Arnaldo Rodriguez-Gonzalez
@aghostinthefigures.bsky.social
Dynamicist, writer, asker of questions. Assistant professor of teaching in the University at Buffalo. Ph.D. in theoretical & applied mechanics from Cornell, B.S. in mechanical engineering from UPR Mayagüez. 🇵🇷 He/him.
Another semester of teaching at UB concluded. Eternally grateful to have the ability to teach—it is not an easy time right now to be a professor, but it is a far harder time to be a student.
They’ve done amazing work in the face of great adversity and fear. What a privilege it is to help them.
They’ve done amazing work in the face of great adversity and fear. What a privilege it is to help them.
May 23, 2025 at 11:09 PM
Another semester of teaching at UB concluded. Eternally grateful to have the ability to teach—it is not an easy time right now to be a professor, but it is a far harder time to be a student.
They’ve done amazing work in the face of great adversity and fear. What a privilege it is to help them.
They’ve done amazing work in the face of great adversity and fear. What a privilege it is to help them.
More fun with computational fluid mechanics:
Here's a 2-D simulation of a layer of air (red) below a layer of water (blue) in a closed container. As air is lighter, buoyant forces cause the interface to be unstable, leading to the complex flow shown here as the air eventually rises to the top.
Here's a 2-D simulation of a layer of air (red) below a layer of water (blue) in a closed container. As air is lighter, buoyant forces cause the interface to be unstable, leading to the complex flow shown here as the air eventually rises to the top.
May 7, 2025 at 2:10 PM
More fun with computational fluid mechanics:
Here's a 2-D simulation of a layer of air (red) below a layer of water (blue) in a closed container. As air is lighter, buoyant forces cause the interface to be unstable, leading to the complex flow shown here as the air eventually rises to the top.
Here's a 2-D simulation of a layer of air (red) below a layer of water (blue) in a closed container. As air is lighter, buoyant forces cause the interface to be unstable, leading to the complex flow shown here as the air eventually rises to the top.
More fun with CFD for my course; here's a simulation of air bubbles being generated in a container of water with a rapidly rotating cylinder inside of it.
As time goes by, the cylinder rotation begins to affect the flow in the whole tank, and droplets start to drift to the right!
As time goes by, the cylinder rotation begins to affect the flow in the whole tank, and droplets start to drift to the right!
April 22, 2025 at 2:40 PM
More fun with CFD for my course; here's a simulation of air bubbles being generated in a container of water with a rapidly rotating cylinder inside of it.
As time goes by, the cylinder rotation begins to affect the flow in the whole tank, and droplets start to drift to the right!
As time goes by, the cylinder rotation begins to affect the flow in the whole tank, and droplets start to drift to the right!
If you look at a front view of the airfoil however, you see a key behavior that only shows up in proper 3-D treatments of flow around an airfoil—vortices!
As the fluid near the airfoil flows past it, it becomes deflected downwards by a pair of counter-rotating vortices off its back end.
As the fluid near the airfoil flows past it, it becomes deflected downwards by a pair of counter-rotating vortices off its back end.
April 17, 2025 at 9:07 PM
If you look at a front view of the airfoil however, you see a key behavior that only shows up in proper 3-D treatments of flow around an airfoil—vortices!
As the fluid near the airfoil flows past it, it becomes deflected downwards by a pair of counter-rotating vortices off its back end.
As the fluid near the airfoil flows past it, it becomes deflected downwards by a pair of counter-rotating vortices off its back end.
It is sometimes claimed that fluid takes the same amount of time traveling across the airfoil via either its top or bottom face; a side view of the flow (and particles traveling through it) confirms this is clearly not the case.
April 17, 2025 at 9:07 PM
It is sometimes claimed that fluid takes the same amount of time traveling across the airfoil via either its top or bottom face; a side view of the flow (and particles traveling through it) confirms this is clearly not the case.
The presence of the airfoil naturally also affects the flow itself; it lowers the flow speed near the surface of the airfoil to zero, and creates a long, angled wake behind it. It can also counter-intuitively /accelerate/ the flow in certain regions, as you can see from the dark red patch below!
