Social 📸 Photography

Khurram Wadee ✅A few days back, I posted some <a href="https://mastodon.org.uk/tags/AnimatedGifs" class="mention hashtag" rel="nofollow noopener" target="_blank">#AnimatedGifs</a> of the exact solution for a large-amplitude undamped, unforced <a href="https://mastodon.org.uk/tags/Pendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#Pendulum</a>. I then thought to complete the study to include the case when it has been fed enough <a href="https://mastodon.org.uk/tags/energy" class="mention hashtag" rel="nofollow noopener" target="_blank">#energy</a> to allow it just to undergo <a href="https://mastodon.org.uk/tags/FullRotations" class="mention hashtag" rel="nofollow noopener" target="_blank">#FullRotations</a>, rather than just <a href="https://mastodon.org.uk/tags/oscillations" class="mention hashtag" rel="nofollow noopener" target="_blank">#oscillations</a>. Well, it turns out that it is “a bit more complicated than I first expected” but I finally managed it.<a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/SpecialFunctions" class="mention hashtag" rel="nofollow noopener" target="_blank">#SpecialFunctions</a> <a href="https://mastodon.org.uk/tags/DynamicalSystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#DynamicalSystems</a> <a href="https://mastodon.org.uk/tags/NonlinearPhenomena" class="mention hashtag" rel="nofollow noopener" target="_blank">#NonlinearPhenomena</a>

Khurram Wadee ✅Here we see three identical pendulums, oscillating independently. The red and purple ones are vibrating with small amplitudes and so their periods are nearly the same. But the blue one is undergoing what would be considered to be very large amplitude oscillations and has a significantly longer period. In fact, as the amplitude approaches π radians, the period increases without bound and approaches infinity.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/ClassicalMechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#ClassicalMechanics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a>

Khurram Wadee ✅The above is only strictly true for “small” amplitudes. If the <a href="https://mastodon.org.uk/tags/pendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#pendulum</a> is subject to large amplitudes, it is no longer governed by the simple linear differential equations alluded to above. Given that the rod remains rigid and no energy is dissipated, the pendulum <a href="https://mastodon.org.uk/tags/equations" class="mention hashtag" rel="nofollow noopener" target="_blank">#equations</a> may still be solved exactly, though they are now <a href="https://mastodon.org.uk/tags/nonlinear" class="mention hashtag" rel="nofollow noopener" target="_blank">#nonlinear</a>.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/ClassicalMechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#ClassicalMechanics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a>

Khurram Wadee ✅In elementary <a href="https://mastodon.org.uk/tags/mechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#mechanics</a> we are taught about the <a href="https://mastodon.org.uk/tags/SimplePendulum" class="mention hashtag" rel="nofollow noopener" target="_blank">#SimplePendulum</a>, which is modelled as a point <a href="https://mastodon.org.uk/tags/mass" class="mention hashtag" rel="nofollow noopener" target="_blank">#mass</a> hanging via a rigid <a href="https://mastodon.org.uk/tags/rod" class="mention hashtag" rel="nofollow noopener" target="_blank">#rod</a> or <a href="https://mastodon.org.uk/tags/string" class="mention hashtag" rel="nofollow noopener" target="_blank">#string</a> of a given length, under uniform <a href="https://mastodon.org.uk/tags/gravity" class="mention hashtag" rel="nofollow noopener" target="_blank">#gravity</a>. This simple model is also useful in introducing <a href="https://mastodon.org.uk/tags/OrdinaryDifferentialEquations" class="mention hashtag" rel="nofollow noopener" target="_blank">#OrdinaryDifferentialEquations</a> and helps us to understand <a href="https://mastodon.org.uk/tags/SimpleHarmonicMotion" class="mention hashtag" rel="nofollow noopener" target="_blank">#SimpleHarmonicMotion</a>.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/ClassicalMechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#ClassicalMechanics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a>

