12 - Hyperbolic PDEs and traffic models57:51 - Predictors for long delays1:08:14 - Extremum seeking1:27:14 - Safe control1:36:30 - Interplay between machine learning and control1:42:28 - Back to the roots: robust adaptive control1:50:50 - On service1:55:54 - AdviceLinksMiroslav's site: https://flyingv.ucsd.edu/Tuning functions paper: https://tinyurl.com/yznv6r9rP. Kokotović: https://tinyurl.com/mwmbm9yhSeparation and swapping:  https://tinyurl.com/y4fre6t8Adaptive nonlinear stabilizers: https://tinyurl.com/4a9wmmvxKKK book: https://tinyurl.com/2kw2b4k6Stochastic nonlinear stabilization: https://tinyurl.com/4td3537aFollow-up with unknown covariance: https://tinyurl.com/4c4n7fd7Boundary state feedbacks for PIDEs: https://tinyurl.com/4e9y4tdrBoundary Control of PDEs: https://tinyurl.com/d8x38bmjStabilization of Navier–Stokes systems: https://tinyurl.com/4a8cbjemTraffic congestion control: https://tinyurl.com/525jphs5Delay compensation: https://tinyurl.com/5yz6uj9pNonlinear predictors for long delays: https://tinyurl.com/7wvce6vyStability of extremum seeking: https://tinyurl.com/mr5cvzd3Nash equilibrium seeking: https://tinyurl.com/yeywrysnInverse optimal safety filters: https://tinyurl.com/9dkrpvkkNeural operators for PDE control: https://tinyurl.com/5yynsp7vBode lecture: https://tinyurl.com/mp92cs9uCSM article: Support the showPodcast infoPodcast website: https://www.incontrolpodcast.com/Apple Podcasts: https://tinyurl.com/5n84j85jSpotify: https://tinyurl.com/4rwztj3cRSS: https://tinyurl.com/yc2fcv4yYoutube: https://tinyurl.com/bdbvhsj6Facebook: https://tinyurl.com/3z24yr43Twitter: https://twitter.com/IncontrolPInstagram: https://tinyurl.com/35cu4kr4Acknowledgments and sponsorsThis episode was supported by the National Centre of Competence in Research on «Dependable, ubiquitous automation» and the IFAC Activity fund. The podcast benefits from the help of an incredibly talented and passionate team. Special thanks to L. Seward, E. Cahard, F. Banis, F. Dörfler, J. Lygeros, ETH studio and mirrorlake . Music was composed by A New Element.

Hello Interactors,Flying provides a great opportunity to catch up on books and podcasts, but it also brings feelings of guilt. My recent trip likely contributed about 136 hot air balloons' worth of CO2 to the atmosphere. Should I feel guilty, or should the responsibility lie with airlines, manufacturers, and oil companies? We all contribute to global warming, but at least our destination was experiencing an unusually cool July. However, globally, the situation is very different and worsening faster than expected. What's to be done? Let's dig in.CLIMATE CONUNDRUMS CONFOUND CALCULATIONSThere are two spots on the planet that are not affected by climate change, and I recently flew over one of them. It's a patch in the ocean just off the coast of Greenland that our plane happened to fly over on a family vacation to Scotland. The other is a small band around the Southern Ocean near Antarctica. I likely won't be visiting that one.I learned this on the plane listening to a podcast interview by the physicist Sean Carroll with climate scientist and Director of NASA's Goddard Institute for Space Studies, Gavin Schmidt. Gavin has been at the forefront of climate science, spearheading efforts to quantify Earth's climatic fluctuations, develop sophisticated models for projecting future climate scenarios, and effectively communicate these findings to the public and policymakers.In this discussion, they talked about the methods currently employed in climate research, while also offering insights into the anticipated climatic shifts and their potential impacts in the coming decades. Gavin is known for bridging gaps between complex science and accessible information. I'm writing this piece to bridge some of my own gaps.For example, there's often mention that climate change has created more extreme swings in temperature — that the weather is increasingly varying from extreme heat to extreme cold. In statistics, this is called variance. Some argue this variance may be hard for us to detect because temperatures have been shifting — a phenomenon known as shifting baseline syndrome.Gavin says there's more to this question than people realize. He notes that it is relatively straightforward to detect changes in the mean temperature because of the law of large numbers. Temperature varies across three dimensions - latitude, longitude, and altitude. We can calculate an average temperature for any two-dimensional slice of this 3D space, resulting in a single representative value for that area.This video is a conceptual simulation showing a 3D volume of temperature readings (warmer toward the ground and cool toward the sky). The 2D plane ‘slices' the cube averaging the values as it encounters them and colors itself accordingly. Source: Author using P5.js with much help from OpenAI.With enough data, it's clear that there has been a significant warming trend almost everywhere on Earth since the 1970s. Approximately 98% of the planet has experienced detectable warming, with a couple exceptions like the ones I mentioned.But determining changes in the variance or spread of temperatures is more complex. Calculating variance requires a comprehensive understanding of the entire distribution of data, which requires a larger dataset to achieve statistical confidence. Schmidt points out that while we have enough data to confirm that the distribution of temperatures has shifted (indicating a change in the mean), we do not yet have sufficient data to conclusively state that the variance has increased.Recent temperature spikes tell this story well. For the last decade or more, many climate scientists have been confident in predicting increased global mean temperatures by looking at past temperatures. The global mean has been predictably increasing within known variances. But in 2023 their confidence was shaken. He said,“Perhaps we get a little bit complacent. Perhaps we then say, 'Okay, well, you know, we know everything.' And for the last 10 years or so, [that's been] on the back of both those long-term trends, which we understand…”He goes on to explain that they've been able to adjust temperature predictions based on past trends and the cyclical variances of El Nino and La Nina. Scientists have boldly claimed,“'Okay, well, it's gonna be a little bit cooler. It's gonna be a little warmer, but the trends are gonna be up. You know, here's the chance of a new record temperature.' And for 10 years that worked out nicely until last year. Last year, it was a total bust, total bust like way outside any of the uncertainties that you would add into such a prediction.”How far outside of known uncertainties? He said,“…we were way off. And we still don't know why. And that's a little disquieting.” He added, “…we ended up with records at the end of last year, August, September, October, November, that were, like they were off the charts, but then they were off the charts in how much they were off the charts. So, they were breaking the records where they were breaking the records by a record-breaking amount as well. So that's record breaking squared, if you like, the second order record breaking. And we don't really have a good answer for that yet.”There is ongoing research into why and some have speculated, but none of them add up.For example, we're currently nearing a solar maximum in the sun's 11-year cycle which increases solar irradiance, but that small increase doesn't fully explain the observed changes. Other factors may be at play. For instance, there have been significant shifts in pollution levels in China, and the shipping industry has transitioned to cleaner fuels, which, as hoped, could be influencing climate patterns.However, Schmidt notes that the quantitative analysis of these factors hasn't yet matched the observed changes. Identifying potential contributors to climate variations is one thing, but precisely quantifying their impacts remains a challenge. Schmidt said climate and planetary scientists hope to convene in December to share and learn more, but the extreme shift remains concerning.CALCULATING CLIMATE'S COUNTLESS COMPONENTSThe amount of data required to model the climate is daunting. In a separate TED talk, Schmidt reveals that understanding climate change requires considering variables that span 14 orders of magnitude, from the microscopic level (e.g., aerosols) to the planetary scale (e.g., atmospheric circulation). These accordingly have their own orders of magnitude on a time scale, from milliseconds of chemical reactions to weather events over days or weeks to long term changes over millennia, like ice ages or long-term carbon cycles.Climate models must integrate processes across these scales to accurately simulate climate dynamics. Early models could only handle a few orders of magnitude, but modern models have significantly expanded this range, incorporating more detailed processes and interactions.Schmidt highlights that climate models reveal emergent properties—patterns that arise from the interactions of smaller-scale processes. For instance, no specific code dictates the formation of cyclones or the wiggles in ocean currents; these phenomena emerge naturally from the model's equations.But there is a staggering amount of data to model. And it all starts with the sun.The sun provides 99% of the Earth's energy, primarily in the visible spectrum, with components in the near-infrared and UV. This energy interacts with the atmosphere, which contains water vapor, greenhouse gases, ozone, clouds, and particles that absorb, reflect, or scatter light.The energy undergoes photolytic reactions. Photolytic reactions are chemical reactions that are initiated or driven by the absorption of light energy which breakdown molecules into smaller units. We couldn't breathe without it. The earth's ozone is decomposed into oxygen in the atmosphere through these reactions, which is initiated by sunlight — especially in the stratosphere. This too must be tracked as the Earth rotates, affecting sunlight exposure.Upon reaching the ground, some sunlight is reflected, by snow for example, or absorbed by oceans and land. This influences temperatures which is then radiated back as infrared energy. This process involves complex interactions with clouds, particles, and greenhouse gases, creating temperature gradients that drive winds and atmospheric motion. These dynamics further affect surface fluxes, water vapor, cloud formation, and associated chemistry, making the entire system highly intricate. And this doesn't even remotely begin to approach the complexity of it all.To simplify Schmidt says they capture what they can in a column roughly 25 kilometers high and wide to study the inherent physics. Most of which he says,“…is just vertical. So, the radiation you can think of as just being a vertical process, to very good order. Convection is also just a vertical process. So, there's a lot of things that you can do in the column that allows you to be quite efficient about how you solve the equations.”  Schmidt adds that “each column [can] sit on a different processor, and so you can do lots of things at the same time, and then they interact via the winds and the waves and those kinds of things.”He said most of the calculations come down to these two sets of equations: Euler and Navier-Stokes. Euler equations are a set of partial differential equations in fluid dynamics that describe the flow of non-viscous and fluids, absent heat exchange. Named after the Swiss mathematician and physicist Leonhard Euler in the 18th century, these simplify the analysis of fluid flow by neglecting viscosity and thermal conductivity, focusing instead on the conservation of mass, momentum, and energy.Navier-Stokes, named after the 19th century French civil engineer Claude-Louis Navier and the Irish physicist George Gabriel Stokes, is based on Euler's work but adds viscosity back into the equation. Schmidt says these equations are sometimes used to measure flows closer to the surface of the earth.This video is a conceptual simulation showing a 3D volume of vectors (randomly changing direction and magnitude) with particles entering the field of vectors. Each particle (e.g. dust, rain, aerosol) gets pushed in the direction of the vector each encounters. You can clearly see the emergent swarming behavior complex adaptive systems, like our atmosphere, can yield. Also present are the apparent challenges that come with measuring and predicting these outcomes. Source: Author using P5.js with much help from OpenAI.These complex computational models are inherently approximations. They are validated against observations but remain simplifications of reality. This inherent uncertainty is a critical aspect of climate science, emphasizing the need for continuous refinement and validation of models.And while human-induce climate change denialists like to say the climate models are wrong and not worth considering, Schmidt has a clever retort,“Models are not right or wrong; they are always wrong, but they are useful.”NAVIGATING NATURE'S NEW NORMALMany wish climate change predictions had the kind of certainty that comes with basic laws of physics. While there are indeed efforts in complexity science to identify such laws, we're still in the foothills of discovery on a steep climb to certainty.For example, to even achieve the current level of climate prediction took approximately 30 years of research, involving multiple methods, replication, and more sophisticated physical modeling. This led to accurate calibration techniques for the paleothermometers that measure ice cores which reveal temperatures from around the planet dating back three million years.While there is some empirical certainty in this — derived from the periodic table, fundamental laws of physics, or observed correlations from spatially dispersed ice core samples — recent extreme variations in global temperatures give reason to question this certainty. These relationships were based on spatial variations observable today, but failed to account for change over time, which behave very differently.Schmidt says, “…it turns out that the things that cause things to change in time are not the same things that cause them to change in space. And so empirical relationships that are derived from data that's available rather than the data that you need can indeed lead you astray.”It begs the question: how far astray are we?We know over the last one hundred years or so the planet has warmed roughly an average of 1.5 degrees Celsius. This is a number that has been contorted in the media to mean some kind of threshold after which “something” “might” happen. But Schmidt cautions there is no way to know when we hit this number, exactly, and it's not going to be obvious. Perhaps it already pushed passed this threshold, or it may not for another decade.He says, “we are going to continue to warm on the aggregates because we are continuing to put carbon dioxide and other greenhouse gases into the atmosphere. Until we get effectively to net zero, so no more addition of carbon dioxide to the atmosphere, temperatures will continue to climb. The less we put in, the slower that will be. But effectively, our best estimate of when global warming will stop is when we get to net zero.”Getting to net zero involves significant and radical changes in energy production, industrial processes, and consumption patterns. Moreover, it will require an unprecedented comprehensive and coordinated worldwide effort across all sectors of the economy, institutions, and governments.This is true even for hypothetical and speculative climate engineering solutions like injecting sulfates into the atmosphere in attempts to cool the planet. According to Schmidt, not only would this require cooperation across borders, so long as we keep spewing emissions into the atmosphere, we'd be forever trying to cool the planet…for eternity or at least until we've exhausted all the planet's fossil fuels.It's hard to imagine this happening in my lifetime, if ever. After all, climate change is already disrupting and displacing entire populations and we're seeing governments, and their citizens, becoming increasingly selfish and isolationist, not collaborative.As Schmidt admits, “We're not on the optimum path. We're not on the path that will prevent further damage and prevent the need for further adaptation. So, we're going to have to be building climate resilience, we're going to have to be adapting, we're going to have to be mitigating, and you have to do all three. You can't adapt to an ever-getting-worse situation, it has to at some point stabilize.”Schmidt says he derives no joy in telling people “that the next decade is going to be warmer than the last decade and it was warmer than the decade before that.” He says, “It gives me no joy to tell people that, oh yeah, we're going to have another record-breaking year this year, next year, whenever. Because I'm not a sociopath. I'm a scientist, yes, but I'm also a person.”Schmidt's words resonate deeply, reminding us that behind the data and predictions are real people—scientists, citizens, and future generations—all grappling with the weight of our changing world. As we stand at this critical juncture, we're not just passive observers but active participants in Earth's unfolding story, a story that's leaving its mark on nearly every corner of our planet.The butterfly effect, as meteorologist Edward Lorenz proposed, isn't just about tornados in Texas being set off by a chain of events from the flap of a butterfly's wings in Brazil; it's a powerful metaphor for our collective impact. Each of us, in our daily choices and actions, creates ripples that extend far beyond our immediate sphere. In a world where only two small patches—one off Greenland's coast and another near Antarctica—remain untouched by climate change, our individual actions carry profound significance.The path to net zero isn't just about grand gestures or technological breakthroughs. It's about millions of small, intentional actions coalescing into a force powerful enough to alter our trajectory. As we face the challenges ahead, let's remember that our individual agency, when combined, has the potential to create tsunamis of change, even in places we may never visit ourselves.In the end, it's not just about preserving a habitable planet — it's about preserving our humanity, our connection to each other and to the Earth that sustains us. As we navigate this critical decade and beyond, let's carry with us the knowledge that every action, no matter how small, contributes to the larger narrative of our planet's future. We are all butterflies, and in a world where climate change-free zones are becoming as rare as a family vacation to Antarctica, our wings have never mattered more. This is a public episode. If you would like to discuss this with other subscribers or get access to bonus episodes, visit interplace.io