April 17, 2025 at 9:07 PM
The presence of the airfoil naturally also affects the flow itself; it lowers the flow speed near the surface of the airfoil to zero, and creates a long, angled wake behind it. It can also counter-intuitively /accelerate/ the flow in certain regions, as you can see from the dark red patch below!
When airfoils are immersed in a background flow—or when an airfoil moves through fluid at constant speed—they're designed to cause a pressure change in the fluid such that pressure drops on its top surface and/or increases on the bottom. That pressure difference causes lift, pushing the airfoil up!
April 17, 2025 at 9:07 PM
When airfoils are immersed in a background flow—or when an airfoil moves through fluid at constant speed—they're designed to cause a pressure change in the fluid such that pressure drops on its top surface and/or increases on the bottom. That pressure difference causes lift, pushing the airfoil up!
More fun with computational fluid dynamics; what does flow past an airfoil look like? #FluidDynamics
A thread:
A thread:
April 17, 2025 at 9:07 PM
More fun with computational fluid dynamics; what does flow past an airfoil look like? #FluidDynamics
A thread:
A thread:
A perk of teaching students how to use finite element/volume methods; making cool animations!
This is "vortex shedding" in 2-D flow past a cylinder; a pair of symmetric counter-rotating vortices develop in the back, which then destabilize to form a long, oscillatory, transient wake. #fluidmechanics
This is "vortex shedding" in 2-D flow past a cylinder; a pair of symmetric counter-rotating vortices develop in the back, which then destabilize to form a long, oscillatory, transient wake. #fluidmechanics
April 12, 2025 at 5:46 PM
A perk of teaching students how to use finite element/volume methods; making cool animations!
This is "vortex shedding" in 2-D flow past a cylinder; a pair of symmetric counter-rotating vortices develop in the back, which then destabilize to form a long, oscillatory, transient wake. #fluidmechanics
This is "vortex shedding" in 2-D flow past a cylinder; a pair of symmetric counter-rotating vortices develop in the back, which then destabilize to form a long, oscillatory, transient wake. #fluidmechanics
Getting back in the habit of making math/physics animations in my spare time. Here's an old favorite of mine; the formation of a homoclinic tangle!
I'm hoping to upload the code for a few of these on GitHub, once I feel it's polished enough for public viewing. Hopefully won't take too long!
I'm hoping to upload the code for a few of these on GitHub, once I feel it's polished enough for public viewing. Hopefully won't take too long!
February 23, 2025 at 4:34 PM
Getting back in the habit of making math/physics animations in my spare time. Here's an old favorite of mine; the formation of a homoclinic tangle!
I'm hoping to upload the code for a few of these on GitHub, once I feel it's polished enough for public viewing. Hopefully won't take too long!
I'm hoping to upload the code for a few of these on GitHub, once I feel it's polished enough for public viewing. Hopefully won't take too long!
My plan was to get students to each simulate an individual test case with random design parameters, compile all data points, and see how they compare to the empirical "surface". Here it is! (The surface is the empirical prediction, dots are data points.)
Matches quite well except when h/r >> 1!
Matches quite well except when h/r >> 1!
February 17, 2025 at 9:40 PM
My plan was to get students to each simulate an individual test case with random design parameters, compile all data points, and see how they compare to the empirical "surface". Here it is! (The surface is the empirical prediction, dots are data points.)
Matches quite well except when h/r >> 1!
Matches quite well except when h/r >> 1!
To estimate this before finite element analysis became commonplace, engineers used empirical relationships between design parameters to find these concentration factors. Here's a complex empirical equation for them in this bar, from "Roark's Formulas for Stress and Strain":
February 17, 2025 at 9:40 PM
To estimate this before finite element analysis became commonplace, engineers used empirical relationships between design parameters to find these concentration factors. Here's a complex empirical equation for them in this bar, from "Roark's Formulas for Stress and Strain":
Stress concentration factors simply indicate the ratio of the largest stress in a structure to the largest stress at the surfaces in which a load is applied. Here, the sharp change in the size of the bar results in a stress spike near/on the fillet larger than the applied stresses at the ends!
