# Ordering My Thoughts (Summer 2013)

In this document I outline my ideas and goals for a handful of projects to work on over the summer (2013). The purpose is to maintain ordered thoughts and to be constructive/directed about tackling problems.

I have abandoned these questions (at least for the meantime) to pursue some more “useful” work.

# Payam & Shura’s Papers

DNA electrophoresis in vicinity of a membrane pore: Rowghanian et al. (2013) and Rowghanian et al. (2013)

Some questions

1. What happens if an interior or bulk DNA piece approaches the pore?

2. In Rowghanian et al. (2013)a page 5, it is claimed that DNA motion will occur in a superposition of longitudinal and transverse orientiations (i.e. along the longest dimension, and perpendicular to it). Is this true? I think one of them will be unstable (though at high $$T$$ the orientation will be frequently randomised).

• Reading: Binder (1939)

• Drag resistances (for elongated/prolate bodies) from Burgers (1995):
For a spherical particle: $$F_{\rm drag}^{\rm sph}=6\pi\eta a V$$.
For an end-on ellipsoid: $$F_{\rm drag}^{\rm ell-e}=\frac{4\pi\eta a V}{\ln(2a/b_0)-0.5}$$, where $$a$$ is half the length and $$b_0$$ is the equatorial radius.
For an end-on cylinder: $$F_{\rm drag}^{\rm cyl-e}=\frac{4\pi\eta a V}{\ln(2a/b)-0.72}$$ (slightly higher than ellipsoid).
For a face-on ellipsoid: $$F_{\rm drag}^{\rm ell-f}\simeq\frac{8\pi\eta a V}{\ln(2a/b)-0.5}$$.
For a face-on cylinder: similar to the face-on ellipsoid, but slightly higher.
As expected, the face-on bodies (where the motion is perpendicular to the long axis) encounter more resistance to motion than the end-on bodies – roughly twice as much, depending on the dimensions. Doesn’t this mean that, in steady-state, all the DNA will move in the longitudinal direction?

• Perhaps this is not a problem: we consider a low-Re regime, where viscous effects are much more important than inertia. Thus the instability or propensity to change orientation might not matter.

• Consider a stick moving through fluid at an angle; what is flow profile and where is the fluid pressure greatest? If the motivating force on the stick acts on the centre, there may still be a torque which aligns the stick in a preferred direction.

3. In Rowghanian et al. (2013) page 5, second column, they use dimensional analysis to justify a steady-state approximation used in the paper: the time taken to establish the fluid velocity profile around a section of DNA must be much shorter than the time taken the DNA to “move past” that fluid.

1. I want to try to do a better job than Payam’s scaling argument: what are the coefficients in the “$$\# \ll 1$$” relationship? This might be interesting because of a problem acknowledged in Payam’s other paper: his predictions do not fit the experimental data.

2. Dimensional analysis argument not fully understood: I guessed $$t_{\rm h}\sim \ell/c_{\rm s}$$ on physical grounds, but Payam says $$t_{\rm h}\sim \ell^2\rho/\eta$$ from dimensional analysis. What is the interpretation of Payam’s $$t_{\rm h}$$?

• The two $$t_{\rm h}$$s are measuring different things. Payam’s $$t_{\rm h}$$ is considering the diffusion of a particle’s momentum via vorticity in the fluid; or the characteristic time in the velocity autocorrelation of neighbouring particles. My $$t_{\rm h}$$ is about information flow via sonic waves: this is much smaller than the vorticity time-scale.
I want to think about which time scale is more appropriate in this context and why?

• So, what should the time-scale be from physical reasoning: what is the physics behind establishing the flow profile $$\vec{v}(\vec{x})$$ in the fluid around the DNA/jet?
Landau et al. (1959) argue $$\vec{v}(\vec{x})$$ purely from momentum conservation / Gauss’ Law, and symmetries of the momentum flux tensor. Time plays no role because the source of the momentum is static with respect to the (“Eulerian”) fluid.
Can we do the same analysis with a moving submerged jet? Say the jet moves through stationary fluid along the jet axis with constant velocity $$U\hat z$$: how do the fluid velocity equations change? (Could use perturbation theory?)

