Author Archives: Richard Sear

Computational physicist at the University of Surrey. My research interests are in COVID-19 transmission, especially masks, soft matter & biological physics

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.

This is a simple formula but whenever I want to calculate a diffusion constant I usually have to get Python running and use it like a calculator. This takes less than a minute, but even so this is irritating.

So (together with LLMs Google Gemini and MS Copilot) I have done a simple Streamlit app that calculates the diffusion constant, given the hydrodynamic radius as input – for water at room temperature. It runs on Streamlit Community Cloud which is a doodle to do as it just pulls the Python/Streamlit code from a GitHub repository.

It is very basic and only does this one thing: compute a diffusion constant. But because of this it is very easy to use. If I find it useful may be I will make it fancier but we will see. If you find it useful or feel that a fancier version (eg where you can vary the temperature, viscosity etc), just let me know.

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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Back to the air conditioning units of Boeing 707 airliners

Above is avery pretty cartoon/schematic* of a virus MS2 that infects bacteria. I think it is used in proof-of-principle studies with viruses, as it is safer/easier/cheaper to work with than viruses that infect us. The virus can become airborne, which brings us to air conditioning units, which typically have filters as part of them. The filters can filter out viruses such as MS2 from the air.

Back in the early 1960s, an engineer called Proschan studied data on breakdown statistics of air conditioning units of early airliners. Now the most basic model for these statistics is that each unit fails at some constant fixed rate (i.e., probability per flight), and then the fraction still working after a given time decays exponentially with that time. Proschan realised that if indeed each unit had a constant failure rate, but that the units were heterogeneous in the sense that this rate varied from one unit to another, then the fraction still working would always decay more slowly than exponentially.

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How common are superspreading events for COVID?

Although good systematic data are lacking there are observations of what are called superspreading events. These are single events at which many people become infected. An example from early in the COVID pandemic is the superspreading even at a choir practice in Skagit Valley (USA) where it is likely that 53 attendees were infected in one event. Transmission is a random process. So just by chance we expect sometimes one person becomes infected at an event, sometimes by chance two or three, or …. So we always would expect at least some superspreading events. But how many?

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