Lords: * Daniel * Andrew Topics: * Hyperthymesia * Animal Well * Bad news for my google alerts: "Twinbeard" is the name of the Helldivers 2 community manager * Dean asks "I know there are many seniors who are looking to start a business during retirement. For those who haven't settled on an idea yet, I'd love to suggest house flipping as a good way to earn income. May I write about this topic in a free article for your website -- specifically, how seniors can get into the house flipping business and become successful at it? I'll be sure to cover all the basics, including how to find the perfect property, how to finance the expenses, how to manage payroll for employees and contractors, etc. I'll also feature a link back to your website." * Should game developers "soak themselves" in games. Referring to this clip of Orson Welles: https://www.youtube.com/watch?v=dg-qaeIcuyI Microtopics: * Doom Guy: Life in First Person * A well full of animals. * A game that you keep hearing is great from people that you trust. * Remembering every detail of your life even though that's a terrible idea. * A cop asking you what you were doing on the night of November 3rd, 2021. * Fnords. * Eidetic memory vs. photographic memory vs. hyperthymesic memory. * Checking your memory of a time against your written records of that time. * Remembering that you're going to have a conversation about your day tonight so you had better do something interesting. * Data mining electronic records to figure out how you met somebody. * Playing a Sam Barlow game with your bank records. * Navier-Stokes fluid dynamics. * Spending a lot of time frowning. * Dark Forest. * A game providing you with a spoiler-free mechanism of conveying where you are in the game. * Shooting people vs. swording people. * The tools and verbs of just being a little guy. * Seeing a dog doing something you didn't expect to see a dog do. * Being unable to turn a crank because there's a blue ghost. * Choosing not to be a video game. * Setting up Google alerts for your game's working title. * Famous racehorse Secret Legend. * Liking some parts but not liking other parts. * How to construct a Google alert for t the video game Tunic. * Computer vision algorithms to increase yield and profitablity of insect farms. * How seniors can get into the house flipping business. * Collaborating with a spammer to write a poem about house flipping. * Exhorting a spammer to join the discord and post their dreams in the announcements channel. * A think tank dedicated to increasing house flipping around the world, measured by total rotation. * House flipping vs. house spinning. * A skateboard with the word "quaternion" written on it in a sick font. * Trying to flip a house but it gets gimbal locked and you lose everything. * Soaking in films. Just marinating in them. * Having virgin eyes in order to make great things. * Inventing video games from first principles. * The value of outsider art. * Shigeru Miyamoto inventing jumping. * Trying to remember the last time you jumped in real life. * Whether a historian has discovered the cave that inspired the Legend of Zelda. * What Stories Untold says about the Orson Welles quote about having virgin eyes. * Trying to prove yourself to a piece of software. * Trying to tell a story about a spaceship that doesn't involve combat. * 50 Years of Text Games. * Having read a book back when it was a blog. * Whether we have time for another topic. * Mixing a Red Bull with a Caffeine Free Coca Cola. * Sunlight patterns that remind you of wandering in the forest as a child * The disco ball as an extrapolation of dappled sunlight in the forest. * Out of focus motes of light. * Figuring out how to make a desert beautiful. * Living that home assistant life. * Going to a thrift store to buy a lamp to fix the terrible lighting in your hotel room. * A motor that's constantly jiggling a pot of water.

Outline00:00 - Intro01:17 - Early Years04:17 - The “Scenic Route” to Control Theory12:44 - Sampled Data Systems22:26 - Linear Parameter Varying (LPV) Identification28:07 - From Distributed Systems and PDEs ...38:59 - ... to Distributed Control of Spatially Invariant Systems49:02 - Taming the Navier-Stokes Equations50:55 - Advice to Future Students1:13:12 - Coherence in Large Scale Systems1:32:28 - On Resistive Losses in Power Systems1:39:00 - Cochlear Instabilities1:50:40 - Stochasticity in Feedback Loops2:00:00 - About Linear and Nonlinear Control2:08:14 - How to Select a Research Problem2:14:21 - Future of Control2:22:06 - OutroLinks- Paper on moment-invariants and object recognition: https://tinyurl.com/26tnks3z- Bassam's PhD Thesis: https://tinyurl.com/3n2274dv- Identification of linear parametrically varying systems: https://tinyurl.com/mryebhhy- Distributed control of spatially invariant systems: https://tinyurl.com/rzszjch2- Shift Operator: https://tinyurl.com/24fwehet- Heat Equation: https://tinyurl.com/57rc6s7h- Navier-Stokes Equations: https://tinyurl.com/45ktrd2e- The impulse response of the Navier-Stokes equations: https://tinyurl.com/4vaausfn- Non-Normal Matrix: https://tinyurl.com/58z4sph8- Coherence in large-scale networks: https://tinyurl.com/ynm5cbay- The Price of Synchrony: https://tinyurl.com/3svzancw- Tinnitus: https://tinyurl.com/yc5hm549- Cochlear Instabilities: https://tinyurl.com/fjespjbj- Stochasticity in Feedback Loops: https://tinyurl.com/yc6aw9xt- Koopman Operator: https://tinyurl.com/3jeu68p8- Carleman Linearization: https://tinyurl.com/yckzrnfh- Mamba Model: https://tinyurl.com/33h59jwj- Spectral Factorization: https://Support the Show.Podcast infoPodcast website: https://www.incontrolpodcast.com/Apple Podcasts: https://tinyurl.com/5n84j85jSpotify: https://tinyurl.com/4rwztj3cRSS: https://tinyurl.com/yc2fcv4yYoutube: https://tinyurl.com/bdbvhsj6Facebook: https://tinyurl.com/3z24yr43Twitter: https://twitter.com/IncontrolPInstagram: https://tinyurl.com/35cu4kr4Acknowledgments and sponsorsThis episode was supported by the National Centre of Competence in Research on «Dependable, ubiquitous automation» and the IFAC Activity fund. The podcast benefits from the help of an incredibly talented and passionate team. Special thanks to L. Seward, E. Cahard, F. Banis, F. Dörfler, J. Lygeros, ETH studio and mirrorlake . Music was composed by A New Element.

