Concentration waves
 

I am trying to model second sound (concentration waves) in liquid helium mixtures (3He-4He). This sound can be thought of as being driven by the osmotic pressure in the solution. In this case, concentration gradients (osmotic pressures) do not give rise to diffusion, but instead a current. In this connection, the osmotic pressure has the character of a gas pressure.

I am uncertain as to which model in CFX would be appropriate for my system. The isotopes "mix at the molecular level", and thus the multicomponent flow with a variable composition mixture would seem good. In the documentation, however, it says that "they share the same mean velocity, pressure and temperature fields, and that mass transfer takes place by convection and diffusion". In my case, the other component is superfluid (zero viscosity) and the two components can have different velocity fields. The other component simply acts as a background for the other. I am not interested in the flow field of the superfluid, so maybe a multicomponent flow could still be used to approximate the system? In a sense, the two isotopes should oscillate in antiphase ("normal" pressure variations in the system remain almost zero).

Can I use the variable composition mixture and simply set the equation of state of the orher component to depend on the concentration in a way, which guarantees a "gas-like behavior" and concentration waves?

The other option is, I guess, to use inhomogeneous multiphase flow with two continuous phases and the mixture model. In this case I am unaware how to take into account the fact that they indeed are mixed at the molecular level. What is the interfacial length scale in this case?

All help is greatly appreciated.

I have done a quick google search but have not found anything to explain to me how concentration waves work. Can you explain how they work?

Regardless, you almost certainly do not want to take a multiphase approach. All the multiphase models assume the substances are mixed at a microscopic level, but not an atomic level. So it is certainly not multiphase.

But as for whether multi component mixtures or a special EOS or an additional variable approach would work I cannot say until I understand the physics.

Thank You for the reply! I hope I will be able to explain shortly some of the characteristics of second sound. Here is a quote from a book: "Second sound in dilute solutions [3He-4He] is a hydrodynamic mode in which there is a counter-oscillation of the normal [3He] and superfluid [4He] mass densities". So second sound is "internal convection carrying energy and entropy but no net transfer of matter".

In addition to concentration waves, second sound can equivalently be considered as a temperature or entropy waves. The superfluid 4He has zero viscosity, zero entropy, and "infinite" thermal conductivity. It acts as a "vacuum" for the 3He atoms, which carry all the entropy of the system. The mean free path of 3He atoms in superfluid 4He is tremendous. Thus, temperature (concentration) is capable of propagating in a wave-like motion instead of diffusively. Second sound in helium mixtures can be thought of as being the first sound (ordinary sound) of the 3He-component. Actually, at low enough temperatures (~< 50 mK), diffusion begins to blur out the hydrodynamics, and second sound gradually vanishes. As I mentioned before, the concentration waves can also be described as being driven by the osmotic pressure in the mixture, where the osmotic pressure acts as a "gas pressure" for the 3He "phase".

Second sound is also observed in pure 4He, where the concentration of the normal component oscillates (at finite temperatures, 4He can be considered to be composed of a mixture of normal 4He and superfluid 4He). In isotopic helium mixtures at low enough temperatures (below ~1 K), the normal component of 4He can be neglected (its fraction is vanishingly small).

Using multiphase sounds appealing, since it is often easier to consider the two components as a two-phase system (vacuum fluid + gas) rather than a two-component system. But apparently, as you say, this model in CFX does not naturally assume mixing at the atomic level, which would be desirable.

Here is another quote from an article: "entropy plays the role of an equation of state; density(3He) ~ entropy(3He)". I was thinking if it is possible to just write this as the equation of state for the 3He-material of the multi component mixture defined in CFX. Of course, the entropy of 3He depends on its concentration x, so the equation of state would be something like rho = rho(T,P,x). But I guess making the equation of state depend on the concentration poses no problems? I am also not familiar enough with CFX to decide between the different choices you mentioned at the end of your post.

For the interested reader, some of the hydrodynamics are (rather briefly) described e.g. in this paper (here in the case of pure 4He):
http://cs.stanford.edu/people/jlhiatt/SecondSound.pdf

CFX cannot model a vacuum. And it sounds like the basic equations of what you are trying to do are not governed by the Navier Stokes equations either. In that case CFX cannot help you.

Or can you ignore the 4He and describe the 3He as a compressible gas (with some strange EOS) and therefore the compressible Navier Stokes equations are applicable?

I thank again for the effort. I think the system should in principle be describable by the Navier-Stokes equations (or something analogous), at least if some approximations are made (e.g. quantized vortices are neglected). The normal component (3He) should obey the Navier-Stokes and the superfluid (4He) could be considered as an Euler fluid (Navier-Stokes with no viscosity). I then have a set of two N-S equations, since the velocity fields of the two components can be different. Perhaps another peculiarity is that the equations of motion (equations of conservation of momenta) include a contribution to the fluid acceleration which is essentially thermal in character (proportional to the gradient of T). So temperature functions almost identically to pressure, except that there is a sign-difference between the two components (temperature gradients cause the two components to move in antiphase).

I probably could ignore the 4He and describe the 3He as a compressible gas with some effective EOS. This would likely yield OK results, but I would lose some detail, for example effects related to compressibility of the entire mixture (first sound).

Is there any simple way to define the actual equations to be solved in CFX? Or, can I somehow define some kind of "partial pressures", which drive the two components in different directions? In some sense, I would like to have the 3He to behave as an almost independent compressible gas phase within the 4He, but such that it experiences two pressures (the "real pressure" of the mixture and the "internal pressure" of 3He). This "internal pressure" would be a negative "internal pressure" for the 4He.

The multi component fluid approach is based on partial pressures. It may be of assistance.

If compressibility is the only important effect the 4He has then you could model the 3He as a compressible gas with a suitable EOS, and use an additional variable to describe the 4He pressure. Not sure if this works (I still do not entirely understand the physics!) but it might spark some ideas.

Thank you. I will try to figure out what is the best approximation and will report it here should I come up with something.

One thing came to my mind. What happens if I use the mixture model in inhomogeneous multiphase, and set the interfacial length scale (mixture length scale) to zero (is this possible)? Does this then describe a continuous atomic mixture of two phases (and two flow fields)?

No. All multiphase models assume mixing on a microscopic scale and are not suitable for atomic scale mixing.