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Feb
3
comment 1-D Fick's first law - partial derivative?
Fick's law also applies to non-steady states. The only requirement is that the $t$ and $x$ dependence is slowly varying compared to collisional time and length scales.
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
I should explain why I am so insistent on this point. I have written a number of papers (such as this one arxiv.org/abs/1404.6843) in which I compute transport coefficients using a variety of microscopic theories. The results violate "frame invariance" (although I had not heard of this concept when I wrote them). That's not a surprise, because "frame invariance" is not a property of Newton's equation, or the Boltzmann equation, or quantum field theory.
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
To me, wikipedia pages like this one en.wikipedia.org/wiki/Objectivity_%28frame_invariance%29 illustrate the problem. The page states what "frame invariance of material response" is, but not why we expect it to be true. Muller (for all his faults) actually gives an explicit argument why we don't expect it to be true.
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
If you want to ponder this further; Take a look at the book by Muller that I found using google (mentioned above). In Insert 8.1 he provides am example for the frame dependence of heat flux which (I think) is completely correct.
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
You keep saying this, but that's not a physical argument. If the microscopic laws of physics look different to an inertial and a rotating observer, why would the laws of fluid dynamics look the same (which is what this principle asserts)?
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
Yes, but Newton's equations are not invariant under rotations of the observer. A rotating observer sees a Coriolis force, a non-rotating observer does not.
Jan
27
comment Does nature really follow the heat equation?
The heat equation is not a primitive ad-hoc equation. It is an exact consequence of energy conservation in the limit of small temperature gradients.
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
To me, the point is this: Why would we expect the equation of continuum (fluid) mechanics to form-invariant (covariant) under rotations, given that the laws of mechanics are not?
Jan
27
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
That's interesting -- I was unaware of this. Some googling shows that there was (and maybe still is) an argument between Truedell, Noll, and others and various members of the physics community about whether such a principle exist. For some particularly nasty comments about Truesdell see Ingo Muller "A History of Thermodynamics."
Jan
26
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
Is that a known (valid?) principle of physics? If it is, it should follow from a symmetry of the laws of physics.
Jan
26
comment Derivation of the diffusion coefficient?
Not sure what you are asking. The book you refer to is called "Mathematical Theory" and it does, indeed, provide detailed, rigorous, derivations of transport coefficients, The page you refer is from an introductory chapter -- just keep reading. (Although the Chapman and Cowling is, admittedly, somewhat heavy going. I provide the a brief summary of the calculation here physics.stackexchange.com/questions/230380/… )
Jan
26
comment Viscous forces with asymmetric gradient velocity in fluid mechanics
This sounds sort of right, but I am not quite sure if the argument is entirely correct. A rotating observer is non-inertial (and may see funny forces). I think the correct version of this argument is that there should not be a stress in a rigidly rotating fluid, because this in an equilibrium state.
Jan
25
comment Yang-Mills theories, confinement and chiral symmetry breaking
Any text book on particle physics will have a short discussion on quarkonia (charmonium, most notably the J/psi, bottomonium, in particular the upsilon). A more technical discussion is in chapter 6 of Yndurain, or reviews like this arxiv.org/abs/1111.0165 .
Jan
24
comment Why do superconductors have a maximum current density?
@busukxuan Indeed, see my answer to this question physics.stackexchange.com/questions/36053/…
Jan
24
comment Why do superconductors have a maximum current density?
In quantum mechanics, in order to get a current, we need $\psi=\psi_0 e^{i\phi}$ with $\phi=kx$. Then $j \sim k |\psi_0|^2$. This is the sense in which the current $j$ is proportional to the gradient $k$ of the phase $\phi$. The Hamiltonian has a term $|\nabla\psi|^2$, so there is a cost $k^2|\psi_0|^2$ to having a current, even in a superconductor.
Jan
20
comment Momentum of slowly spinning (viscous) fluid
I added the derivation to the answer.
Jan
20
comment Momentum of slowly spinning (viscous) fluid
I think I finally figured out what is wrong with your equation, see my comment below.
Jan
20
comment Momentum of slowly spinning (viscous) fluid
Wow, this had me stumped for a while (could not find the mistake in what you did). We consider force balance (in $\hat{e}_\theta$ direction) on a cylindrical fluid element. The force/area on the angular sections is $\tau_{\theta r}\hat{e}_\theta$, and force balance gives the result you discuss. However, there is also a force/area on the radial sections $\tau_{\theta r}\hat{e}_r$, and this contributes an extra term because of $\partial \hat{e}_r/(\partial \theta)=\hat{e}_\theta$.
Jan
20
comment Momentum of slowly spinning (viscous) fluid
I can't quite figure out what you did there, but it clearly not right (you can just look up the Laplacian on a vector function). First of all, the equation of motion does not come from radial force balance (radial force balance determines the pressure), it comes from angular force balance.
Jan
20
comment Momentum of slowly spinning (viscous) fluid
The viscous drag is $\nu\nabla^2 v$. In cylindrical coordinates this must give Bessel functions.