could we know the amount of energy dissipated by turbulence in the bulk of turbulent flow? could we estimate the rate of energy transfer from large eddies to small ones and vice versa?

Kolmogorov came up with a theory to answer them in 1941, and this theory seems at least approximately true, but it’s a complicated business which I don’t understand nearly as well as I’d like to.

]]>Huh, I misspelled “Bénard” in my previous comment.

]]>The real killer is “linear phenomenological laws”: it means your system has to respond in a linear way to any change in boundary conditions.

I thought “linear phenomenological laws” means that thermodynamical fluxes depend linearly on thermodynamical forces. Thus it is really crucial what to call fluxes and forces. Ultimately one can say that Newtonian fluid is always governed by linear phenomenological laws — Fourier’s law for heat conduction and Newton’s model for molecular stresses. In the usual formulation of the minimum entropy production it is said it is valid for steady states and that’s why it fails when Bernard’s cells collapse, though the laws remain linear.

But anyway, let’s assume to be the force and heat flux to be the flux. If you state that Bernard’s cells correspond to a linear regime, than it seems there are two linear regimes! The first one when the heat is transported due to heat conduction alone rather than by any type of convection. I’m not an expert to claim that the regime with pure heat conductivity is possible, but Prigogine himself in his books claims many times that it is true. It is a canonical example of self-organization — pure heat conduction -> Bernard’s cells.

Does minimal entropy production theorem implies that at some point heat conduction produces more entropy then Bernard’s convection?

]]>Suppose that we are in a steady-state with an ensemble of particles; and have established that the probability density function is in a maximum entropy configuration given the constraints.

I think it is worth emphasizing that maximization of the entropy given certain constraints does not necessary lead to the true probability density function. Although this result can be used to construct the true one.

I’d like to know is there any cases when MaxEnt gives the true probability density function for a steady non-equilibrium state?

]]>I don’t really mean to deprive the Navier-Stokes equations of the glory that flows display, but at the same time I believe the focus put on the equations blinds to… what similarly paired images can in a sense directly teach about flows.

]]>Anyway, high energy physicists are proud to say that the standard model (or quantum electrodynamics, to be more precise) is the most successful physical theory known today, with the most convincing agreement of theory and experiment. I was half-jokingly thinking about the possibility that – depending on your reference frame – the Navier-Stokes equations may be more worthy of this praise, because they successfully predict a lot more phenomena on a wider range of length scales, and have a lot of applications in engineering, too.

]]>the thermal spectrum of the Earth,

mixing a general explanation and some particulars about TPW determination. In general there are two important contributions:

* the * real * thermal spectrum of the Earth, which exists because of the insolation;

* the reflected sunlight spectrum (including near-infrared) which is a more direct response to the solar radiation;

but for TPW the latter of the two is important.

For those who have read Tim van Beek’s post A Quantum of Warmth this should come as no surprize.

]]>