Pushing on fluids at points far from, and very near, walls

This is just a post on me teaching myself a bit about the flow patterns that occur when you apply a small force at a point in a fluid. Small here means that the flows are sufficiently small and slow that the flow is Stokes flow, i.e., the Reynolds number is close to 0.

I used LLMs a fait bit, mainly MS Copilot. The LLMs helped with elements of the problem I am interested in, but could not do the whole thing. The code to work through the flow patterns is on GitHub if you’re interested. In that sense LLMs are – very roughly speaking! – currently at the level of say a final year undergraduate lecture course on Stokes flow, but no higher. Perhaps because their training sets contains a few fluid mechanics textbooks but not enough examples beyond that. To be fair on Copilot etc clear descriptions of the problem are hard to find in the literature …

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A toy app to calculate Stokes-Einstein diffusion constants

The one formula that I use above all others in my research is that for the diffusion constant of a dilute species in a fluid, called the Stokes-Einstein (or sometimes Stokes-Einstein-Sutherland) equation:

DSE = kT/(6 π η RH)

The diffusion constant DSE of something with a size (hydrodynamic radius) RH is just the thermal energy kT divided by 6π times the product of the viscosity of the fluid, η, and the hydrodynamic radius, RH. All it tells you is that large species in viscous fluids diffuse more slowly than small species in less viscous fluids.

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Jumping on the LLM bandwagon with a RAGbot to answer questions on my lecture notes

Unless you have just emerged from a cave you will have heard of ChatGPT, Google Gemini, Claude, etc – collectively LLMs (Large Language Models, also called Generative AI models). Wikipedia calls Google Gemini a generative artificial intelligence chatbot. They are fancy and powerful, and touching many parts of the world including education. My employer (as I am sure many universities are) is trying to keep up.

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Flows near a wall

The schematics above shows the flows in a liquid (eg water) which is being pushed into motion at point (shown by red and green arrows*). The flows are shown by what are called streamlines (in blue) which show the paths water molecules in the water follow as they are moved by the flowing liquid. The two flows are what are called Stokes flows because they obey Stokes’ equation – which is just the low inertia limit of the Navier-Stokes equation all simple liquids like water follow. Inertia is irrelevant for small scale (say millimetres or smaller) flows. The difference between the two flows is that there is a wall (in black) on the left, while on the right the flowing liquid is in the middle of the liquid, far from any wall.

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Adding variability to the Wells-Riley model of airborne disease transmission

In the plot above, the black circles are data for the probability of infection with COVID, as a function of the time exposed to an infected person. The data are from the NHS app many of us in the UK used during part of the pandemic, and analysed by people at the UK Health Security Agency (HSA) and Oxford University. It was published by Ferretti et al.. The orange line is a power law fit to the data, with exponent (= slope on this log-log plot) of 0.47. It is a decent but not perfect fit to the data. The fit can be viewed as a purely empirical function that just describes the data. But it can also be – after the fact – justified by saying that if you combine the standard Wells-Riley model for disease transmission with a power law distribution of transmission rates, then you can recover this power law.

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How much second-hand air do you breathe indoors?

Above is plotted a slightly odd function called an exceedance, the y axis is the probability that the value on the x axis is exceeded. The data is from a large dataset of measurements of indoor air by a Chinese company called gams Environmental Monitoring*. The x axis is the estimated** fraction of air in the room that is second hand, i.e., has been breathed out by another person (or by you). For example, the probability is close to 0.1 = 10% for a value along the x axis of 0.03 = 3 %. So, for this dataset, about 10% of the time the room air is at least 3% second hand. Another way of approaching is to say that the median fraction of the air that is second hand is around half a %.

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How many need to become infected, to declare air filtering a success?

This post is, basically, a part II to the previous post on the study of Brock et al. on the effect of installing room air filters, on COVID transmission in hospitals. Brock et al. found that the air filtering “was associated with a non-significant trend of lower hazard for SARS-CoV-2 infection”. The “non-significant” here is a bit sad. The study was of a total of 229 hospital-acquired infections and the conclusion was that this number is too small to draw a definite conclusion. So, how big a study do you need to come to a definite conclusion?

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Minimising hospital-acquired COVID in Cambridge

Hospitals are full of both infected people, and very susceptible people – people whose immune systems are, due to age or illness, very weak. This is a terrible combination but unavoidable. And it is why it is important to try and minimise the spread of infectious diseases such as COVID, but also why it is hard to do this, especially for airborne diseases like COVID. With airborne diseases you can reduce transmission via (good, eg FFP2) mask wearing. But masks are not very comfortable, so a more palatable solution is to improve the quality of the air itself, either by improved ventilation (ie more fresh air), or by filtration. Brock et al. at Cambridge Hospitals recently (2024) published a study of the effect of air filtration on COVID transmission in hospital wards.

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