david jon furbish"The "Correlation" Between Statistics and Eugenics" (2024) 2/2 <a href="https://lareviewofbooks.org/article/the-correlation-between-statistics-and-eugenics/" rel="nofollow noopener" translate="no" target="_blank">https://lareviewofbooks.org/article/the-correlation-between-statistics-and-eugenics/</a> <a href="https://mastodon.online/tags/statistics" class="mention hashtag" rel="nofollow noopener" target="_blank">#statistics</a> <a href="https://mastodon.online/tags/philosophy" class="mention hashtag" rel="nofollow noopener" target="_blank">#philosophy</a> <a href="https://mastodon.online/tags/philosophyofscience" class="mention hashtag" rel="nofollow noopener" target="_blank">#philosophyofscience</a> <a href="https://mastodon.online/tags/mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#mathematics</a> <a href="https://mastodon.online/tags/appliedmathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#appliedmathematics</a> <a href="https://mastodon.online/tags/eugenics" class="mention hashtag" rel="nofollow noopener" target="_blank">#eugenics</a> <a href="https://a.gup.pe/u/philosophy" class="u-url mention" rel="nofollow noopener" target="_blank">@philosophy@a.gup.pe</a> <a href="https://newsmast.community/@philosophy" class="u-url mention" rel="nofollow noopener" target="_blank">@philosophy@newsmast.community</a>

david jon furbishI cannot think of an applied mathematics that is more beautiful and far-reaching, or philosophically wilder, than probability. No, nonlinear dynamics and chaos people, it’s not even close 🤣<a href="https://mastodon.online/tags/probability" class="mention hashtag" rel="nofollow noopener" target="_blank">#probability</a> <a href="https://mastodon.online/tags/mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#mathematics</a> <a href="https://mastodon.online/tags/appliedmathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#appliedmathematics</a> <a href="https://mastodon.online/tags/philosophy" class="mention hashtag" rel="nofollow noopener" target="_blank">#philosophy</a> <a href="https://mastodon.online/tags/philosophyofscience" class="mention hashtag" rel="nofollow noopener" target="_blank">#philosophyofscience</a> <a href="https://newsmast.community/@philosophy" class="u-url mention" rel="nofollow noopener" target="_blank">@philosophy@newsmast.community</a> <a href="https://a.gup.pe/u/philosophy" class="u-url mention" rel="nofollow noopener" target="_blank">@philosophy@a.gup.pe</a>

Khurram Wadee ✅We’re having a <a href="https://mastodon.org.uk/tags/Soliton" class="mention hashtag" rel="nofollow noopener" target="_blank">#Soliton</a> sale over here! Get yours quick! Seriously, though, this is a quick <a href="https://mastodon.org.uk/tags/animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#animation</a> of the <a href="https://mastodon.org.uk/tags/SolitonInteraction" class="mention hashtag" rel="nofollow noopener" target="_blank">#SolitonInteraction</a> surface – the solution of the <a href="https://mastodon.org.uk/tags/Korteweg" class="mention hashtag" rel="nofollow noopener" target="_blank">#Korteweg</a>-deVries <a href="https://mastodon.org.uk/tags/Eauation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Eauation</a> showing (x,t,u(x,t)) giving you an all-round view.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Mathematic" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematic</a> <a href="https://mastodon.org.uk/tags/Phyiscs" class="mention hashtag" rel="nofollow noopener" target="_blank">#Phyiscs</a> <a href="https://mastodon.org.uk/tags/KdV" class="mention hashtag" rel="nofollow noopener" target="_blank">#KdV</a> <a href="https://mastodon.org.uk/tags/KdVEquation" class="mention hashtag" rel="nofollow noopener" target="_blank">#KdVEquation</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅I posted on <a href="https://mastodon.org.uk/tags/Soliton" class="mention hashtag" rel="nofollow noopener" target="_blank">#Soliton</a> interactions in the <a href="https://mastodon.org.uk/tags/Korteweg" class="mention hashtag" rel="nofollow noopener" target="_blank">#Korteweg</a>-deVries <a href="https://mastodon.org.uk/tags/Equation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Equation</a> yesterday with an <a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a> of how two <a href="https://mastodon.org.uk/tags/SolitaryWaves" class="mention hashtag" rel="nofollow noopener" target="_blank">#SolitaryWaves</a> interact. The <a href="https://mastodon.org.uk/tags/PhaseShift" class="mention hashtag" rel="nofollow noopener" target="_blank">#PhaseShift</a> experienced by both can also be seen in a static <a href="https://mastodon.org.uk/tags/3D" class="mention hashtag" rel="nofollow noopener" target="_blank">#3D</a> plot with time, t, plotted on one axis. You should be able to see a “dog-leg” in the trajectory of each wave after it interacts with the other.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅A subtlety probably difficult to spot in the animation is that the interaction of the two waves leaves them phase shifted, with the taller wave gaining position, while the shorter one loses it. This further animation shows the interactions again (purple) but I’ve also shown what would happen if each wave moved without interaction with the other (red and blue).<a href="https://mastodon.org.uk/tags/Animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#Animation</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a>