Après l'équation Navier-Stokes et les explications de notre cher Bill François autour des secrets de la nage des larves de poissons, nous plongeons cette semaine dans le monde des gros poissons ! Thons, espadons, requins mako et autres flèches des océans (de 70 à 120 km/h !) auraient peut-être plus en commun avec les avions qu'on pourrait le penser ... Là où les humains "poussent" l'eau autour d'eux pour avancer, ces gros poissons-là utilisent, comme les engins volants, le principe de portance pour atteindre leurs folles vitesses. Toutes ces explications, et bien d'autres encore, à retrouver dans l'épisode de cette semaine !______Grande nouveauté dans la sphère de PPDP : les émissions seront désormais présentées par des équipiers des Baleine sous Gravillon. Pour cette fin de saison 2024, c'est Arthur Hannoun, le nouveau rédacteur en chef de l'équipe BSG, qui se chargera de l'enregistrement et du partage des épisodes.Pour la fin de la 3e saison, PPDP renoue avec une vieille connaissance Bill François, qui a publié à la fin de l'année dernière (octobre 2023) son dernier livre, Les génies des mers (Flammarion).______On aime ce qui nous a émerveillé … et on protège ce qu'on aime.______PARTAGERSous notre Gravillon vous trouverez… 4 podcasts, 1 site, 1 compte Instagram, 1 page + 1 groupe Facebook et 1 asso. Il nous serait très précieux et utile que vous partagiez ce lien :https://baleinesousgravillon.com/liens-2Pour nous aider, vous pouvez vous abonner et partager nos liens, et nous laisser des étoiles et surtout un avis sur Apple Podcast et Spotify. Ainsi, nous serons ainsi plus visibles et mieux recommandés. Merci :)_______SOUTENIRTous nos podcasts sont faits bénévolement. Ils sont gratuits, sans pub et accessibles à tous. Vous pouvez faire un don sur Helloasso (ou sur Tipeee), adhérer à l'asso BSG, ou installer gratuitement le moteur de recherche Lilo et nous reverser vos gouttes :https://bit.ly/helloasso_donsUR_BSGhttps://bit.ly/lien_magq_lilo_BSGhttp://bit.ly/Tipeee_BSG_______TRAVAILLER ENSEMBLEVous voulez créer un podcast ? Nous vous formons et/ou accompagnons !Nous proposons des Fresques de la biodiversité, des conférences et animons des tables rondes. Nous cherchons des sponsors et des partenaires : contact@baleinesousgravillon.comHébergé par Ausha. Visitez ausha.co/politique-de-confidentialite pour plus d'informations.

L'équation Navier-Stokes est l'un des sept problèmes du millénaire, réputés insolubles dans le monde mathématique. Elle représente un enjeu majeur dans le monde de la mécanique des fluides, permettant de calculer comment l'eau se met en mouvement sous l'effet des forces que l'on y applique. Et notamment la nage des poissons dans l'eau ! Comment un phénomène d'apparence si simple et anodine peut-elle receler, encore de nos jours, autant de mystères ? C'est l'enjeu de l'épisode cette semaine.______Grande nouveauté dans la sphère de PPDP : les émissions seront désormais présentées par des équipiers des Baleine sous Gravillon. Pour cette fin de saison 2024, c'est Arthur Hannoun, le nouveau rédacteur en chef de l'équipe BSG, qui se chargera de l'enregistrement et du partage des épisodes.Pour la fin de la 3e saison, PPDP renoue avec une vieille connaissance Bill François, qui a publié à la fin de l'année dernière (octobre 2023) son dernier livre, Les génies des mers (Flammarion).Photo : Fabrice Guérin______On aime ce qui nous a émerveillé … et on protège ce qu'on aime.______PARTAGERSous notre Gravillon vous trouverez… 4 podcasts, 1 site, 1 compte Instagram, 1 page + 1 groupe Facebook et 1 asso. Il nous serait très précieux et utile que vous partagiez ce lien :https://baleinesousgravillon.com/liens-2Pour nous aider, vous pouvez vous abonner et partager nos liens, et nous laisser des étoiles et surtout un avis sur Apple Podcast et Spotify. Ainsi, nous serons ainsi plus visibles et mieux recommandés. Merci :)_______SOUTENIRTous nos podcasts sont faits bénévolement. Ils sont gratuits, sans pub et accessibles à tous. Vous pouvez faire un don sur Helloasso (ou sur Tipeee), adhérer à l'asso BSG, ou installer gratuitement le moteur de recherche Lilo et nous reverser vos gouttes :https://bit.ly/helloasso_donsUR_BSGhttps://bit.ly/lien_magq_lilo_BSGhttp://bit.ly/Tipeee_BSG_______TRAVAILLER ENSEMBLEVous voulez créer un podcast ? Nous vous formons et/ou accompagnons !Nous proposons des Fresques de la biodiversité, des conférences et animons des tables rondes. Nous cherchons des sponsors et des partenaires : contact@baleinesousgravillon.comHébergé par Ausha. Visitez ausha.co/politique-de-confidentialite pour plus d'informations.

We discuss the basics of fluid mechanics. What is a Newtonian fluid? Can fluids be compressed? How do hydraulics work? What are fluid statics vs fluid dynamics? What is Pascal's Law? What are some of the formulas used in fluid mechanics? Can a fluid be squishy? What are the Navier–Stokes equations? How do you design a submarine and a jet engine? What is laminar flow vs turbulent flow?

Στην 7η σεζόν αναλύουμε θέματα που δεν γνωρίζουμε καλά! Πώς περιγράφουμε τη συμπεριφορά των ρευστών (υγρών και αερίων;) Εξισώσεις Navier-Stokes Τι είναι η τυρβώδης ροή; Γιατί είναι τόσο δύσκολο να βρούμε λύσεις; Post-show: Remarkable e-ink tablet, 1 month review Επικοινωνία hello@notatop10.fm @notatop10  @timaras@mstdn.social @giorgos.dimop 

Here's something weird to think about. Can fluids think? No, we're not talking about a liquid metal shape-shifting creature like what we saw in Terminator 2. We're asking, can fluid systems make computations? In this episode, we chat with Professor Eva Miranda, head of the Laboratory of Geometry and Dynamical Systems at the Polytechnic University of Catalonia in Spain. She's on a mission exploring one of math's most famous open problems – the existence of solutions of the Navier-Stokes equation, which governs the flow of fluids such as water and air. Since the 1800's, researchers interested in how fluids flow have turned to Navier-Stokes. To investigate Navier-Stokes, Eva and her colleagues have constructed an abstract mathematical machine, a theoretical fluid computer, if you will. In this podcast, Professor Miranda tells us how a post from Australian Mathematician Terrence Tao inspired her, how she used geometry to construct her fluid computer, and why she believes that Navier-Stokes might not always be physically valid. Our host for this episode is Dr James Nichols.See omnystudio.com/listener for privacy information.

Visste du at grunnlaget for Norges velstand er en ligning? Mange steder i verden kan man nærmest stikke et rør ned i bakken, og så spruter olja opp, men ute i Nordsjøen hadde vi ikke kommet langt uten Navier- Stokes -ligningen. Og nå, som vi skal bygge både havvindmøller og bedre klimamodeller, er det igjen Navier-Stokes vi må lene oss på. Det er bare et tilsynelatende problem: Vi klarer ikke å løse ligningen. Vi vet ikke en gang om en skikkelig løsning eksisterer. Abels tårn har møtt alle tre matematikerne som er de i verden som har kommet nærmest å forstå ligningen. En av dem var denne uka i Oslo der han blei tildelt Abelprisen av Hans Majestet Kongen. Hør episoden i appen NRK Radio

Stephen Wolfram answers questions from his viewers about the history science and technology as part of an unscripted livestream series, also available on YouTube here: https://wolfr.am/youtube-sw-qa Questions include: You have published several other books after NKS. Has publishing technology and quality changed in the intervening time? - Would you like to provide a history of fluid mechanics, for example how the Navier–Stokes equations were discovered and how they work? - Given the recent hearings on and history of UFOs, do you have any thoughts on this subject? - In general, how do you engage with conspiracy theories or "alternative science"? I'm curious because most scientists in institutions are immediately dismissive of anything outside mainstream thought, but to me this seems just as intellectually dishonest as ascribing absolute certainty to any given conspiracy theory. - Isn't there a suspicious correlation between a surge in UFO sightings since the 1950s and a surge of UFO movies during that same period? - What about the Phoenix Lights event where thousands of people saw the same exact thing? - What are some other notable phenomena that people thought they observed that never were proven to have scientific validity (e.g. alchemy)? - How has your view of the future changed over the past 40 years? - Most surprising is that so many people are using the internet for watching cat videos instead of doing useful things. - Maybe cats are the aliens.

Fysikk handler ikke bare om det største og det minste. Det er mye spennende fysikk på mer dagligdagse størrelsesskalaer også, for eksempel, på kjøkkenet. Der er det mye som renner - kaffe, champagne, pannekakerøre og oppvaskvann. Væsker beskrives av et sett av matematiske ligninger kalt Navier-Stokes ligningene. De ser ganske uskyldige ut, men har mange overraskelser å by på. Dagens gjest er fluidmekaniker og bruker eksempler fra kjøkkenet for å formidle fagfeltet.

Welcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Book Review: Worlds of Flow, published by remember on January 16, 2023 on LessWrong. This work was written at Conjecture. “Worlds of Flow,” a history of 19th and early 20th-century hydrodynamics by Oliver Darrigol, concludes: What distinguishes the history of hydrodynamics from that of other physical theories is not so much the tremendous effect of challenges from phenomenal worlds, but rather it is the slowness with which these challenges were successfully met. Nearly two centuries elapsed between the first formulation of the fundamental equations of the theory and the deductions of laws of fluid resistance in the most important case of large Reynolds numbers.The reasons for this extraordinary delay are easily identified a posteriori. They are the infinite number of degrees of freedom and the nonlinear character of the fundamental equations, both of which present formidable obstacles to obtaining solutions in concrete cases. Moreover, instability often deprives the few known exact solutions of any physical relevance. These difficulties have barred progress along purely mathematical lines. They have also made physical intuition a poor guide, and a source of numerous paradoxes. Hydrodynamicists therefore sought inspiration in concrete phenomena. Engagement with and challenges from the real worlds of flow were essential to the development of the above-mentioned strategies. The challenged theorists strove to find new solutions and to develop new methods of approximation. Experience indicated some general properties of the motion, such as the existence of boundary layers, the random character of turbulence, the sudden character of the Reynolds transition, or the formation of trailing vortices.Altogether, there were many ways in which practical concerns oriented theorists in the conceptual maze of fluid dynamics. The evolution from a paper theory to an engineering tool thus depended on transgressions of the limits between academic hydrodynamics and applied hydrodynamics. This quote captures one of the most significant lessons in the book: the study of concrete phenomena was critical in overcoming many of the difficulties in hydrodynamics. Roughly two upstream problems required interacting with concrete phenomena to solve. The first is summarized nicely by the quote above: theorizing and abstract thinking alone was not enough to solve the problems posed by hydrodynamics. The second is a subtler point: early mathematical and theoretical tools weren't adapted to understanding hydrodynamics. Much of the necessary mathematical machinery existed quite early in the 1800s, but people hadn't built the physico-mathematical tools or intuitions to tell us what they physically meant. Contact with reality forced scientists to confront the inadequacies of their theories while guiding the adaption of physico-mathematical tools. While the book is organized along rough problems that hydrodynamics faced (waves, viscosity, vortices, instability, etc.), this review will focus on broader scientific lessons. First on the two big themes I think are most important, then briefly on the other themes at the end. Practice and theory in hydrodynamics The first problem was that abstract thinking and theorizing proved unable to solve many of the problems of hydrodynamics. A great example of this comes from the discovery of Reynold's number, which predicts whether flow is turbulent or laminar. Reynold's number could have potentially been hypothesized as a consequence of Navier-Stokes, which describes viscous flow behavior. But Navier-Stokes is not analytically solvable, so Reynold's number doesn't come automatically. Making this more difficult is that turbulent flow, such as after submerged propellers, is generally invisible. Instead of reasoning his way there from first principles, Osborne Reynolds firs...

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Book Review: Worlds of Flow, published by remember on January 16, 2023 on LessWrong. This work was written at Conjecture. “Worlds of Flow,” a history of 19th and early 20th-century hydrodynamics by Oliver Darrigol, concludes: What distinguishes the history of hydrodynamics from that of other physical theories is not so much the tremendous effect of challenges from phenomenal worlds, but rather it is the slowness with which these challenges were successfully met. Nearly two centuries elapsed between the first formulation of the fundamental equations of the theory and the deductions of laws of fluid resistance in the most important case of large Reynolds numbers.The reasons for this extraordinary delay are easily identified a posteriori. They are the infinite number of degrees of freedom and the nonlinear character of the fundamental equations, both of which present formidable obstacles to obtaining solutions in concrete cases. Moreover, instability often deprives the few known exact solutions of any physical relevance. These difficulties have barred progress along purely mathematical lines. They have also made physical intuition a poor guide, and a source of numerous paradoxes. Hydrodynamicists therefore sought inspiration in concrete phenomena. Engagement with and challenges from the real worlds of flow were essential to the development of the above-mentioned strategies. The challenged theorists strove to find new solutions and to develop new methods of approximation. Experience indicated some general properties of the motion, such as the existence of boundary layers, the random character of turbulence, the sudden character of the Reynolds transition, or the formation of trailing vortices.Altogether, there were many ways in which practical concerns oriented theorists in the conceptual maze of fluid dynamics. The evolution from a paper theory to an engineering tool thus depended on transgressions of the limits between academic hydrodynamics and applied hydrodynamics. This quote captures one of the most significant lessons in the book: the study of concrete phenomena was critical in overcoming many of the difficulties in hydrodynamics. Roughly two upstream problems required interacting with concrete phenomena to solve. The first is summarized nicely by the quote above: theorizing and abstract thinking alone was not enough to solve the problems posed by hydrodynamics. The second is a subtler point: early mathematical and theoretical tools weren't adapted to understanding hydrodynamics. Much of the necessary mathematical machinery existed quite early in the 1800s, but people hadn't built the physico-mathematical tools or intuitions to tell us what they physically meant. Contact with reality forced scientists to confront the inadequacies of their theories while guiding the adaption of physico-mathematical tools. While the book is organized along rough problems that hydrodynamics faced (waves, viscosity, vortices, instability, etc.), this review will focus on broader scientific lessons. First on the two big themes I think are most important, then briefly on the other themes at the end. Practice and theory in hydrodynamics The first problem was that abstract thinking and theorizing proved unable to solve many of the problems of hydrodynamics. A great example of this comes from the discovery of Reynold's number, which predicts whether flow is turbulent or laminar. Reynold's number could have potentially been hypothesized as a consequence of Navier-Stokes, which describes viscous flow behavior. But Navier-Stokes is not analytically solvable, so Reynold's number doesn't come automatically. Making this more difficult is that turbulent flow, such as after submerged propellers, is generally invisible. Instead of reasoning his way there from first principles, Osborne Reynolds firs...

Link to original articleWelcome to The Nonlinear Library, where we use Text-to-Speech software to convert the best writing from the Rationalist and EA communities into audio. This is: Book Review: Worlds of Flow, published by remember on January 16, 2023 on LessWrong. This work was written at Conjecture. “Worlds of Flow,” a history of 19th and early 20th-century hydrodynamics by Oliver Darrigol, concludes: What distinguishes the history of hydrodynamics from that of other physical theories is not so much the tremendous effect of challenges from phenomenal worlds, but rather it is the slowness with which these challenges were successfully met. Nearly two centuries elapsed between the first formulation of the fundamental equations of the theory and the deductions of laws of fluid resistance in the most important case of large Reynolds numbers.The reasons for this extraordinary delay are easily identified a posteriori. They are the infinite number of degrees of freedom and the nonlinear character of the fundamental equations, both of which present formidable obstacles to obtaining solutions in concrete cases. Moreover, instability often deprives the few known exact solutions of any physical relevance. These difficulties have barred progress along purely mathematical lines. They have also made physical intuition a poor guide, and a source of numerous paradoxes. Hydrodynamicists therefore sought inspiration in concrete phenomena. Engagement with and challenges from the real worlds of flow were essential to the development of the above-mentioned strategies. The challenged theorists strove to find new solutions and to develop new methods of approximation. Experience indicated some general properties of the motion, such as the existence of boundary layers, the random character of turbulence, the sudden character of the Reynolds transition, or the formation of trailing vortices.Altogether, there were many ways in which practical concerns oriented theorists in the conceptual maze of fluid dynamics. The evolution from a paper theory to an engineering tool thus depended on transgressions of the limits between academic hydrodynamics and applied hydrodynamics. This quote captures one of the most significant lessons in the book: the study of concrete phenomena was critical in overcoming many of the difficulties in hydrodynamics. Roughly two upstream problems required interacting with concrete phenomena to solve. The first is summarized nicely by the quote above: theorizing and abstract thinking alone was not enough to solve the problems posed by hydrodynamics. The second is a subtler point: early mathematical and theoretical tools weren't adapted to understanding hydrodynamics. Much of the necessary mathematical machinery existed quite early in the 1800s, but people hadn't built the physico-mathematical tools or intuitions to tell us what they physically meant. Contact with reality forced scientists to confront the inadequacies of their theories while guiding the adaption of physico-mathematical tools. While the book is organized along rough problems that hydrodynamics faced (waves, viscosity, vortices, instability, etc.), this review will focus on broader scientific lessons. First on the two big themes I think are most important, then briefly on the other themes at the end. Practice and theory in hydrodynamics The first problem was that abstract thinking and theorizing proved unable to solve many of the problems of hydrodynamics. A great example of this comes from the discovery of Reynold's number, which predicts whether flow is turbulent or laminar. Reynold's number could have potentially been hypothesized as a consequence of Navier-Stokes, which describes viscous flow behavior. But Navier-Stokes is not analytically solvable, so Reynold's number doesn't come automatically. Making this more difficult is that turbulent flow, such as after submerged propellers, is generally invisible. Instead of reasoning his way there from first principles, Osborne Reynolds firs...

The Navier–Stokes equations are important for science and engineering, since they describe the motion of fluids.  However, these equations can not describe the physical responses of fluids with a complex microstructure. Michal Bathory, Miroslav Bulíček, and Josef Málek, Charles University, Czech Republic, have developed a robust mathematical theory for viscoelastic fluids. Which could serve as an analytical framework, to quantify errors between exact and computed solutions for these models.Read more in Research OutreachRead some of their latest work here: https://doi.org/10.1515/anona-2020-0144

Has it ever crossed your mind how would our discipline look like, if we did not have Fire Dynamics Simulator? Maybe you had an opportunity to discuss CFD with colleagues from other disciplines, to find their faces in shock and awe that the fire community actually has its own, FREE AND OPEN SOURCE, validated and fully recognized solver? A testimony to the impact of FDS may be the citation count on its user guide, which has recently exceeded 5.000 citations! The FDS code is something special and our little scientific community can feel proud that a tool of this magnitude was built just for us!But it did not build itself. There is a history of giants paving the route with their low-Mach number approximation of Navier-Stokes. There is my today's guest - Dr Kevin McGrattan who saw the need and built the first iteration of FDS (and still leads its development 20 years later). And there is a team of brilliant scientists at NIST, VTT and other parts of the world, who shared this dream of a robust, open-access fire simulator, and volunteered their hard work into making this dream a reality.In this podcast episode, we focus on the very early days of FDS (or even before it came to life). My mission was to learn how FDS was built, what the landscape looked like back then and how it was growing. When particular important sub-models came into existence and what triggered that. We also learn what were the goals for the tool development and how did they evolve over the years as the project got more and more serious.  You may be surprised by some very simple explanations beyond some very tough design decisions!I hope you enjoy this episode. Please appreciate dr Kevin McGrattan and the hard work done at NIST, VTT and other places, that enabled us to have something really special. Our solver. Fine tuned for fire and open to all.Learn more about FDS at the project website at NIST or the GitHub repository.

