Particles in Fluids

I’m back – posts will be much shorter and more paper-centered from now on, as classes and research continue to consume my life.

Three really cool papers recently, all dealing with particles in some kind of flow:

1. Effects of particles in chaotic flow (Ouellette et al., PRL 2008)
Small tracer particles are often used to ‘visualize’ fluid flows, by seeding them into the fluid. If the particles are small enough and have low enough density to match the fluid, they can be considered as infinitesimal fluid elements to a good approximation. This breaks down if the particles are (i) too large, or (ii) too dense. While the effect of having small but dense particles is pretty well studied (since the particles can be taken to be pointlike), the case of large particles is more complicated – one has to solve the relevant Navier-Stokes equation over the surface of each particle. How does a large tracer particle perturb fluid flow?

By imaging the motion of tracer particles of different sizes in a chaotic fluid flow, Ouellette et al. study the flow field around a large tracer particle as well as its own motion. (The smallest particles act as the ‘ideal’ infinitesimal fluid elements that follow the flow well.) The effect of tracer particles being too large or too dense is often thought to be captured by the Stokes number where and are the particle and fluid densities, and are the particle radius and characteristic flow length scale, and is the fluid Reynolds number. It is surprising, then, that the data in these experiments does not seem to solely depend on the Stokes or Reynolds numbers – these dimensionless parameters don’t appear to capture all of the physics associated with inertial effects. Weird.

2. Effects of particles in turbulent flow (Tanaka and Eaton, PRL 2008)
Ok, so the previous paper dealt with non-turbulent flow. This one deals with the case of how particles in a turbulent flow affect the turbulence. Do they make it more turbulent, make it less turbulent, or (unlikely) don’t affect the flow? Can these effects be captured by the Stokes or Reynolds number, unlike the previous case?

Tanaka and Eaton looked at data from many different experiments on this subject, finding (as in Ouellette et al.’s experiments) no systematic dependence on the Stokes or Reynolds numbers. Hm. Instead, they use some very beautiful dimensional analysis to come up with a new dimensionless parameter, what they call the particle momentum number Pa, which seems to capture more of the physics here – for very large and very small values of Pa, the particles augment turbulence, while for an intermediate range of Pa turbulence is attenuated. (Instead of attempting to write the two forms of Pa out, I’m just going to refer the reader to equations 14 and 15 in the paper). This is cool – finally a parameter that yields information about the physics of the situation!

Physically, is there a simple way of seeing what Pa actually means, versus just being a combination of Re, St, and various relevant variables? (The Reynolds number, for example, can be understood as telling one about the relative importance of inertial forces versus viscous forces on a tracer particle; the Stokes number on the other hand tells one about how ‘impactable’ a tracer particle is – it describes how independently the particle can move from the carrier flow.) I wasn’t fully able to decipher this.

Secondly, and I’m not sure if this even makes sense or not, but could this have any relevance to Ouellette et al.’s experiments, in which St or Re on their own were not enough to account for the effects of perturbations due to tracer particles? I did some mindless playing around with Ouellette et al.’s data from figure 4c-d, plotting it as a function of two such possible parameters. The first, ‘Pa1′, is inspired by Tanaka and Eaton’s particle momentum number, and is defined as ; the second, ‘Pa2′, is Tanaka and Eaton’s equation 15: . This is what I’m showing here:

Perhaps unsurprisingly, the two curves (for two tracer particle sizes) still don’t fall on a single curve. Oh well. Again, I’m not entirely sure if it makes sense to ask this question, but is there some combination of St and Re (similar to Pa) that is a more relevant dimensionless parameter for Ouellette et al.’s experiments?

3. Phonons in a 1D microfluidic ‘crystal’ (Beatus et al., Nature Physics 2008)
This is a cute paper that touches on many, many interesting ideas. The basic idea is straightforward: Beatus et al. produced a continuously-flowing array of uniformly-spaced oil drops in a microfluidic channel, surrounded by a continuous oil phase. The drops are disc-like in shape (they are confined in the z-direction), unconstrained in the x-direction (the direction of flow), and the constraint in y (i.e. the width of the channel) is varied, thus varying the friction on the drops.

The cool thing is that these researchers see interesting longitudinal and transverse fluctuations (it’s worth looking at the supplementary movies), and by fourier-transforming their data, they pull out dispersion relations that surprisingly show acoustic phonon propagation. The phonon propagation speed is much smaller than the speed of sound in the surrounding fluid, which leads them to hypothesize that these collective modes arise from dipole-like hydrodynamic interactions between droplets. Very pretty stuff.