Industrial mathematics is a rather loose term, nowadays it seems to cover almost any application of mathematics in a practical context. The research group at CRM has four main focus areas and expertise in many others.

**Carbon capture/Contaminant removal by sorption**

Since the industrial revolution the concentration of atmospheric CO2 has been rising exponentially, at a rate of about 0.17% per year, primarily due to the combustion of fossil fuels. The central aim of the Paris Agreement, December 2015, was to limit global warming to below 2C of pre-industrial levels, with a key component being the reduction of greenhouse gas emissions. Despite this world emissions have continued to steadily rise and so have temperatures. It is claimed that, in 2017, none of the major industrialized nations had met their pledged emission reduction targets and even if they had, the sum of all member pledges (as of 2016) would not keep global temperature rise “well below 2C". The IPCC has estimated that by mid-century the world will need to use carbon capture technology to remove an average of 10 billion tons of CO2 a year from the atmosphere.

Our most recent research branch began with an investigation of carbon capture via column adsorption. The mathematical formulation is virtually identical for the removal of contaminant so we have extended the work to include this as well.

The mathematical model describes the flow of a fluid mixture through a porous, adsorbing (or absorbing) media. One or more components are removed by the porous media. Our initial work has highlighted a number of errors standard to the literature. Results show excellent agreement with experimental data.

Our aim here is to start collaborations with local experimental groups and help to optimise contaminant removal (and carbon in particular) processes.

Myers T.G., Font F., Hennessy M.G. Mathematical modelling of carbon capture in a packed column by adsorption. Applied Energy, 278, 2020. DOI: j.apenergy.2020.115565

Myers T.G., Font F. Mass transfer from a fluid flowing through a porous media. Int. J. Heat Mass Trans., 163, 2020. DOI: j.ijheatmasstransfer.2020.120374

**Phase change**

Phase transitions occur in a multitude of natural and industrial situations such as in ice formation, metal formation from the molten state, computer disk manufacture, chocolate coating and many more. To model phase transitions requires studying heat flow in the different phases, which are defined over an unknown, moving domain. The problem may be further complicated since liquid and gas layers may also flow. Mathematically equivalent problems occur in the study of diffusion, porous media flow, financial mathematics and viscous flow.

Groups members have worked on phase change problems for many years, in the context of aircraft icing (models are now used in commercial software and termed “the Myers model”, I am trying to promote this designation); ice removal in fuel cell flow channels; Leidenfrost (droplet evaporating on a very hot surface); contact melting and more recently at the nanoscale.

T. Myers, M. Hennessy, M. Calvo. The Stefan problem with variable thermophysical properties and phase change temperature. International Journal of Heat and Mass Transfer. 149, 2020.

M. Calvo, T. Myers, M. Hennessy. The one-dimensional Stefan problem with non-Fourier heat conduction. International Journal of Thermal Sciences. Elsevier. 150, 2020.

M. Hennessy, M. Calvo, T. Myers. Modelling ultra-fast nanoparticle melting with the Maxwell–Cattaneo equation. Applied Mathematical Modelling. 69, 2019.

F Font, TG Myers, SL Mitchell. A mathematical model for nanoparticle melting with density change. Microfluidics and Nanofluidics 18 (2), 233-243, 2015

TG Myers, JPF Charpin, SJ Chapman. The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Physics of Fluids 14 (8), 2788-2803, 2002

TG Myers. Extension to the Messinger model for aircraft icing. AIAA journal 39 (2), 211 – 218, 2001.

**Mathematics in Nanotechnology**

Nanotechnology is a rapidly growing interdisciplinary area with a broad range of applications. Lying at the heart of nanotechnology is the nanoparticle (NP), a unit of matter with a critical diameter between 1 and 100nm. The wide range of current and potential uses of NPs, which include medicine, manufacturing, environment, and energy, is reflected in the international research drive. For example, in the 2014 financial year, the U.S. government’s National Nanotechnology Initiative provided approximately $1.5 billion in funding, while funding in the EU, China and Japan was of a similar order of magnitude. Despite extensive research into NPs, many breakthroughs and advances are the result of trial and error, since the mathematical framework and appropriate solution techniques which could provide a theoretical understanding often do not exist.

The group's work in nanotechnology involves a number of strands: the fundamentals of heat flow (in collaboration with the Nano Transport Group at UAB); nanoscale phase change; nanocrystal growth (in collaboration with the Insitut Català de Nanociència i Nanotecnologia, Inorganic Nanoparticles Group); enhanced water flow through carbon nanotubes; optics at the nanoscale (yes, it can actually work!) (with Prof. Wolfgang Bacsa, Centre d'Elaboration de Matériaux et d'Etudes Structurales).

W. Bacsa, R. Bacsa, T.G. Myers. Optics Near Surfaces and at the Nanometer Scale. ISBN 978-3-030-58983-7. Monograph, 2020 Springer-Nature. BUY IT NOW!!!

C. Fanelli, F. Font, V. Cregan, T. Myers. Modelling nanocrystal growth via the precipitation method. International Journal of Heat and Mass Transfer 165, 2021.