February 17, 2025 at 9:40 PM
Stress concentration factors simply indicate the ratio of the largest stress in a structure to the largest stress at the surfaces in which a load is applied. Here, the sharp change in the size of the bar results in a stress spike near/on the fillet larger than the applied stresses at the ends!
Tried a fun exercise with my finite elements students a few weeks ago:
Do the empirical equations we use for finding the stress concentration factor in the bar below match the results we'd get from doing finite element analysis?
Do the empirical equations we use for finding the stress concentration factor in the bar below match the results we'd get from doing finite element analysis?
February 17, 2025 at 9:40 PM
Tried a fun exercise with my finite elements students a few weeks ago:
Do the empirical equations we use for finding the stress concentration factor in the bar below match the results we'd get from doing finite element analysis?
Do the empirical equations we use for finding the stress concentration factor in the bar below match the results we'd get from doing finite element analysis?
Been writing a handbook on finite element methods for a bit now; unsure why. Hope to make it, and a few algorithms within it, free and publicly accessible once I'm done.
Tough to feel as if this sort of thing matters in the new Age. Logic fights volition and electrochemistry. Let's see what wins.
Tough to feel as if this sort of thing matters in the new Age. Logic fights volition and electrochemistry. Let's see what wins.
January 29, 2025 at 9:32 PM
Been writing a handbook on finite element methods for a bit now; unsure why. Hope to make it, and a few algorithms within it, free and publicly accessible once I'm done.
Tough to feel as if this sort of thing matters in the new Age. Logic fights volition and electrochemistry. Let's see what wins.
Tough to feel as if this sort of thing matters in the new Age. Logic fights volition and electrochemistry. Let's see what wins.
I obtained a copy of Physics of Continuous Matter in December—was incredibly impressed by what I consider to be a singular work in continuum physics—and thought I would send a “thank you” e-mail to the author, Benny Lautrup, after the semester had started. I just learned he passed away on the 3rd.
January 22, 2025 at 3:43 PM
I obtained a copy of Physics of Continuous Matter in December—was incredibly impressed by what I consider to be a singular work in continuum physics—and thought I would send a “thank you” e-mail to the author, Benny Lautrup, after the semester had started. I just learned he passed away on the 3rd.
Replayed Portal 2 for the holidays; pleasantly surprised to find some decorative fluid mechanics on the whiteboards!
(Not sure if that Reynolds transport theorem statement is correct, though.)
(Not sure if that Reynolds transport theorem statement is correct, though.)
December 26, 2024 at 5:16 PM
Replayed Portal 2 for the holidays; pleasantly surprised to find some decorative fluid mechanics on the whiteboards!
(Not sure if that Reynolds transport theorem statement is correct, though.)
(Not sure if that Reynolds transport theorem statement is correct, though.)
This flow, however, is irrotational—viscous friction is neglected. Due to this, it's impossible for the flow to satisfy the no-slip condition on the sphere's surface, and so the particles that get close to the sphere just slip right past it.
This is one of the key issues with irrotational flow!
This is one of the key issues with irrotational flow!
December 7, 2024 at 9:21 PM
This flow, however, is irrotational—viscous friction is neglected. Due to this, it's impossible for the flow to satisfy the no-slip condition on the sphere's surface, and so the particles that get close to the sphere just slip right past it.
This is one of the key issues with irrotational flow!
This is one of the key issues with irrotational flow!
What does flow past a sphere look like when viscous friction dominates versus when it's neglected? Take a look for yourself! #physics
The flow below is creeping flow, where frictional effects dominate. Particles that get close to the sphere slow down drastically due to the no-slip condition.
The flow below is creeping flow, where frictional effects dominate. Particles that get close to the sphere slow down drastically due to the no-slip condition.
December 7, 2024 at 9:21 PM
What does flow past a sphere look like when viscous friction dominates versus when it's neglected? Take a look for yourself! #physics
The flow below is creeping flow, where frictional effects dominate. Particles that get close to the sphere slow down drastically due to the no-slip condition.