Khurram Wadee ✅Here is an example taken from Solitons: an introduction by Drazin & Johnson, where taller wave overtakes a shorter one moving in the same direction.4/4<a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a>

Khurram Wadee ✅When you start looking at <a href="https://mastodon.org.uk/tags/NonlinearWaves" class="mention hashtag" rel="nofollow noopener" target="_blank">#NonlinearWaves</a>, some of these principles no longer apply. For example, in the Korteweg-de Vries equation, which has <a href="https://mastodon.org.uk/tags/Soliton" class="mention hashtag" rel="nofollow noopener" target="_blank">#Soliton</a> or solutions, you can no longer simply add two solutions together as the resulting function would not be a solution of the governing equation. Waves of different heights travel at different velocities, with the taller waves moving faster than the shorter ones. Instead, they interact <a href="https://mastodon.org.uk/tags/nonlinearly" class="mention hashtag" rel="nofollow noopener" target="_blank">#nonlinearly</a>.3/4<a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a>

Khurram Wadee ✅Also, waves of different heights travel at different velocities, with the taller waves moving faster than the shorter ones.2/4<a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a>

Khurram Wadee ✅In <a href="https://mastodon.org.uk/tags/FluidMechanics" class="mention hashtag" rel="nofollow noopener" target="_blank">#FluidMechanics</a>, things like <a href="https://mastodon.org.uk/tags/SurfaceWaves" class="mention hashtag" rel="nofollow noopener" target="_blank">#SurfaceWaves</a> are often modelled using <a href="https://mastodon.org.uk/tags/LinearEquations" class="mention hashtag" rel="nofollow noopener" target="_blank">#LinearEquations</a>, which give rise to phenomena such as the principle of <a href="https://mastodon.org.uk/tags/LinearSuperposition" class="mention hashtag" rel="nofollow noopener" target="_blank">#LinearSuperposition</a>, where you can take two or more distinct solutions and simply add them together to get a new solution. Solutions are also periodic and can be decomposed into sinusoidal functions. 1/4<a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Physics</a>

Her Maths Story“My PhD was all about understanding what happens to blood flow in collapsible blood vessels like the giraffe jugular vein. In my postdoc I was investigating how to optimise ventilator settings for patients in ICU and then how to deliver inhaled therapies into the lungs. Since then, my focus has been in trying to understand how diseases like Asthma and other respiratory diseases originate and then progress. This involves incorporating biology and physics into mathematical and computational models, using approaches from different areas of applied maths. More recently I have started to look into the mechanisms that could lead to a rare lung disease called lymphangioleiomyomatosis (LAM) and Long Covid.” - Bindi Brook➡️ <a href="https://hermathsstory.eu/bindi-brook/" rel="nofollow noopener" translate="no" target="_blank">https://hermathsstory.eu/bindi-brook/</a><a href="https://mathstodon.xyz/tags/Academia" class="mention hashtag" rel="nofollow noopener" target="_blank">#Academia</a> <a href="https://mathstodon.xyz/tags/PhD" class="mention hashtag" rel="nofollow noopener" target="_blank">#PhD</a> <a href="https://mathstodon.xyz/tags/Professor" class="mention hashtag" rel="nofollow noopener" target="_blank">#Professor</a> <a href="https://mathstodon.xyz/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mathstodon.xyz/tags/MathematicalBiology" class="mention hashtag" rel="nofollow noopener" target="_blank">#MathematicalBiology</a> <a href="https://mathstodon.xyz/tags/MathematicalMedicine" class="mention hashtag" rel="nofollow noopener" target="_blank">#MathematicalMedicine</a> <a href="https://mathstodon.xyz/tags/UnconsciousBias" class="mention hashtag" rel="nofollow noopener" target="_blank">#UnconsciousBias</a> <a href="https://mathstodon.xyz/tags/WomenInMaths" class="mention hashtag" rel="nofollow noopener" target="_blank">#WomenInMaths</a> <a href="https://mathstodon.xyz/tags/WomenInSTEM" class="mention hashtag" rel="nofollow noopener" target="_blank">#WomenInSTEM</a> <a href="https://mathstodon.xyz/tags/HerMathsStory" class="mention hashtag" rel="nofollow noopener" target="_blank">#HerMathsStory</a>