Stephen Wolfram answers general questions from his viewers about science and technology as part of an unscripted livestream series, also available on YouTube here: https://wolfr.am/youtube-sw-qa Questions include: Can you talk about turbulence and why its the greatest unsolved problem of classical mechanics? - ​Hello, can you talk a little bit about turbulence. Is it true turbulence can't be predicted from the underlying physical equations (Navier-Stokes)? - how about intelligent fluids that are about to take over the world? - Which self-driving technology is better: Imaging or LIDAR? - What are finite fields and why are they important?

Tim Palmer is Royal Society Research Professor in Climate Physics, and a Senior Fellow at the Oxford Martin Institute.He is interested in the predictability and dynamics of weather and climate, including extreme events.He was involved in the first five IPCC assessment reports and was co-chair of the international scientific steering group of the World Climate Research Programme project (CLIVAR) on climate variability and predictability.After completing his DPhil at Oxford in theoretical physics, Tim worked at the UK Meteorological Office and later the European Centre for Medium-Range Weather Forecasts. For a large part of his career, Tim has developed ensemble methods for predicting uncertainty in weather and climate forecasts.In 2020 Tim was elected to the US National Academy of Sciences.Steve, Corey Washington, and Tim first discuss his career path from physics to climate research and then explore the science of climate modeling and the main uncertainties in state-of-the-art models.In this episode, we discuss:00:00 Introduction1:48 Tim Palmer's background and transition from general relativity to climate modeling15:13 Climate modeling uncertainty46:41 Navier-Stokes equations in climate modeling53:37 Where climate change is an existential risk1:01:26 Investment in climate researchLinks:Tim Palmer (Oxford University)https://www.ox.ac.uk/news-and-events/find-an-expert/professor-tim-palmerThe scientific challenge of understanding and estimating climate change (2019)https://www.pnas.org/doi/pdf/10.1073/pnas.1906691116ExtremeEarthhttps://extremeearth.eu/Physicist Steve Koonin on climate changehttps://infoproc.blogspot.com/2021/04/how-physicist-became-climate-truth.htmlMusic used with permission from Blade Runner Blues Livestream improvisation by State Azure.–Steve Hsu is Professor of Theoretical Physics and of Computational Mathematics, Science, and Engineering at Michigan State University. Previously, he was Senior Vice President for Research and Innovation at MSU and Director of the Institute of Theoretical Science at the University of Oregon. Hsu is a startup founder (SafeWeb, Genomic Prediction, Othram) and advisor to venture capital and other investment firms. He was educated at Caltech and Berkeley, was a Harvard Junior Fellow, and has held faculty positions at Yale, the University of Oregon, and MSU.Please send any questions or suggestions to manifold1podcast@gmail.com or Steve on Twitter @hsu_steve.

Heute reden wir wirklich mal etwas über die Fluiddynamik und deren Lösungen über die Navier-Stokes-Gleichungen. Dafür bekommen wir quasi eine Mathe-Geplänkel Folge umsonst mit dazu, denn diese Gleichungen sind auch ein Teil der Millennium-Probleme. Wie immer überall, wo es Podcasts gibt. Viel Vergnügen! #navier #stokes #gasgleichung #fluiddynamik #strömung #flüssigkeit #newton ********** Anmerkungen, Fragen, Kritik oder interessante Themenvorschläge bitte an physikgeplaenkel@gmail.com ********** Unsere Instragram Seite: https://www.instagram.com/physikgeplaenkel/ Unsere Facebook Seite: https://www.facebook.com/Physik-Geplänkel-1153934681433003/ Unser Youtube Channel: https://www.youtube.com/channel/UCD1CT-nTdEagwMF16P6gIKQ/ Folgt uns unter "Physik-Geplänkel" auf Spotify, iTunes, Deezer, PocketCasts oder als Amazon Alexa Skill. Oder am besten direkt unter https://physik-geplaenkel.podigee.io/

Can we write a compiler that will Target a fluid system that can be described as the navier-stokes equation --- Send in a voice message: https://anchor.fm/stre/message

Join host Pranet Sharma as he sits down with guest Dr. Ashwin Ramchandran, a biomedical engineer and a postdoc at Princeton University, to discuss the intersection of astronomy and biomedical engineering. Topics include: the interdisciplinary nature of engineering, how engineering is essential for maintaining spacecraft integrity, the ways that understanding human perception is necessary for journeys in space, why the core biology of humans is necessary to be understood for astrophysical endeavors, how biology and astronomy share their core in the understanding of microscopic and macroscopic ideals respectively, how fluid dynamics is the cornerstone connecting biomedical engineering and astrophysics, the various dimensionality of fluids and their applicability, Navier-Stokes, the ways that physics is essential for biology's development, the importance of considering science as a large field on its own, and why one should always be fearless in following one's passions. Space Agencies:www.nasa.govwww.esa.intwww.isro.gov.inhttps://global.jaxa.jp/ For any questions about the show, visit www.skysimplified.com/contact. Thank you for listening, and as always, clear skies! SkySimplified Twitter: @skysimplifiedSkySimplified Instagram: @skysimplified

Conservation laws (This is episode 32) The assumption of a fluid continuum allows problems in aerodynamics to be solved using fluid dynamics conservation laws. Three conservation principles are used: Conservation of mass Conservation of mass requires that mass is neither created nor destroyed within a flow; the mathematical formulation of this principle is known as the mass continuity equation. Conservation of momentum The mathematical formulation of this principle can be considered an application of Newton's Second Law. Momentum within a flow is only changed by external forces, which may include both surface forces, such as viscous (frictional) forces, and body forces, such as weight. The momentum conservation principle may be expressed as either a vector equation or separated into a set of three scalar equations (x,y,z components). Conservation of energy The energy conservation equation states that energy is neither created nor destroyed within a flow, and that any addition or subtraction of energy to a volume in the flow is caused by heat transfer, or by work into and out of the region of interest. Together, these equations are known as the Navier-Stokes equations, although some authors define the term to only include the momentum equation(s). The Navier-Stokes equations have no known analytical solution and are solved in modern aerodynamics using computational techniques. Because computational methods using high speed computers were not historically available and the high computational cost of solving these complex equations now that they are available, simplifications of the Navier-Stokes equations have been and continue to be employed. The Euler equations are a set of similar conservation equations which neglect viscosity and may be used in cases where the effect of viscosity is expected to be small. Further simplifications lead to Laplace's equation and potential flow theory. Additionally, Bernoulli's equation is a solution in one dimension to both the momentum and energy conservation equations. The ideal gas law or another such equation of state is often used in conjunction with these equations to form a determined system that allows the solution for the unknown variables. --- This episode is sponsored by · Anchor: The easiest way to make a podcast. https://anchor.fm/app

Holography explains why black hole horizons have thermodynamic and hydrodynamic properties and inspires researchers to re-visit foundations and explore limits of relativistic hydrodynamics Since the work of Bekenstein, Hawking and others in the early 1970s, it was known that the laws of black hole mechanics are closely related if not identical to the laws of thermodynamics. A natural question to ask, then, is whether this analogy or the correspondence extends beyond the equilibrium state. The affirmative answer, given by various authors during the 1980s and 90s, became known as the "black hole membrane paradigm". It was shown that black hole horizons can be viewed as being endowed with fluid-like properties such as viscosity, thermal conductivity and so on, whose values remained mysterious. The development of holography 15-20 years ago clarified many of these issues and has led to the quantitative correspondence between Navier-Stokes and Einstein equations. It became possible to study the long-standing problems such as thermalization and turbulence by re-casting them in the dual gravity language. We review those developments focusing, in particular, on the issue of the "unreasonable effectiveness" of hydrodynamic description in strongly interacting quantum systems. Final remarks, Prof Julia Yeomans FRS, Head of Rudolf Peierls Centre for Theoretical Physics

Spanners and Trumpets are joined by ace race caller Chris ‘Catman’ Turner as they turn over all the rocks looking for the latest F1 gold. From shorter practices to new start times, from the new bossman to the best driver pairings, no Navier-Stokes equation goes uncalculated in this, the latest episode of Missed Apex Podcast.Spanners Ready Spanners

