Tag: mathematics

Optimal Bubble Clusters
With a bubble wand, it’s quite easy to create clusters of two or more soap bubbles. These clusters seem to instantly find the lowest energy state, forming a shape that minimizes the cluster’s surface area (including interior walls) for the volume of air they enclose. But mathematicians have struggled for thousands of years to prove that this is actually the case.
In 1995, mathematician John Sullivan had a breakthrough conjecture, at least for some types of bubble clusters. A proof for double bubble clusters quickly followed. But then progress stalled out, with the triple bubble version seemingly out of reach. But now a duo of mathematicians have published proofs for Sullivan’s bubble clusters in triple and quadruple clusters. Learn their story over at Quanta. (Image credit: N. Franz; via Quanta Magazine)
March 3, 2023
Squeeze or Splatter?
Many a white shirt has met the disaster of a nearly-empty condiment bottle. One moment, you’re carefully squeezing out ketchup, and the next — sppplltlttt — you’re covered in red splatters. This messy phenomenon of gas displacing a liquid is widespread, showing up in condiments, some volcanic eruptions, and even the reinflation of a collapsed lung. Researchers have now constructed a mathematical model to fully capture and explain the process.
When you squeeze a container with both air and a liquid — like ketchup — in it, the air is easily compressed but the liquid is not. The extra pressure of the air creates a driving force that pushes the liquid out, despite its viscous resistance. Most of the time, these two forces are balanced, and the ketchup flows smoothly out of the container. But when the volume of ketchup is small compared to the air, squeezing can overpressurize the air, driving the ketchup out in an uncontrolled burst.
Luckily, the mathematics also suggest a solution to this problem: squeeze more slowly and double the size of the nozzle. You can also, they note, simply remove the top to avoid splatter. (Image credit: Rodnae Productions; research credit: C. Cuttle and C. MacMinn; via Ars Technica; submitted by Kam-Yung Soh)
December 27, 2022
Searching for Stability
At present, there is no theory of relativistic fluid dynamics, which is problematic for those studying black holes, neutron star mergers, and heavy-ion collisions, where fluids may wind up moving at near-light speeds. Many current models for these systems allow energy to dissipate using equations that permit faster-than-light speeds. A new study shows that these assumptions lead to problematic results.
The paper shows that, if the mathematical equations allow for faster-than-light speeds — thereby breaking causality — then the fluid system will behave stably to one observer and unstably to an observer in a different reference frame. In other words, there will always be a frame of reference where disturbances grow exponentially and destroy the system. That’s clearly not ideal.
Fortunately, the paper also offers an important solution: if causality holds, the stability (or instability) of a system is the same regardless of reference frame. That’s incredibly powerful for researchers because it means that they only have to show the stability of the system in one reference frame to know that the result applies to all reference frames, so long as they’re not breaking causality. (Image credit: A. Pal; research credit: L. Gavassino; via APS Physics; submitted by Kam-Yung Soh)
November 29, 2022
Blowing Up Euler
The mathematics of fluid dynamics still have many unknowns, which makes them an attractive playground for mathematicians of all stripes. One perennial area of interest is the Euler equations, which describe an ideal (i.e., zero viscosity), incompressible fluid. Mathematicians suspect that these equations may produce impossible answers — vortices with infinite velocities, for example — under just the right circumstances, but so far no one has been able to prove the existence of such singularities.
A recent Quanta article delves into this issue and the race between researchers using traditional methods and those using new deep learning techniques. Will the singularities be found and who will get there first? It’s well worth a read, whether theoretical mathematics is your thing or not. (Image credit: S. Wilkinson; see also Quanta; submitted by Jo V.)
May 23, 2022
Why Masks Cut COVID-19 Transmission So Well
Face masks are an important tool for curtailing disease transmission, and this video explains how even imperfect masks do a much better job of protecting people than you may think. Strictly speaking, this video is not fluid dynamical — fluid dynamics plays more of a role in the details of what makes a mask effective — but the video is so good and so timely that I just have to share it. Given it a watch and then go explore the interactive essay to get an even better handle on mask mathematics. (Image and video credit: Minute Physics; see also The Multiplicative Power of Masks)
December 3, 2020
Lava Barriers
Inspired by protecting people and property from lava flows, researchers investigated how viscous fluids flow downhill past large obstacles. As seen above, when the obstacle is tall enough that the flow does not overtop it, there’s substantial deflection of the fluid both up- and downstream. Upstream of the barrier, the flow gets deeper, and downstream there’s a dry region left behind.