The flow below is creeping flow, where frictional effects dominate. Particles that get close to the sphere slow down drastically due to the no-slip condition.
In irrotational flow—where friction is neglected—it is impossible for the fluid to satisfy the no-slip condition on the surface of the sphere, so the particles that get close to the sphere simply slide right past it.
This is one of the key problems of using irrotational flow in fluid mechanics.
This is one of the key problems of using irrotational flow in fluid mechanics.
December 7, 2024 at 9:05 PM
In irrotational flow—where friction is neglected—it is impossible for the fluid to satisfy the no-slip condition on the surface of the sphere, so the particles that get close to the sphere simply slide right past it.
This is one of the key problems of using irrotational flow in fluid mechanics.
This is one of the key problems of using irrotational flow in fluid mechanics.
Long nights and low lights at UB as the semester winds down and the first few snowfalls come in.
I’m so lucky I get to do what I do.
I’m so lucky I get to do what I do.
December 7, 2024 at 1:50 AM
Long nights and low lights at UB as the semester winds down and the first few snowfalls come in.
I’m so lucky I get to do what I do.
I’m so lucky I get to do what I do.
Now compare this to what happens if I replace the second boundary condition with x(3π/4) = 1. This does have an analytical solution!
As a result, the max. abs. value of the numerical approx. converges onto a single number, the max. abs. value of the analytical solution, as the grid becomes finer.
As a result, the max. abs. value of the numerical approx. converges onto a single number, the max. abs. value of the analytical solution, as the grid becomes finer.
November 27, 2024 at 9:05 PM
Now compare this to what happens if I replace the second boundary condition with x(3π/4) = 1. This does have an analytical solution!
As a result, the max. abs. value of the numerical approx. converges onto a single number, the max. abs. value of the analytical solution, as the grid becomes finer.
As a result, the max. abs. value of the numerical approx. converges onto a single number, the max. abs. value of the analytical solution, as the grid becomes finer.
That—the absence of convergence onto something—is the proverbial canary that indicates if a BVP solved numerically actually has no solution.
Consider this plot of the "solution's" max. abs. value vs. the no. of grid points for this problem. The value just keeps going up as the grid becomes finer!
Consider this plot of the "solution's" max. abs. value vs. the no. of grid points for this problem. The value just keeps going up as the grid becomes finer!
November 27, 2024 at 9:05 PM
That—the absence of convergence onto something—is the proverbial canary that indicates if a BVP solved numerically actually has no solution.
Consider this plot of the "solution's" max. abs. value vs. the no. of grid points for this problem. The value just keeps going up as the grid becomes finer!
Consider this plot of the "solution's" max. abs. value vs. the no. of grid points for this problem. The value just keeps going up as the grid becomes finer!
Increasing the no. of grid points generates something tantalizingly like a sine function, and it fits the required boundary conditions (although it's hard to see on the graph).
But this can't possibly be converging onto a solution as the resolution increases, because there is no solution!
But this can't possibly be converging onto a solution as the resolution increases, because there is no solution!
November 27, 2024 at 9:05 PM
Increasing the no. of grid points generates something tantalizingly like a sine function, and it fits the required boundary conditions (although it's hard to see on the graph).
But this can't possibly be converging onto a solution as the resolution increases, because there is no solution!
But this can't possibly be converging onto a solution as the resolution increases, because there is no solution!
One particularly dangerous aspect of this is that, even if we analytically catch that there's no solution to such a BVP, numerical methods based on finite differences can spit out a "solution" no problem!
Here's the numerical "solution" a computer determines for this BVP on a five-point grid:
Here's the numerical "solution" a computer determines for this BVP on a five-point grid:
November 27, 2024 at 9:05 PM
One particularly dangerous aspect of this is that, even if we analytically catch that there's no solution to such a BVP, numerical methods based on finite differences can spit out a "solution" no problem!
Here's the numerical "solution" a computer determines for this BVP on a five-point grid:
Here's the numerical "solution" a computer determines for this BVP on a five-point grid: