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
Mathematics in Nanotechnology
Continuum theory may be applied when there is a sufficiently large sample size to ensure that statistical variation of material quantities, such as density, is small. For fluids the variation is often quoted as 1%. This leads to a critical dimension of the order 10 and 90nm for liquids and gases respectively. Nanoscale is typically described as involving materials with at least one dimension below 100nm, so there is clearly a range of sizes where continuum theory may be applied to nano phenomena.
The group's work in nanotechnology involves enhanced water flow through carbon nanotubes, melting of nanoparticles and the heat transfer properties of nanofluids, see [1,2,3,11]. We have just started a project on Ostwald ripening of nanoparticles in conjunction with the Insitut Catalá de Nanotecnologia.
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 in the context of aircraft icing; ice removal in fuel cell flow channels; Leidenfrost (droplet evaporating on a very hot surface); contact melting, see [2,4,5,7,8,9,10,12,13,14,18,19].
Thin film flows
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, see [15,16,18,19,20].
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, see [4,15,16].
Selected publications
[1] MacDevette M., Myers T.G., Wetton B.R. Boundary layer analysis and heat transfer of a nanofluid. Microfluidics and Nanofluidics, 2014 DOI 10.1007/s10404-013-1319-1.
[2] Font F., Myers T.G. Spherically symmetric nanoparticle melting with a variable phase change temperature. J. Nanoparticle Res. 2013, 15:2086 DOI 10.1007/s11051-013-2086-3
[3] Myers T.G., MacDevette M.M. and Ribera H. A time dependent model to determine the thermal conductivity of a nanofluid. J. Nanoparticle Res. 15:1775 2013, DOI 10.1007/s11051-013-1775-2
[4] Myers T.G., Low J. Modelling the solidification of a power-law fluid flowing through a narrow pipe. Int. J. Thermal Sci., 2013, http://dx.doi.org/10.1016/j.ijthermalsci.2013.03.021
[5] Font F., Mitchell S.L., Myers T.G. One-dimensional solidification of supercooled melts. Int. J. Heat Mass Trans. 62, 411-421, 2013. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.02.070
[6] Myers T.G., S.L. Mitchell. A mathematical analysis of the motion of an in-flight soccer ball. Sports Engineering, 1-13, 2013. DOI 10.1007/s12283-012-0105-8
[7] Myers T.G. , Mitchell S.L., Font. F. Energy conservation in the one-phase supercooled Stefan problem Int. Comm. Heat Mass Trans. 39, 2012 http://dx.doi.org/10.1016/j.icheatmasstransfer.2012.09.005
[8] MacDevette M.M., Myers T.G.. Contact melting of a three-dimensional phase change material on a flat substrate. Int. J. Heat Mass Trans., 55, 2012 http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.06.087
[9]Myers T.G., Low J. An approximate mathematical model for solidification of a flowing liquid in a microchannel. Microfluid. Nanofluid. 11 (4), 417-428, 2011. DOI 10.1007/s10404-011-0807-4
[10] Myers T.G., Mitchell S.L. Application of the combined integral method to Stefan problems Appl. Math. Model. 35 (9), 4281-4294 2011. doi:10.1016/j.apm.2011.02.049
[11] T.G. Myers. Why are slip lengths so large in carbon nanotubes? Microfluid. Nanofluid. 10 (5), 1141-1145, 2011. DOI 10.1007/s10404-010-0752-7
[12] S.L. Mitchell & T.G. Myers The application of standard and refined heat balance integral methods to one-dimensional Stefan problems. SIAM Review 52 (1), 57-86 32 2010. DOI. 10.1137/080733036
[13] Myers T.G. & Charpin J.P.F. A mathematical model of the Leidenfrost effect on an axisymmetric droplet. Physics of Fluids 2009 DOI: 10.1063/1.3155185.
[14] Myers T.G., Mitchell S.L. & Muchatibaya G. Unsteady contact melting of a rectangular cross-section phase change material on a flat plate. Phys. Fluids 20, 103101, 2008 DOI:10.1063/1.2990751
[15] Charpin J.P.F., Lombe M., Myers T.G. Spin coating of non-Newtonian fluids with a moving front. Phys Rev E76, 2007 DOI: 10.1103/PhysRevE.76.016312.
[16] Balmforth N., Ghadge S. & Myers T.G. Surface tension driven fingering of a viscoplastic film. J. non-Newtonian Fluid Mech. 143- 149, March 2007 DOI: 10.1016/j.jnnfm.2006.07.011.
[17] Myers T.G. The application of non-Newtonian models to thin film flow. Physical Rev. E, 72: 066302 -1-11, 2005.
[18] Myers T.G., Charpin J.P.F. & Chapman S.J. The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Physics of Fluids 14(8) pp2788-2803 2002.
[19] Myers T.G., Charpin J.P.F. & Thompson C.P. Slowly accreting glaze ice due to supercooled droplets impacting on a cold substrate. Physics of Fluids 14(1) pp240-256 2002.
[20] Myers T.G. Thin films with high surface tension. SIAM Review 40(3) pp441-462 1998.