M. Beardo; M Calvo; J. Camacho; T.G. Myers. Hydrodynamic Heat Transport in Compact and Holey Silicon Thin Films Physical Review Applied. 11-3, 2019.

TG Myers, C Fanelli. On the incorrect use and interpretation of the model for colloidal, spherical crystal growth. Journal of colloid and interface science 536, 98-104, 2019.

M Calvo-Schwarzwälder, MG Hennessy, P Torres, TG Myers, FX Alvarez. Effective thermal conductivity of rectangular nanowires based on phonon hydrodynamics. International Journal of Heat and Mass Transfer 126, 1120-1128, 2018.

M Calvo-Schwarzwälder, MG Hennessy, P Torres, TG Myers, FX Alvarez. A slip-based model for the size-dependent effective thermal conductivity of nanowires. International Communications in Heat and Mass Transfer 91, 57-63, 2018.

TG Myers. Why are slip lengths so large in carbon nanotubes? Microfluidics and nanofluidics 10 (5), 1141-1145, 2011 (okay, it is old, but I like it).

**Nanofluid heat transfer and energy generation**

There exists a wide experimental literature concerning the heat transfer properties of nanofluids. In the past remarkable increases in thermal conductivity, viscosity and heat transfer coefficient were reported with the addition of a very small volume fraction of nanoparticles to a base fluid. However, a remarkable spread in the experimental data prompted a benchmark study by 34 laboratories around the world. One of their main conclusions was that “no anomalous enhancement of thermal conductivity was observed in the limited set of nanofluids tested in this exercise”. This result is backed up by other recent studies. Despite this there exist hundreds of mathematical papers proving the remarkable abilities of nanofluids.

The IM group provided unequivocal proof that nanofluids do not give the miraculous enhancements in heat transfer predicted by many theoreticians. However, due to the increased scattering of light by nanoparticles they may be used for Direct Absorption Solar Cells. This is an active branch of our work, with researchers in Ireland (where, due to constant rain, they are unable to verify any results).

G. O'Keeffe; S. Mitchell; T. Myers; V. Cregan. Nanofluid based direct absorption solar collectors Nanofluids and their engineering applications. Taylor and Francis Group. 2020.

GJ O’Keeffe, SL Mitchell, TG Myers, V Cregan. Modelling the efficiency of a nanofluid-based direct absorption parabolic trough solar collector. Solar Energy 159, 44-54, 2018.

TG Myers, H Ribera, V Cregan. Does mathematics contribute to the nanofluid debate? International Journal of Heat and Mass Transfer 111, 279-288, 2017.

V Cregan, TG Myers. Modelling the efficiency of a nanofluid direct absorption solar collector. International Journal of Heat and Mass Transfer 90, 505-514, 2015.

**General Industrial Mathematics**

In recent years we have worked on developing a strategy for legally selling rhino horn; making clutch components; beer bottle labelling; modelling the cardiovascular system; sound proofing, cooling concrete, spontaneous combustion in sugar cane piles and many more.

F. Font Martinez; T. Myers (Eds). Multidisciplinary Mathematical Modelling - Applications of Mathematics to the Real World. Springer Nature, 2021 (this book is a bit overpriced in my opinion, but buy it anyway!).

**Miscellaneous research topics …**

*Thin film flow*

The definition of “thin” is perhaps rather ambiguous. Thin film flows can cover the motion of lubricants, paints, water running down a window, the air supporting a rapidly rotating computer hard drive or the motion of lava or a glacier.

Mathematical modeling of thin film flows can lead to a rich variety of behaviour and obviously has many applications. We have published extensively in this field and used thin film models in a variety of applications, such as aircraft icing, nanofluid flow etc.

TG Myers. Thin films with high surface tension. SIAM review 40 (3), 441-462, 1998.

*Heat Balance Integral Method*

This is an approximate solution technique largely neglected by the mathematical community, possibly due to the arbitrary nature of the choice of approximating function. It is particularly useful for studying Stefan problems, for which only one practically useful exact solution exisits. IM group members have improved the method, by an order of magnitude in some cases, such that it can be more accurate than the second order perturbation.

TG Myers. Optimal exponent heat balance and refined integral methods applied to Stefan problems. International Journal of Heat and Mass Transfer 53 (5-6), 1119-1127, 2010.

TG Myers. Optimizing the exponent in the heat balance and refined integral methods. International Communications in Heat and Mass Transfer 36 (2), 143-147, 2009.

*Non-Newtonian fluid flows*

A Newtonian fluid has a constant viscosity. Water is the most obvious example. However, most practically interesting fluids have a variable viscosity. For example, paints and oils are shear thinning (they become less viscous when a shear force is applied). Certain fluids such as toothpaste, molten chocolate or ketchup behave as a solid until sufficient force is applied. Most liquid food products and biological fluids are non-Newtonian, for example, blood is shear thinning, but its behaviour also depends on the size of the blood vessel.

There is therefore great interest in the modeling of non-Newtonian fluids as well as the application of non-Newtonian fluid models to practical situations.

TG Myers. Application of non-Newtonian models to thin film flow. Physical Review E 72 (6), 066302, 2015.