• Does Payam assume incompressible fluid? Yes – he follows Landau et al. (1959) pp. 86-88 in setting up the submerged jet; they assume zero divergence in the velocity field & use stress tensor for incompressible flow. (The derivation in Broman et al. (2010) also assumes incompressibility.) This means that the sound speed in his model will be infinite, and sonic information transfer will be irrelevant. This is why there is only one possible time-scale in the theory.

• Should I re-visit the submerged/elongated jet derivation with compressibility / sound included? Method 1: I would have to derive the formulae for the submerged jet in a compressible fluid, and perhaps solve the time-dependent Navier-Stokes equation to boot.
Method 2: tack some ad-hoc changes onto the existing work – e.g. treat the DNA/jet as an extended source of spherical waves. This will be inconsistent and crude, but probably easier than method 1.
In any case, a model with “sound” propagation might be a more accurate model (see references below). It may also be more illuminating as to how the velocity profile is established, and the consequences of interactions between neighbouring DNA molecules.

• Which process (sound waves or vorticity diffusion) carries more momentum? Read Landau et al. (1959) chapter 8 (p. 245), on sound. Specifically: how is a sound wave generated; how much momentum does it carry / how does it affect the surrounding medium; how does it compare with vorticity?
Sound waves are a higher-order effect than vorticity propagation. Read more.

• Another thing to consider: does it even matter? The time-scale they use in the paper is much larger than the sound-time, and places a more stringent test on the steady-state assumption. So my objection would only strengthen the conclusion.
The vorticity moves slowest, and the fluid velocity profile might only assume the anticipated form once this final contribution has had time to influence the entire profile.

• Some reading about propagation of hydrodynamic information:
Hinch (1975) application of Langevin equation to fluid suspensions; includes fluid inertia and interaction with suspension.
Ladd (1993) short-time motion of colloidal particles: theory.
Español (1995), Español et al. (1995) for discussion of how HD disturbances propagate, and evidence for faster-than-vorticity interaction. One considers incompressible and the other considers compressible; the vorticity time-scale is not the only important time-scale.
Clercx (1997) scaling of diffusion coefficient.
Kao et al. (1993) high-speed experiment finds interaction between colloids is faster than the vorticity timescale. Single-particle theory simply scales with parameters to account for interactions at various volume fractions.
Henderson et al. (2002): high-speed observations of two interacting colloids. Sonic effects.
Padding et al. (2006) coarse-graining approach; Langevin does a poor job.

3. Payam has (equation 12) $$v_{\rm DNA}=F_{\rm ext} \ln(\ell/d)/\eta\ell - \lambda E\ln(1+r_{\rm D}/d)/\eta$$, which we use to calculate $$t_{\rm D}$$. He finds $$t_{\rm D}\sim \ell\eta/E\lambda$$ – whence this simplification?

4. Is the elongated jet derivation valid? (Since integration along length of the DNA may break L&L’s “weak momentum injection” assumption.)
Yes it is valid – this is actually addressed on pp. 2-3: the criterion for the submerged jet formulae is the same as the L&L criterion for linear fluid velocity dependence.

5. Is there anything to be done about the discrepancy between experiment and observations in the DNA capture paper?

• The onset of the diffusion-limited regime occurs later than predicted when varying the DNA length (figure 4a). But when varying the potential difference, the crossover happens earlier (figure 4b).

• The approximations and order-of-magnitude guesses in the paper mean that we have no reason to expect high-fidelity predictions. They actually do surprisingly well.

• But there might be a question of internal consistency: does the fact that there are two discrepancies (figures 4a and 4b) in “different” directions point to some problem with the theory? I should read the paper more thoroughly and try to understand the significance of the different crossover regions.

• I could improve the work by putting error bars on the predictions – perhaps this will boost faith in the results, or point to some hitherto unaccounted-for issue.

6. Is there anything to learn from the inevitable energy dissipation as the DNA moves through the viscous fluid? Could predict this from hydrodynamic considerations. How much work is done by the electric field? Biological implications?

7. Where does the nonequilibrium stuff happen? There is no mention of linear response or anything like that.