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### What do we mean by "multiscale" and "non-continuum" flows?

Duncan Lockerby and I are leading a “Special Interest Group” (SIG) within the EPSRC-funded UK Fluids Network. The name of this SIG is “Multiscale and Non-Continuum Flows”.

In practice, this means we are funded to arrange two meetings annually, for the next three years, to bring together UK researchers in multiscale and non-continuum flows with the aim of building links among the community that lead to joint projects in the future. Duncan and I both work on theoretical and numerical fluid dynamics, so we are very aware that in these SIG meetings we need to link with experimentalists and industrialists too.

We had our first SIG meeting in Edinburgh in early May this year, with more than 35 participants from around the UK. However, I don’t want to write a report on that here …

Instead, I found that this SIG meeting gave me the opportunity to think about what we actually mean by “multiscale” and “non-continuum” in the context of flows - and how this may be different from how other fluid dynamicists understand the terms. After all, in order to explain our work to others, we need to agree on and be clear in our own use of language.

So I thought I would jot down in this blog some general ideas from my perspective, which could form the basis for discussion.

In **non-continuum** flows there are not enough discrete fluid material elements, i.e. particles or molecules, for a fluid element (that is small compared to the scale of the process) to be assumed continuous and indefinitely divisible. While the mathematical models of continuum fluid mechanics may therefore not be applicable, the language of fluid mechanics can still in some cases provide us with a useful shorthand that highlights the key challenges in the physics of these flows. What do I mean by this? For instance, the “enhancement factor” is frequently used to describe the performance of nanotubes in membrane-type applications. It is the ratio of the measured fluid flowrate to the no-slip Hagen-Poiseuille (pipe) theoretical prediction. Nanotubes can show enhancement factors of several 100s or 1000s, which indicate how different the non-continuum water flow in the nanotubes is from the flow in a conventional pipe.

Fluid dynamic models can be modified to include some of the features of non-continuum flow behaviour, but to what extent is this helpful? For water flow in nanotubes, molecular pre-simulations can provide constitutive and boundary data that means some non-continuum features, like slip, can be captured in a fluid dynamic model. But some cannot - for example, the molecular layering that leads to density oscillations close to surfaces at the nanoscale. So we have to be very careful how far we can push a continuum fluid model for a flow application that is strictly non-continuum. (Borg, Reese (2017) MRS Bulletin 42:294-299; Holland, Borg, Lockerby, Reese (2015) Comput. Fluids 115:46-53)

The terms “non-equilibrium” and “non-continuum” often seem to be used interchangeably in the fluid dynamics literature. But I don’t think they are the same thing. In some cases, e.g. molecular dynamics simulations, **non-equilibrium** simply means the fluid is moving or flowing. In our context, it should more accurately mean that while there may be enough molecules for averaging over fluid elements, they do not collide often enough in flow timescales to ensure local thermodynamic equilibrium. This leads to a breakdown of the conventional linear constitutive relations and no-slip boundary conditions. Non-equilibrium is most often seen in gas flows at the microscale, or in high-speed or high-altitude aerodynamics. (Lockerby, Patronis, Borg, Reese (2015) J. Comput. Phys. 284:261-272)

So what are **multiscale** flows? “Multiscale” describes an analysis where we identify different models for distinct components of the flow. Turbulence is clearly multiscale in nature, but the scales are inextricable. A multiscale analysis requires some separation in the space and time scales of the effects being modelled, and a consequent simplification of the scale interactions. So, for example, parametric models (e.g. Newton’s law of viscosity, slip conditions) can be regarded as a type of multiscale analysis. Another type are hybrid models: in one small part of the flow domain there are fine flow details that are modelled in one way, and in some separate (larger) region the details are much coarser and modelled differently. (Borg, Lockerby, Reese (2015) J. Fluid Mech. 768:388-414)

If you have a different or complementary perspective on these issues, come along and share your views at the next SIG meeting at the end of September in Warwick University. You are very welcome! (Please email either Duncan or me for more details.)