Khurram Wadee ✅Yesterday, I posted an image of the <a href="https://mastodon.org.uk/tags/LorenzAttractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#LorenzAttractor</a> showing the evolution of three trajectories (shown in red, green and blue) starting close together. Here, I’ve made it into a little animation to show how the paths initially stay close to each other but after about a quarter of the duration plotted, they <a href="https://mastodon.org.uk/tags/diverge" class="mention hashtag" rel="nofollow noopener" target="_blank">#diverge</a> from each other irrevocably (i.e. become uncorrelated) but remain part of the <a href="https://mastodon.org.uk/tags/ChaoticAttractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#ChaoticAttractor</a>.<a href="https://mastodon.org.uk/tags/DynamicalSystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#DynamicalSystems</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>

Khurram Wadee ✅Just messing about a bit. Here is the famous <a href="https://mastodon.org.uk/tags/LorenzAttractor" class="mention hashtag" rel="nofollow noopener" target="_blank">#LorenzAttractor</a> plotted using <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a>. The three <a href="https://mastodon.org.uk/tags/trajectories" class="mention hashtag" rel="nofollow noopener" target="_blank">#trajectories</a>, shown in red, green and blue are for three fairly nearby <a href="https://mastodon.org.uk/tags/InitialConditions" class="mention hashtag" rel="nofollow noopener" target="_blank">#InitialConditions</a>.<a href="https://mastodon.org.uk/tags/DynamicalSystems" class="mention hashtag" rel="nofollow noopener" target="_blank">#DynamicalSystems</a> <a href="https://mastodon.org.uk/tags/ChaoticAttractors" class="mention hashtag" rel="nofollow noopener" target="_blank">#ChaoticAttractors</a> <a href="https://mastodon.org.uk/tags/StrangeAttractors" class="mention hashtag" rel="nofollow noopener" target="_blank">#StrangeAttractors</a> <a href="https://mastodon.org.uk/tags/NumericalSolutions" class="mention hashtag" rel="nofollow noopener" target="_blank">#NumericalSolutions</a> <a href="https://mastodon.org.uk/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Mathematics</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a>

Khurram Wadee ✅Here, each frame is a different <a href="https://mastodon.org.uk/tags/wing" class="mention hashtag" rel="nofollow noopener" target="_blank">#wing</a> profile, obtained this time by keeping the <a href="https://mastodon.org.uk/tags/imaginary" class="mention hashtag" rel="nofollow noopener" target="_blank">#imaginary</a> eccentricity of the generating <a href="https://mastodon.org.uk/tags/circle" class="mention hashtag" rel="nofollow noopener" target="_blank">#circle</a> constant and varying only the <a href="https://mastodon.org.uk/tags/real" class="mention hashtag" rel="nofollow noopener" target="_blank">#real</a> part.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Aeronautics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aeronautics</a> <a href="https://mastodon.org.uk/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a>

Khurram Wadee ✅Following on from yesterday’s posting of flow around a <a href="https://mastodon.org.uk/tags/ZhukovskyAerofoil" class="mention hashtag" rel="nofollow noopener" target="_blank">#ZhukovskyAerofoil</a>, I’ve added <a href="https://mastodon.org.uk/tags/equipotential" class="mention hashtag" rel="nofollow noopener" target="_blank">#equipotential</a> lines (blue) in addition to <a href="https://mastodon.org.uk/tags/streamlines" class="mention hashtag" rel="nofollow noopener" target="_blank">#streamlines</a> (black). The two families are <a href="https://mastodon.org.uk/tags/contours" class="mention hashtag" rel="nofollow noopener" target="_blank">#contours</a> of the <a href="https://mastodon.org.uk/tags/real" class="mention hashtag" rel="nofollow noopener" target="_blank">#real</a> and <a href="https://mastodon.org.uk/tags/imaginary" class="mention hashtag" rel="nofollow noopener" target="_blank">#imaginary</a> parts of an <a href="https://mastodon.org.uk/tags/AnalyticFunction" class="mention hashtag" rel="nofollow noopener" target="_blank">#AnalyticFunction</a> and therefore have the property of intersecting <a href="https://mastodon.org.uk/tags/orthogonally" class="mention hashtag" rel="nofollow noopener" target="_blank">#orthogonally</a>. The plots aren’t ideal as there are some anomalous features where the equipotential lines are very close to the <a href="https://mastodon.org.uk/tags/aerofoil" class="mention hashtag" rel="nofollow noopener" target="_blank">#aerofoil</a>.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/Aeronautics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aeronautics</a> <a href="https://mastodon.org.uk/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a>

Khurram Wadee ✅Using the mapping w → z+1/z, you get something that looks remarkably like an <a href="https://mastodon.org.uk/tags/aerofoil" class="mention hashtag" rel="nofollow noopener" target="_blank">#aerofoil</a>.It is a <a href="https://mastodon.org.uk/tags/ConformalMapping" class="mention hashtag" rel="nofollow noopener" target="_blank">#ConformalMapping</a> meaning that angles are preserved during the mapping. In this animation, I’ve varied the imaginary part of the the eccentricity, while keeping the real part the same. With a zero <a href="https://mastodon.org.uk/tags/AngleOfAttack" class="mention hashtag" rel="nofollow noopener" target="_blank">#AngleOfAttack</a>, you can see the change in the airflow around the <a href="https://mastodon.org.uk/tags/wing" class="mention hashtag" rel="nofollow noopener" target="_blank">#wing</a> as its shape changes.<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/Aeronautics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aeronautics</a> <a href="https://mastodon.org.uk/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a> <a href="https://mastodon.org.uk/tags/LaminarFlow" class="mention hashtag" rel="nofollow noopener" target="_blank">#LaminarFlow</a>

Khurram Wadee ✅The <a href="https://mastodon.org.uk/tags/Zhukovsky" class="mention hashtag" rel="nofollow noopener" target="_blank">#Zhukovsky</a> <a href="https://mastodon.org.uk/tags/Aerofoil" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerofoil</a> (sometimes transliterated as <a href="https://mastodon.org.uk/tags/Joukowsky" class="mention hashtag" rel="nofollow noopener" target="_blank">#Joukowsky</a> from <a href="https://mastodon.org.uk/tags/Russian" class="mention hashtag" rel="nofollow noopener" target="_blank">#Russian</a>), is a 2D model of <a href="https://mastodon.org.uk/tags/streamlined" class="mention hashtag" rel="nofollow noopener" target="_blank">#streamlined</a> <a href="https://mastodon.org.uk/tags/Airflow" class="mention hashtag" rel="nofollow noopener" target="_blank">#Airflow</a> past a <a href="https://mastodon.org.uk/tags/wing" class="mention hashtag" rel="nofollow noopener" target="_blank">#wing</a>. It uses <a href="https://mastodon.org.uk/tags/ComplexVariable" class="mention hashtag" rel="nofollow noopener" target="_blank">#ComplexVariable</a> and is an <a href="https://mastodon.org.uk/tags/AnalyticFunction" class="mention hashtag" rel="nofollow noopener" target="_blank">#AnalyticFunction</a> (i.e. <a href="https://mastodon.org.uk/tags/Differentiable" class="mention hashtag" rel="nofollow noopener" target="_blank">#Differentiable</a> everywhere, save at isolated <a href="https://mastodon.org.uk/tags/Singularities" class="mention hashtag" rel="nofollow noopener" target="_blank">#Singularities</a>). Take a circle in the <a href="https://mastodon.org.uk/tags/ComplexPlane" class="mention hashtag" rel="nofollow noopener" target="_blank">#ComplexPlane</a> which is not quite centred at the <a href="https://mastodon.org.uk/tags/origin" class="mention hashtag" rel="nofollow noopener" target="_blank">#origin</a> but passes through the <a href="https://mastodon.org.uk/tags/coordinate" class="mention hashtag" rel="nofollow noopener" target="_blank">#coordinate</a> (1,0) or (z=1+0i).<a href="https://mastodon.org.uk/tags/MyWork" class="mention hashtag" rel="nofollow noopener" target="_blank">#MyWork</a> <a href="https://mastodon.org.uk/tags/CCBYSA" class="mention hashtag" rel="nofollow noopener" target="_blank">#CCBYSA</a> <a href="https://mastodon.org.uk/tags/AppliedMathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#AppliedMathematics</a> <a href="https://mastodon.org.uk/tags/WxMaxima" class="mention hashtag" rel="nofollow noopener" target="_blank">#WxMaxima</a> <a href="https://mastodon.org.uk/tags/FreeSoftware" class="mention hashtag" rel="nofollow noopener" target="_blank">#FreeSoftware</a> <a href="https://mastodon.org.uk/tags/Aeronautics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aeronautics</a> <a href="https://mastodon.org.uk/tags/Aerodynamics" class="mention hashtag" rel="nofollow noopener" target="_blank">#Aerodynamics</a> <a href="https://mastodon.org.uk/tags/LaminarFlow" class="mention hashtag" rel="nofollow noopener" target="_blank">#LaminarFlow</a>

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