In dieser Folge spricht Gudrun mit Ayca Akboyraz und Alejandro Castillo. Beide sind im Masterstudiengang Chemieingenieurwesen bzw. Bioingenieurwesen am KIT eingeschrieben und haben 2019 das Projektorientierte Softwarepraktikum in Gudruns Arbeitsgruppe absolviert. Das Gespräch dreht sich um ihre Erfahrungen in dieser Lehrveranstaltung. Ayca stammt aus der Türkei und Alejandro ist in Mexico aufgewachsen. Beide haben in ihren Heimatländern deutsche Schulen besucht. Anschließend haben sie sich jeweils um ein Studium in Deutschland beworben. Ayca hatte sich zunächst für Wirtschaftsingenieurwesen entschieden, hat aber nach einiger Zeit gemerkt, dass ihr Chemieingenieurwesen viel mehr liegt. Das Projektorientierte Softwarepraktikum wurde 2010 als forschungsnaher Lernort konzipiert. Studierende unterschiedlicher Studiengänge arbeiten dort ein Semester lang an konkreten Strömungssimulationen. Es wird regelmäßig im Sommersemester angeboten. Seit 2014 liegt als Programmiersprache die Open Source Software OpenLB zugrunde, die ständig u.a. in der Karlsruher Lattice Boltzmann Research Group weiter entwickelt wird. Außerdem wird das Praktikum seit 2012 vom Land Baden-Württemberg gefördert als eine Möglichkeit für Studierende, sich im Studium schon an Forschung zu beteiligen. Konkret läuft das Praktikum etwa folgendermaßen ab: Die Studierenden erhalten eine theoretische Einführung in Strömungsmodelle und die Idee von Lattice-Boltzmann-Methoden und finden sich für ein einführendes kleines Projekt in Zweiergruppen zusammen. Anschließend wählen sie aus einem Katalog eine Frage aus, die sie bis zum Ende des Semesters mit Hilfe von Computersimulationen gemeinsam beantworten. Diese Fragen sind Teile von Forschungsthemen der Gruppe, z.B. aus Promotionsprojekten oder Drittmittelforschung. Während der Projektphase werden die Studierenden von dem Doktoranden/der Doktorandin der Gruppe, die die jeweilige Frage gestellt haben, betreut. Am Ende des Semesters werden die Ergebnisse in Vorträgen vorgestellt und diskutiert. Hier ist die ganze Arbeitsgruppe beteiligt. In einer Ausarbeitung werden außerdem die Modellbildung, die Umsetzung in OpenLB und die konkreten Simulationsergebnisse ausführlich dargelegt und in den aktuellen Forschungsstand eingeordnet. Diese Ausarbeitung wird benotet. Die Veranstaltung wird mit 4 ECTS angerechnet. In der klassischen Theorie der Strömungsmechanik werden auf der Grundlage der Erhaltung von Masse, Impuls und Energie und unter berücksichtigung typischer Materialeigenschaften die Navier-Stokes-Gleichungen als Modell für das Strömungsverhalten von z.B. Wasser hergeleitet. Die beiden unbekannten Größen in diesem System partieller Differentialgleichungen sind das Geschwindigkeitsfeld und der Druckgradient. Wenn geeigneten Rand- und Anfangsbedingungen für die Geschwindigkeit vorgeschrieben werden, liegt im Prinzip die Lösung des Gleichungssystem fest. Sie kann aber in der Regel nur numerisch angenähert berechnet werden. Eine wichtige Ausnahme ist die Strömung durch einen Zylinder mit kreisförmigem Querschnitt. Wenn am Rand des Zylinders als Randbedingung vorgeschrieben wird, dass dort das Fluid anhaftet, also die Geschwindigkeit ganz am Rand Null ist, dann stellt sich eine zeitlich unveränderliche (stationäre) Strömung ein, die am Rand des Zylinders still steht und in der Mitte am schnellsten ist. Der Verlauf zwischen diesen beiden Extremen entspricht genau dem einer Parabel. Diese Lösung heißt Poiseuille-Strömung. Der Durchfluss ergibt sich dann aus dem Druckgradienten. Wenn der Querschnitt des Kanals nicht genau kreisförmig ist, lässt sich das Prinzip noch übertragen, aber in der Regel ist die Lösung dann nicht mehr analytisch berechenbar. Die Poiseuille-Strömung ist ein häufiges Test- oder Benchmark-Problem in der numerischen Strömungsmechanik, zumal diese Strömungskonfiguration einer der wenigen Fälle der Navier-Stokes-Gleichungen ist, die analytisch gelöst werden können. Der Sinn des Tests besteht darin, zunächst sicherzustellen, dass die Berechnung mit Hilfe von OpenLB, eine gewisse Genauigkeit aufweist. Zweitens wird die Genauigkeit der Methode überprüft, indem analysiert wird, wie der numerische Fehler mit der Gitterverfeinerung skaliert. Ayca und Alejandro haben in ihrem Projekt diesen Benchmark vollzogen und dafür Simulationen im 2D und 3D Fall mit verschiedenen Randbedingungen, die in der Lattice Boltzmann Methode vorkommen (und in OpenLB implementiert vorliegen), und Gitterverfeinerungen mit Auflösung von 25, 50, 75, 100 Unterteilungen durchgeführt. Obwohl die Randbedingungen in numerischen Verfahren die gleichen grundlegenden Ziele wie im analytischen Fall haben, entwickeln sie sich entlang konzeptionell degenerativer Linien. Während analytische Randbedingungen die zugehörige Lösung aus einer Schar von zulässigen Lösungen der Gleichungen auswählen, wirken die Randbedingungen im Lattice Boltzmann Modell dynamisch mit. Sie sind ein Teil des Lösungsprozesses, der für die Änderung des Systemzustands in Richtung der Lösung zuständig ist. Eine der häufigsten Ursachen für die Divergenz der numerischen Lösung ist die falsche Umsetzung von Randbedingungen. Daher ist es für die Genauigkeit und Konvergenz sehr wichtig, dass die geeigneten Randbedingungen für die untersuchte Geometrie und den Strömungsfall ausgewählt werden. Es gibt eine große Familie Randbedingungen, die für die Lattice Boltzmann Methode entwickelt wurden. Für das Praktikum liegt der Fokus für die Wand auf den Randbedingungen "bounce-back" (Haftbedingung), "local", "interpolated" und "bouzidi". Alle genannten Randbedingungen erzeugen ein parabolisches Strömungsprofil passend zur analytischer Lösung. Unterschiede zeigen sich darin, wie groß die numerische Fehler ist, und in welchem Maß sich der numerische Fehler durch Gitterverfeinerung reduzieren lässt. Der graphische Vergleich der Simultionsergebnisse mit der analytischen Lösung hat gezeigt, dass bouzidi Randbedingung den kleinsten numerischen Fehler und die höchste Konvergenzordnung für den 3D-Fall erzeugt, während local und interpolated Randbedingungen für den 2D-Fall bessere Ergebnisse liefern. Zu beachten ist aber, dass mit erhöhter Gitterverfeinerung die Unterschiede zwischen diesen Randbedingungen verschwinden. Bei der Auswahl der Randbedingung sollte dementsprechend ein Kompromiss zwischen Aufwand und Güte der Lösung gefunden werden. Literatur und weiterführende Informationen T. Krüger e.a.: The Lattice Boltzmann Method. Graduate Texts in Physics. Springer, 2017. M. Portinari: 2D and 3D Verification and Validation of the Lattice Boltzmann Method. Master Thesis, Montréal 2015. C.J. Amick: Steady solutions of the Navier-Stokes equations in unbounded channels and pipes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4, 473–513 (1977). A. Akboyraz und A. Castillo, Ausarbeitung Softwarepraktikum 2019. M.J. Krause e.a.: OpenLB Release 1.3: Open Source Lattice Boltzmann Code. Podcasts L. Dietz, J. Jeppener, G. Thäter: Flache Photobioreaktoren - Gespräch im Modellansatz Podcast, Folge 213, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2019. T. Hoffmann, G. Thäter: Luftspalt, Gespräch im Modellansatz Podcast, Folge 153, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2017.

In the inaugural episode of “Highly Variable”, I talk about the purpose of the podcast, Independence Day in America, my research on black holes, Millennium Prize Problems in mathematics, and Navier-Stokes equations.

Jenn Stroud Rossmann is a fiction writer and an engineer. Her first novel, The Place You're Supposed to Laugh, is out now from 7.13 Books. She writes the essay series An Engineer Reads a Novel at Public Books. We talk about the ways in which society perceives us, the challenge of pursuing two passions at the same time—and the Navier Stokes equations, of course! As always, if you're a reader, writer, creative type, someone with something to say, you can always get in touch with me using losingtheplotpodcast [at] gmail [dot] com. I look forward to hearing from you! Marshall, who provided Losing the Plot’s intro music, has a new EP out now! Check out “Emerald Shitty” here! https://emeraldshitty714.bandcamp.com/releases

In this episode Gudrun talks with her new colleague Xian Liao. In November 2018 Xian has been appointed as Junior Professor (with tenure track) at the KIT-Faculty of Mathematics. She belongs to the Institute of Analysis and works in the group Nonlinear Partial Differential Equations. She is very much interested in Dispersive Partial Differential Equations. These equations model, e.g., the behaviour of waves. For that it is a topic very much in the center of the CRC 1173 - Wave phenomena at our faculty. Her mathematical interest was always to better understand the solutions of partial differential equations. But she arrived at dispersive equations through several steps in her carreer. Originally she studied inhomogeneous incompressible fluids. This can for example mean that the fluid is a mixture of materials with different viscosities. If we have a look at the Navier-Stokes equations for materials like water or oil, one main assumption therein is, that the viscosity is a material constant. Nevertheless, the equations modelling their flows are already nonlinear and there are a few serious open questions. Studying flows of inhomogneous materials brings in further difficulties since there occur more and more complex nonlinearities in the equations. It is necessary to develop a frame in which one can characterise the central properties of the solutions and the flow. It turned out that for example finding and working with quantities which remain conserved in the dynamics of the process is a good guiding line - even if the physical meaning of the conserved quantitiy is not always clear. Coming from classical theory we know that it makes a lot of sense to have a look at the conservation of mass, energy and momentum, which translate to conserved quantities as combinations of velocity, its derivatives, pressure and density. Pressure and density are not independent in these simplified models but are independent in the models Xiao studies. In the complex world of inhomogeneous equations we lose the direct concept to translate between physics and mathematics but carry over the knowledge that scale invarance and conservation are central properties of the model. It is interesting to characterize how the complex system develops with a change of properties. To have a simple idea - if it is more developing in the direction of fast flowing air or slow flowing almost solid material. One number which helps to see what types of waves one has to expect is the Mach number. It helps to seperate sound waves from fluid waves. A mathematical/physical question then is to understand the process of letting the Mach number go to zero in the model. It is not that complicated to make this work in the formulae. But the hard work is done in proving that the solutions to the family of systems of PDEs with lower and lower Mach number really tend to the solutions of the derived limit system. For example in order to measure if solutions are similar to each other (i.e. they get nearer and nearer to each other) one needs to find the norms which measure the right properties. Xian was Undergraduate & Master student at the Nanjing University in China from 2004 to 2009, where she was working with Prof. Huicheng Yin on Partial Differential Equations. She succeeded in getting the scholarship from China Scholarship Council and did her PhD within the laboratory LAMA (with Prof. Raphaël Danchin on zero-Mach number system). She was member of the University Paris-Est but followed many master courses in the programs of other Parisian universities as well. In 2013 she spent 8 months at the Charles University in Prague as Postdoc within the research project MORE. There she collaborated with Prof. Eduard Feireisl and Prof. Josef Málek on understanding non-Newtonian fluids better. After that period she returned to China and worked two years at the Academy of Mathematics & Systems Science as Postdoc within the research center NCMIS. With Prof. Ping Zhang she was working on density patch problems. Before her appointment here in Karlsruhe she already returned to Europe. 2016-2018 she was Postdoc at the University Bonn within the CRC 1060. She was mainly working with Prof. Herbert Koch on Gross-Pitaevskii equations - a special topic within dispersive equations. References Short Interview with the CRC 1173 Wave phenomena X. Liao, R. Danchin: On the wellposedness of the full low-Mach number limit system in general Besov spaces. Commun. Contemp. Math.: 14(3), 1250022, 2012. X. Liao: A global existence result for a zero Mach number system. J. Math. Fluid Mech.: 16(1), 77-103, 2014. X. Liao, E. Feireisl and J. Málek: Global weak solutions to a class of non-Newtonian compressible fluids. Math. Methods Appl. Sci.: 38(16), 3482-3494, 2015. X. Liao: On the strong solutions of the nonhomogeneous incompressible Navier-Stokes equations in a thin domain. Differential Integral Equations: 29, 167-182, 2016. X. Liao, P. Zhang: Global regularities of 2-D density patches for viscous inhomogeneous incompressible flow with general density: high regularity case, 2016.

