By rediscovering the original, unaltered version of a century-old mathematical formula, a University of Warwick researcher has developed a more accurate model for tracking the movement of dangerous, irregularly shaped nanoparticles that can cause serious health problems.
Pollutants such as soot, dust, and microplastics can enter the bloodstream after inhalation, leading to conditions like cancer, heart disease, and stroke. Yet most modern mathematical models struggle to represent these irregular particles, assuming instead that they are perfect spheres — a simplification that fails to reflect their real-world behavior.
Nanoparticle Pollution
On average, humans breathe in millions of nanoparticles each day. Although we rarely notice them, their cumulative effects can be harmful over time. Acute events such as industrial leaks pose even greater risks, requiring precise monitoring. However, the mismatch between idealized spherical models and the irregular shapes of actual particles has long limited scientists’ ability to predict their motion and impact accurately.
That gap has finally been bridged. In a new study published in the Journal of Fluid Mechanics, Professor Duncan Lockerby of the University of Warwick’s School of Engineering revisited the earliest version of a key mathematical model to improve how scientists calculate the drag and motion of airborne particles.
“The motivation was simple: if we can accurately predict how particles of any shape move, we can significantly improve models for air pollution, disease transmission, and even atmospheric chemistry,” Lockerby said. “This new approach builds on a very old model – one that is simple but powerful – making it applicable to complex and irregular-shaped particles.”
The Cunningham Correction Factor
The foundation of aerosol science dates back to 1910, when Ebenezer Cunningham developed what became known as the Cunningham correction factor. His work explored how the drag on tiny particles differs from predictions based on classical fluid dynamics. That same year, Cunningham’s colleague Robert Millikan helped verify his findings, later revising the formula in 1920.
However, Millikan’s revision introduced a limitation: it could only account for the movement of perfectly spherical particles, overlooking the broader applications of Cunningham’s simpler original correction.
Lockerby returned to Cunningham’s initial work and built upon it by creating a correction tensor — a mathematical tool that adapts equations for spheres to fit objects of virtually any shape. By combining this tensor with Cunningham’s original insights, Lockerby produced a formula capable of accurately modeling the movement of non-spherical particles without requiring additional parameters.
“This paper is about reclaiming the original spirit of Cunningham’s 1910 work,” Lockerby said. “By generalising his correction factor, we can now make accurate predictions for particles of almost any shape — without the need for intensive simulations or empirical fitting.
“It provides the first framework to accurately predict how non-spherical particles travel through the air, and since these nanoparticles are closely linked to air pollution and cancer risk, this is an important step forward for both environmental health and aerosol science,” Lockerby added.
Applying Lockerby’s Work
Lockerby’s revised model offers a powerful foundation for applications across air quality research, climate science, nanotechnology, and medicine. In emergency situations, it could help responders predict how pollutants, volcanic ash, or wildfire smoke spread through the atmosphere. In industrial and medical settings, it may enable manufacturers and pharmaceutical researchers to refine production processes and drug-delivery systems with greater precision.
The Warwick School of Engineering now plans to construct a state-of-the-art aerosol generation system to further test and refine Lockerby’s model. This facility will allow researchers to generate and measure non-spherical particles under controlled conditions, bridging theory and experimentation.
“This new facility will allow us to explore how real-world airborne particles behave under controlled conditions,” said Professor Julian Gardner of Warwick’s School of Engineering, “helping translate this theoretical breakthrough into practical environmental tools.”
The paper, “A Correction Tensor for Approximating Drag on Slow-Moving Particles of Arbitrary Shape and Knudsen Number,” appeared in Nature Astronomy on October 29, 2025.
Ryan Whalen covers science and technology for The Debrief. He holds an MA in History and a Master of Library and Information Science with a certificate in Data Science. He can be contacted at ryan@thedebrief.org, and follow him on Twitter @mdntwvlf.