The researchers modeled these flows numerically, leading to equations designers can use to predict the necessary height, strength, and shape of barrier necessary to protect areas from encroaching lava. (Image and research credit: E. Hinton et al.)
October 12, 2020
Breaking the Euler Equations
Mathematicians like to break things. Or, more exactly, they like to know when the equations we use to describe physics break down. One popular target in fluid mechanics are the Euler equations, which describe the motion of frictionless, incompressible flows. Mathematicians have been on the hunt for centuries for situations where these equations predict singularities, points where the velocity or vorticity of a fluid change infinitely quickly. Since that can’t happen in reality (at least as far as we understand it), these singularities indicate weaknesses in our mathematical description and may help uncover fundamental flaws in our understanding.
Despite centuries of effort, the Euler equations withstood mathematical assault… until recently. Since 2013, a series of mathematicians have been successfully chipping away at the Euler equations’ seeming perfection with a series of scenarios that seem to lead to singularities. One is similar to stirring a cup of tea, except that you stir the upper part of the cup in one direction and the bottom half in the opposite. As the flow develops, a singularity occurs where the secondary flows of these two stirring motions collide. For more, check out these two articles over at Quanta. (Image credit: L. Fotios; see also Quanta Magazine 1, 2)
January 27, 2020
An Introduction to the Reynolds Number
For those who’d like an overview of the mathematics involved in fluid dynamics, Numberphile has a lovely introduction, given by our friend Tom Crawford. The governing equations in fluid dynamics, the Navier-Stokes equations, are quite complicated, but that’s just been inspiration for scientists and mathematicians to come up with clever ways to simplify them. And, ultimately, that’s what the Reynolds number is — a way to help us judge which forces, and therefore which mathematical terms, are the most important in a given problem. (Video credit: Numberphile; submitted by COMPLETE)
December 11, 2019
Finding New Shapes in Foam
In the summer of 2018, a group of researchers announced they’d discovered a new geometrical shape, the scutoid. They found the scutoid, a sort of twisted prism, in the shape of epithelial cells packed between curved surfaces. Having heard of this new geometry, a different group of physicists wondered if they could find scutoids elsewhere, specifically, in the cells of a foam. As shown in the picture above, they did.
To visualize a scutoid, first image a prism. Take two polygons with an equal number of sides and connect them. But if you imagine packing such prisms between two curved surfaces, you’ll quickly see that it won’t work. They just don’t fit together. Instead, one face may adopt, say, six sides, while the other takes on five. To join those two end faces, one of the sides will have to have a Y-shaped junction and a triangular face. This is a scutoid.
You can see two such shapes in the image above. In the left bubble, the far side forms a pentagon, while the near face is a hexagon. On the right, the bubble has six faces in the background and eight in the foreground. And between them, you can just see the triangular face that connects the two scutoids.
It’s not only exciting to find scutoids in a new, non-biological medium; it suggests a physical mechanism behind their formation. Foams are a well-known example of energy minimization. The fact that scutoids are found in a curved foam suggests that the shape itself is connected to energy minimization, something that could help us understand how biological scutoids grow and form. (Image and research credit: A. Mughal et al.; via Physics World; submitted by Kam-Yung Soh)
January 7, 2019
An Introduction to Turbulence
With some help from Physics Girl and her friends, Grant Sanderson at 3Blue1Brown has a nice video introduction to turbulence, complete with neat homemade laser-sheet illuminations of turbulent flows. Grant explains some of the basics of what turbulence is (and isn’t) and gives viewers a look at the equations that govern flow – as befits a mathematics channel!
There’s also an introduction to Kolmogorov’s theorem, which, to date, has been one of the most successful theoretical approaches to understanding turbulence. It describes how energy is passed from large eddies in the flow to smaller ones, and it’s been tested extensively in the nearly 80 years since its first appearance. Just how well the theory holds, and what situations it breaks down in, are still topics of active research and debate. (Video and image credit: G. Sanderson/3Blue1Brown; submitted by Maria-Isabel C.)
November 12, 2018