## A problem of fluid flowing through a hole

Posted in Cute Problems with tags , , , , on December 19, 2017 by telescoper

I’m sure you’re all already as bored of Christmas as I am so I thought I’d do you all a favour by giving you something interested to do to distract you from the yuletide tedium,
The cute problem of the water tank I posted a while ago seemed to provide a diversion for many – although only about 10% of respondents go it right – so here’s a similar one. It’s not multiple choice so you will have to write your answers to the two parts in the comments box. As a hint, I’ll  say that this is from some notes on dimensional analysis, and it’s one of the harder problems I have in that file!

An incompressible fluid flows through a small hole of diameter d in a thin plane metal sheet. The volume flow rate R depends on d, on the fluid viscosity η and density ρ, and on the pressure difference p between the two sides of the she

(a) Find the most general possible relationship between the quantities  R, d, η,  ρ, and p.

(b) Measurement of the flow rate R1  through this the hole for a pressure difference p1 is made using a particular fluid. What can be predicted for a fluid of twice the density and one-third the viscosity?

## Ned Wright’s Dark Energy Piston

Posted in The Universe and Stuff with tags , , , , on April 29, 2015 by telescoper

Since Ned Wright picked up on the fact that I borrowed his famous Dark Energy Piston for my talk I thought I’d include it here in all its animated glory to explain a little bit better why I think it was worth taking the piston.

The two important things about dark energy that enable it to reconcile apparently contradictory observations within the framework of general relativity are: (i) that its energy-density does not decrease with the expansion of the Universe (as do other forms of energy, such as radiation); and (ii) that it has negative pressure which, among other things, means that it causes the expansion of the universe to accelerate. The Dark Energy Piston (above) shows how these two aspects are related. Suppose the chamber of the piston is filled with “stuff” that has the attributes described above. As the piston moves out the energy density of dark energy does not decrease, but its volume does, so the total amount of energy in the chamber must increase. Since the system depicted here consists only of the piston and the chamber, this extra energy must have been supplied as work done by the piston on the contents of the chamber. For this to have happened the stuff inside must have resisted being expanded, i.e. it must be in tension. In other words it has to have negative pressure.

Compare the case of “ordinary” matter, in the form of an ideal gas. In such a case the stuff inside the piston does work pushing it out, and the energy density inside the chamber would therefore decrease.

If it seems strange to you that something that is often called “vacuum energy” has the property that its density does not decrease when it subjected to expansion, then just consider that a pretty good definition of a vacuum is something that, when you do dilute it, you don’t any less!

So how does this dark vacuum energy stuff with negative pressure cause the expansion of the Universe to accelerate?

Well, here’s the equation that governs the dynamical evolution of the Universe: I’ve included a cosmological constant term (Λ) but ignore this for now. Note that if the pressure p is small (e.g. how it would be for cold dark matter) and the energy density ρ is positive (which it is for all forms of energy we know of) then in the absence of Λ the acceleration is always negative, i.e. the universe decelerates. This is in accord with intuition: because gravity always pulls we expect the expansion to slow down by the mutual attraction of all the matter. However, if the pressure is negative, the combination in brackets can be negative so can imply accelerated expansion.

In fact if dark energy stuff has an equation of state of the form p=-ρc2 then the combination in brackets leads to a fluid with precisely the same effect that a cosmological constant would have, so this is the simplest kind of dark energy.

When Einstein introduced the cosmological constant in 1915/6 he did it by modifying the left hand side of his field equations, essentially modifying the law of gravitation. This discussion shows that he could instead have modified the right hand side by introducing a vacuum energy with an equation of state p=-ρc2. A more detailed discussion of this can be found here.

Anyway, which way you like to think of dark energy the fact of the matter is that we don’t know how to explain it from a fundamental point of view. The only thing I can be sure of is that whatever it is in itself, dark energy is a truly terrible name for it.

I’d go for “persistent tension”…