Archive for Navier-Stokes equations

Classical Fluid Analogies for Schrödinger-Newton Systems

Posted in The Universe and Stuff with tags , , , , on July 16, 2025 by telescoper
Stock viscosity image: Photo by Fernando Serrano on Pexels.com

I thought I’d mention here a paper now on arXiv that I co-wrote with my PhD student Aoibhinn Gallagher. Here is the abstract:

The Schrödinger-Poisson formalism has found a number of applications in cosmology, particularly in describing the growth by gravitational instability of large-scale structure in a universe dominated by ultra-light scalar particles. Here we investigate the extent to which the behaviour of this and the more general case of a Schrödinger-Newton system, can be described in terms of classical fluid concepts such as viscosity and pressure. We also explore whether such systems can be described by a pseudo-Reynolds number as for classical viscous fluids. The conclusion we reach is that this is indeed possible, but with important restrictions to ensure physical consistency.

arXiv:2507.08583

It is based on work that his in her now-completed PhD thesis, along with another paper mentioned here. I have been interested for many years in the Schrödinger-Newton system (or, more specifically, the Schrödinger-Poisson system in the case where self-gravitational forces are involved). In its simplest form this involves a wave-mechanical representation, in the form of an effective Schrödinger equation, of potential flow described classically by an Euler equation. More recently we got interested in the extent to which such an approach could be used to model viscous fluids represented by a Navier-Stokes equation rather than an Euler equation. That was largely because the effective Planck constant that arises in this representation has the same dimensions as kinematic viscosity (but there’s more to it than that).

In the paper we explored a limited aspect of this, by looking at situations where there is no vorticity (so still a potential flow) but there is viscosity. There aren’t many examples of fluid flow in which there is viscosity but no vorticity, and most of those that do exist are about one-dimensional flow along channels or pipes with boundary conditions that don’t really apply to astrophysics, but one example we did look at in detail was the dissipiation of longitudinal waves in such a fluid.

One upshot of this work is that one can indeed describe some aspects of quantum-mechnical fluids such as ultra-light scalar matter in terms of classical fluid properties, such as viscosity, but you have to be careful. For more information, read the paper!

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Dond’escono quei vortici?

Posted in Education, Opera, The Universe and Stuff with tags , , , , on November 10, 2012 by telescoper

Just time for a quickie today. I seem to be writing that virtualy every day at this time, in fact. Anyway, yesterday I gave the last of a series of lectures on Fluid Dynamics during which I talked a little bit about the Navier-Stokes equation, and introduced the concept of turbulence, topic that Richard Feynman described as “the most important unsolved problem in classical physics”. Given that the origin of turbulence is so poorly understood, I had to cover it all fairly qualitatively but did at least explain that its onset is associated with high values of the Reynold’s Number, an interesting dimensionless number that characterizes the properties of viscous fluid flow in such a way as to bring out the dynamical similarity inherent in the equations. The difficulty is that there is no exact theory that allows one to calculate the critical value of the Reynold’s number and in any particular situation; that has to be determined by experiments, such as this one which shows turbulent vortices (or “eddies”) forming downstream of a cylindrical obstacle placed in flowing fluid. The (laminar) flow upstream, and in regions far from the cylinder, has no vorticity.

What happens is obviously extremely complicated because it involves a huge range of physical scales – the vorticity is generated by very small-scale interactions between the fluid elements and the boundary of the object past which they flow. It’s a very frustrating thing for a physicist, actually, because one’s gut feeling is that it should be possible to figure it out. After all, it’s “just” classical physics. It’s also of great practical importance in a huge range of fields. Nevertheless, despite all the progress in “exotic” field such as particle physics and cosmology, it remains an open question in many respects.

That’s why it’s important to teach undergraduates about it. Physics isn’t just about solved problems. It’s a living subject, and it’s important for students to know those fields where we don’t really know that much about what is going on…

PS. The title is a quotation from the libretto of Mozart’s opera, Don Giovanni, uttered by the eponymous Count as he is dragged down to hell. It translates as “Whence come these vortices?” Pretentious, moi?

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