Turbulence, the splintering of smooth streams of fluid into chaotic vortices, doesn't just make for bumpy plane rides. It also throws a wrench into the very mathematics used to describe atmospheres, oceans and plumbing. Turbulence is the reason why the Navier-Stokes equations—the laws that govern fluid flow—are so famously hard that whoever proves whether or not they always work will win a million dollars from the Clay Mathematics Institute.

Two mathematicians prove that under certain extreme conditions, the Navier-Stokes equations output nonsense.

Martina Hofmanová has been working as a professor at the University of Bielefeld since October 2017. Previously, she was a Junior Professor at TU Berlin from February 2016 onwards, and before that an Assistant Lecturer, there. She studied at the Charles University in Prague, and got her PhD in 2013 at the École normale supérieure de Cachan in Rennes. Her time in Germany started in 2013 when she moved to the Max Planck Institute for Mathematics in the Sciences in Leipzig as a postdoc.Gudrun and Martina talk about randomness in the modeling of fluid motion. This topic is connected to the study of turbulent flow. Of course, we observe turbulence all around us, i.e. chaotic behaviour of the pressure and the velocity field in fluid flow. One example is the smoke pattern of a freshly extinguished candle. Its first part is laminar, then we observe transitional turbulent flow and fully turbulent one the further away the smoke travels. A second example is Rayleigh Bénard convection. Under the influence of a temperature gradient and gravity, one observes convection rolls when the temperature difference between bottom and top becomes large enough. If we look more closely, one can prescribe the motion as a mean flow plus random fluctuations. These fluctuations are difficult to measure but their statistical properties are reproduced more easily. A general procedure in physics and science is to replace expensive time averages by ensemble averages, which can be calculated together on a parallel computer. The concept why this often works is the so-called ergodic hypothesis. To justify this from the mathematical side, the main problem is to find the right measure in the ensemble average. In the model problem one can see that the solution is continuously dependent on the initial condition and the solution operator has a semigroup property. For random initial conditions, one can construct the solution operator correspondingly. Already with this toy problem one sees that the justification of using ensemble averages is connected to the well-posedness of the problem. In general, this is not apriori known. The focus of Martina's work is to find the existence of steady solutions for the compressible flow system, including stochastic forces with periodic boundary conditions (i.e. on the torus). At the moment, we know that there are global weak solutions but only local (in time) strong solutions. It turned out that the right setting to study the problem are so-called dissipative martingale solutions: Unfortunately, in this setting, the velocity is not smooth enough to be a stochastic process. But the energy inequality can be proved. The proof rests on introducing artificial dissipation in the mass conservation, and a small term with higher order regularity for the density. Then, the velocity is approximated through a Faedo-Galerkin approximation and a lot of independent limiting processes can be carried out successfully. The project is a collaboration with Dominic Breit and Eduard Feireisl. References M. Hofmanová: Stochastic partial differential equations, Lecture notes, Technical University of Berlin, 2016. D. Breit, E. Feireisl, M. Hofmanová, B. Maslowski: Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces, preprint, 2016. D. Breit, E. Feireisl, M. Hofmanová: Local strong solutions to the stochastic compressible Navier-Stokes system, preprint, 2016. D. Breit, E. Feireisl, M. Hofmanová: Compressible fluids driven by stochastic forcing: The relative energy inequality and applications, Comm. Math. Physics 350, 443-473, 2017. D. Breit, M. Hofmanová: Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ. Math. J. 65 (4), 1183-1250, 2016. Podcasts N. Vercauteren: Lokale Turbulenzen, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 144, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2017. http://modellansatz.de/lokale-turbulenzen B.Valsler, D. Ansell: The Science of Turbulence, The Naked Scientists Podcast, 2010.

Martina Hofmanová has been working as a professor at the University of Bielefeld since October 2017. Previously, she was a Junior Professor at TU Berlin from February 2016 onwards, and before that an Assistant Lecturer, there. She studied at the Charles University in Prague, and got her PhD in 2013 at the École normale supérieure de Cachan in Rennes. Her time in Germany started in 2013 when she moved to the Max Planck Institute for Mathematics in the Sciences in Leipzig as a postdoc.Gudrun and Martina talk about randomness in the modeling of fluid motion. This topic is connected to the study of turbulent flow. Of course, we observe turbulence all around us, i.e. chaotic behaviour of the pressure and the velocity field in fluid flow. One example is the smoke pattern of a freshly extinguished candle. Its first part is laminar, then we observe transitional turbulent flow and fully turbulent one the further away the smoke travels. A second example is Rayleigh Bénard convection. Under the influence of a temperature gradient and gravity, one observes convection rolls when the temperature difference between bottom and top becomes large enough. If we look more closely, one can prescribe the motion as a mean flow plus random fluctuations. These fluctuations are difficult to measure but their statistical properties are reproduced more easily. A general procedure in physics and science is to replace expensive time averages by ensemble averages, which can be calculated together on a parallel computer. The concept why this often works is the so-called ergodic hypothesis. To justify this from the mathematical side, the main problem is to find the right measure in the ensemble average. In the model problem one can see that the solution is continuously dependent on the initial condition and the solution operator has a semigroup property. For random initial conditions, one can construct the solution operator correspondingly. Already with this toy problem one sees that the justification of using ensemble averages is connected to the well-posedness of the problem. In general, this is not apriori known. The focus of Martina's work is to find the existence of steady solutions for the compressible flow system, including stochastic forces with periodic boundary conditions (i.e. on the torus). At the moment, we know that there are global weak solutions but only local (in time) strong solutions. It turned out that the right setting to study the problem are so-called dissipative martingale solutions: Unfortunately, in this setting, the velocity is not smooth enough to be a stochastic process. But the energy inequality can be proved. The proof rests on introducing artificial dissipation in the mass conservation, and a small term with higher order regularity for the density. Then, the velocity is approximated through a Faedo-Galerkin approximation and a lot of independent limiting processes can be carried out successfully. The project is a collaboration with Dominic Breit and Eduard Feireisl. References M. Hofmanová: Stochastic partial differential equations, Lecture notes, Technical University of Berlin, 2016. D. Breit, E. Feireisl, M. Hofmanová, B. Maslowski: Stationary solutions to the compressible Navier-Stokes system driven by stochastic forces, preprint, 2016. D. Breit, E. Feireisl, M. Hofmanová: Local strong solutions to the stochastic compressible Navier-Stokes system, preprint, 2016. D. Breit, E. Feireisl, M. Hofmanová: Compressible fluids driven by stochastic forcing: The relative energy inequality and applications, Comm. Math. Physics 350, 443-473, 2017. D. Breit, M. Hofmanová: Stochastic Navier-Stokes equations for compressible fluids, Indiana Univ. Math. J. 65 (4), 1183-1250, 2016. Podcasts N. Vercauteren: Lokale Turbulenzen, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 144, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2017. http://modellansatz.de/lokale-turbulenzen B.Valsler, D. Ansell: The Science of Turbulence, The Naked Scientists Podcast, 2010.

From our first breath of the day to brushing our teeth to washing our faces to our first sip of coffee, and even in the waters of the rivers we have built cities upon since antiquity, we find ourselves surrounded by fluids. Fluids, in this context, mean anything that can take the shape of its container. Physically, that means anything that has molecules that can move past one another, but mathematics has, as always, a slightly different view. This view is seen by some as more nuanced, others as more statistical, but by all as a challenge. This definition cannot fit into an introduction, and I’ll be picking away at it for the remainder of this episode. So what is a fluid? What can we learn from it? And how could learning from it be worth a million dollars? --- This episode is sponsored by · Anchor: The easiest way to make a podcast. https://anchor.fm/app Support this podcast: https://anchor.fm/breakingmathpodcast/support

Peer Kunstmann hat in Kiel Mathematik studiert und 1995 promoviert. In seiner Zeit an der Fakultät für Mathematik in Karlsruhe hat er sich 2002 habilitiert. Er arbeitet als Akademischer Oberrat dauerhaft in der Arbeitsgruppe Angewandte Analysis an unserer Fakultät. Gudrun hat das Gespräch über ein für beide interessantes Thema - das Stokesproblem - gesucht, weil beide schon über längere Zeit mit unterschiedlichen Methoden zu dieser Gleichung forschen. Das Stokesproblem ist der lineare Anteil der Navier-Stokes Gleichungen, dem klassischen Modell für Strömungen. Sie haben eine gewisse Faszination, da sie einfach genug erscheinen, um sie in ihrer Struktur sehr eingehend verstehen zu können, zeigen aber auch immer wieder, dass man sie nicht unterschätzen darf in ihrer Komplexität. Peers Interesse wurde zu Beginn seiner Zeit in Karlsruhe durch Matthias Hieber geweckt, der inzwischen an der TU Darmstadt tätig ist. Es zeigte sich seit damals als sehr aktives Forschungsgebiet, weshalb er auch immer wieder neu zu diesen Fragestellungen zurückgekehrt ist. Mit den klassischen Randbedingungen (konkret, wenn auf dem Rand vorgeschrieben wird, dass die Lösung dort verschwindet = homogene Dirichletbedingung) ist das Stokesproblem auffassbar als Laplaceoperator, der auf Räumen mit divergenzfreien Vektorfeldern agiert. Der Laplaceoperator ist sehr gut verstanden und die Einschränkung auf den abgeschlossenen Unterraum der Vektorfelder mit der Eigenschaft, dass ihre Divergenz den Wert 0 ergibt, lässt sich mit einer Orthogonalprojektion - der Helmholtzprojektion - beschreiben. Im Hilbertraumfall, d.h. wenn die Räume auf einer L^2-Struktur basieren und der Raum deshalb auch ein Skalarprodukt hat, weiß man, dass diese Projektion immer existiert und gute Eigenschaften hat. Für andere Räume ohne diese Struktur (z.B. -basiert für q nicht 2) hängt die Antwort auf die Frage, für welche q die Projektion existiert, von der Geometrie des Gebietes ab. Für beschränkte Gebiete geht vor allem die Glattheit des Randes ein. Das spiegelt sich auch auf der Seite des Laplaceproblems, wo die Regularität im Innern des Gebietes relativ elementar gezeigt werden kann, aber in der Nähe des Randes und auf dem Rand gehen in die Argumente direkt die Regularität des Randes ein. Mathematisch wird das Gebiet dabei mit Kreisen überdeckt und mit Hilfe einer sogenannten Zerlegung der Eins anschließend die Lösung für das ganze Gebiet zusammengesetzt. Für die Kreise, die ganz im Innern des Gebietes liegen, wird die Lösung auf den ganzen Raum mit dem Wert 0 fortgesetzt, weil die Behandlung des ganzen Raumes sehr einfach ist. Für Kreise am Rand, wird der Rand lokal glatt gebogen zu einer geraden Linie und (ebenfalls nach Fortsetzung mit 0) ein Halbraum-Problem gelöst. Natürlich liegt es in der Glattheit des Randes, ob das "gerade biegen" nur kleine Fehlerterme erzeugt, die sich "verstecken" lassen oder ob das nicht funktioniert. Für einen Rand, der lokal durch zweimal differenzierbare Funktion dargestellt werden kann, funktioniert diese Technik gut. Für Gebiete, die einen Rand haben, der lokal durch Lipschitzstetige Funktionen darstellbar ist, werden z.B. Randintegraldarstellungen direkt untersucht. Dort existiert die Helmholtzzerlegung für alle q im Intervall (wobei vom Gebiet abhängt). Durch die kleinen Fehlerterme, die in der Technik entstehen, wird es auch nötig, die Gleichung für die Divergenz zu untersuchen, wo keine 0 sondern eine beliebige Funktion (natürlich mit den entsprechenden Regularitätseigenschaften) als rechte Seite erlaubt ist. Ein Begriff, der eine wichtige Eigenschaft von partiellen Differentialgleichungen beschreibt, ist der der maximalen Regularität. Einfach ausgedrückt heißt das, wenn ich die rechte Seite in einem Raum vorgebe, ist die Lösung genau so viel regulärer, dass nach Anwendung des Differentialoperators das Ergebnis die Regularität der rechten Seite hat. Für das Laplaceproblem wäre also die Lösung v für jedes vorgegebene f so, dass und f im gleichen Raum sind. Diese Eigenschaft ist z.B. wichtig, wenn man bei nichtlinearen Problemen mit Hilfe von Fixpunktsätzen arbeitet, also z.B. den Operators iterativ anwenden muss. Dann sichert die maximale Regularität, dass man immer im richtigen Raum landet, um den Operator erneut anwenden zu können. Im Zusammenhang mit der Frage der maximalen Regularität hat sich der -Kalkül als sehr nützlich erwiesen. Ein anderer Zugang wählt statt der Operatorformulierung die schwache Formulierung und arbeitet mit Bilinearformen und Ergebnissen der Funktionalanalysis. Hier kann man vergleichsweise wenig abstrakt und in diesem Sinn elementar auch viel für das Stokes- und das Navier-Stokes Problem zeigen. Es gibt ein vorbildliches Buch von Hermann Sohr dazu. Literatur und weiterführende Informationen M. Geißert, P.C. Kunstmann: Weak Neumann implies H^infty for Stokes, Journal Math. Soc. Japan 67 (no. 1), 183-193, 2015. P.C. Kunstmann: Navier-Stokes equations on unbounded domains with rough initial data, Czechoslovak Math. J. 60(135) no. 2, 297–313, 2010. H. Sohr: The Navier-Stokes Equations. An Elementary Functional Analytic Approach Birkhäuser, 2001. M. Cannone: Ondelettes, Paraproduits et Navier-stokes, Diderot Editeur, 1995. G. Thäter, H. Sohr: Imaginary powers of second order differential operators and $L^q$ -Helmholtz decomposition in the infinite cylinder, Mathematische Annalen 311(3):577-602, 1998. P.C. Kunstmann, L. Weis: Maximal L_p-regularity for parabolic equations, Fourier multiplier theorems and H^infty-calculus, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer Lecture Notes 1855, 65-311, 2004. P.C. Kunstmann, L. Weis: New criteria for the H^infty-calculus and the Stokes operator on bounded Lipschitz domains, Journal of Evolution Equations, March 2017, Volume 17, Issue 1, pp 387-409, 2017. G.P. Galdi: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems. Springer Tracts in Natural Philosophy, 38. Springer-Verlag, New York, 1994. Podcasts J. Babutzka: Helmholtzzerlegung, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 85, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. M. Steinhauer: Reguläre Strömungen, Gespräch mit G. Thäter im Modellansatz Podcast, Folge 113, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016.

Das Universum ist voll mit Sternen, Galaxien, Planeten und jeder Menge anderer cooler Dinge. Jedes davon hat seine Geschichten und die Sternengeschichten erzählen sie. Der Podcast zum Blog "Astrodicticum Simplex"

Matematikforskare Denis Gaidashev försöker knäcka ett av matematikens viktigaste olösta problem. En grupp ekvationer för mer exakta modeller av till exempel turbulens, tornados och havsströmmar. Amerikanska Clay Mathematics institute har lovat en miljon dollar till de matematiker som lyckas knäcka Navier-Stokes ekvationerna. En lösning på ekvationerna kommer vara användbart vid till exempel tillverkning av nya fartyg och flygplan. Att arbeta med ekvationer är en kamp och kampen kan pågå under flera månader, men slutligen känner man ett stort behag: problemet är löst.Niklas Zachrisson niklas.zachrisson@sverigesradio.se

Turbulence is one of the most widespread phenomena in nature occurring in fluids and plasmas at all scales - from the blood flow in the human body, via the Earth's atmosphere to the remnants of supernovas at astrophysical scales. Despite its frequent occurrence, constructing a theory of turbulent motion, which provides reliable quantitative predictions, represents one of the unsolved problems of classical physics. Most of the research efforts in the past have been focused on studying the Navier-Stokes model of simple fluids and trying to understand the Fourier spectrum of velocity fluctuations. Due to this common restriction to the Navier-Stokes equations, turbulence is usually associated with power-law spectra of universal form which arise only at scales where both driving and dissipation mechanisms are inactive. However, recent studies reveal that many active systems which do not possess a true inertial range can, nevertheless, exhibit power-law spectra. Furthermore, those spectra are not of universal form which contradicts the classical theory of turbulence. One of the turbulent models we shall consider in this work derives from the Kuramoto-Sivashinsky equation. It describes a simple one-dimensional active system where energy is injected at large scales and dissipated at small scales. Based on observations from plasma physics we modify the linear part of the equation such that the large scales remain practically intact but the damping rate at high wave numbers approaches a constant. We construct a semi-analytical approximation for the modified equation which predicts a power-law form for the energy spectrum in the range where the ratio between the characteristic linear and nonlinear frequencies is scale-independent. Furthermore, we conclude that the steepness of this power law is not universal but depends on the frequency ratio. These results are confirmed by numerical simulations. Our analysis could also be relevant for kinetic Alfven-wave turbulence in the solar wind where similar conditions might occur. Further in this work we present the first systematic study of another active system which provides a continuum model aimed at the coarse-grained description of the dynamics observed in dense bacterial suspensions. The model extends the framework of the familiar Navier-Stokes equations by including additional linear and nonlinear terms in order to emulate energy injection and dissipation as well as the flocking tendency of bacteria. The resulting dynamics has been described as 'low-Reynolds-number turbulence' and the corresponding energy spectrum exhibits nonuniversal power laws at large scales. With the aid of extensive numerical simulations we study the scale-to-scale energy flow in spectral space. The physical insight gained this way helped us to develop an approximation for the spectral energy balance equation. Its solution provides an energy spectrum of a power-law form at small wave numbers. Furthermore, we derive a functional dependence of the steepness of this power law on the system parameters. A comparison with data from numerical simulations verifies our results.

Wirosoetisno, D (Durham University) Wednesday 04 December 2013, 14:00-14:45

Gibbon, J (Imperial College London) Monday 02 December 2013, 10:15-11:00

Equazioni di Navier-Stokes

equazioni di Navier-Stokes

Glatt-Holtz, N (Virginia Polytechnic Institute and State University) Thursday 31 October 2013, 10:10-10:45

Liu, Y (Universität Regensburg) Tuesday 09 April 2013, 15:30-16:00

Wilkinson, M (University of Oxford) Monday 08 April 2013, 15:30-16:00

We propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus natually imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabilic equations.

Stents are expandable tubes that are inserted into blocked or damaged blood vessels. They offer a practical way to treat coronary artery disease, repairing vessels and keeping them open so that blood can flow freely. When stents work, they are a great alternative to radical surgery, but they can deteriorate or become dislodged. Mathematical models of blood vessels and stents are helping to determine better shapes and materials for the tubes. These models are so accurate that the FDA is considering requiring mathematical modeling in the design of stents before any further testing is done, to reduce the need for expensive experimentation. Precise modeling of the entire human vascular system is far beyond the reach of current computational power, so researchers focus their detailed models on small subsections, which are coupled with simpler models of the rest of the system. The Navier-Stokes equations are used to represent the flow of blood and its interaction with vessel walls. A mathematical proof was the central part of recent research that led to the abandonment of one type of stent and the design of better ones. The goal now is to create better computational fluid-vessel models and stent models to improve the treatment and prediction of coronary artery disease the major cause of heart attacks. For More Information: Design of Optimal Endoprostheses Using Mathematical Modeling, Canic, Krajcer, and Lapin, Endovascular Today, May 2006.

Stents are expandable tubes that are inserted into blocked or damaged blood vessels. They offer a practical way to treat coronary artery disease, repairing vessels and keeping them open so that blood can flow freely. When stents work, they are a great alternative to radical surgery, but they can deteriorate or become dislodged. Mathematical models of blood vessels and stents are helping to determine better shapes and materials for the tubes. These models are so accurate that the FDA is considering requiring mathematical modeling in the design of stents before any further testing is done, to reduce the need for expensive experimentation. Precise modeling of the entire human vascular system is far beyond the reach of current computational power, so researchers focus their detailed models on small subsections, which are coupled with simpler models of the rest of the system. The Navier-Stokes equations are used to represent the flow of blood and its interaction with vessel walls. A mathematical proof was the central part of recent research that led to the abandonment of one type of stent and the design of better ones. The goal now is to create better computational fluid-vessel models and stent models to improve the treatment and prediction of coronary artery disease the major cause of heart attacks. For More Information: Design of Optimal Endoprostheses Using Mathematical Modeling, Canic, Krajcer, and Lapin, Endovascular Today, May 2006.

Mahalov, A (Arizona State) Tuesday 27 March 2007, 15:30-16:15