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Cato edited Payam and DNA.tex
almost 11 years ago
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\item So, {\it what should the time-scale be from }physical {\it reasoning: what is the physics behind establishing the flow profile $\vec{v}(\vec{x})$ in the fluid around the DNA/jet?}\\
\citet{1959flme.book.....L} 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 {\it moving} submerged jet?
\item {\it Does Payam assume incompressible fluid?} Yes -- he follows \citet{1959flme.book.....L} 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 \citet{Broman_Rudenko_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.
\item {\it 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.\\ 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
{\it inconsistent} inconsistent and crude, but probably easier than method 1.\\
In any case, a model with
sound ``sound'' propagation might be a more accurate model (see references below). It may also be more illuminating as to how the velocity profile is
set up, established, and the
nature consequences of interactions between neighbouring DNA molecules.
\item {\it Which process (sound
waves or
vorticity) vorticity diffusion) carries more momentum?} Read \citet{1959flme.book.....L} chapter 8 (p. 245), on sound. Specifically: how
does is a sound wave
get 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.
\item Another thing to consider: {\it 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 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.
\item
{\it Could the problems in Payam's other paper conceivably be explained by this phenomenon?
}
\item Some reading about propagation of hydrodynamic information:\\
\citet{Hinch_1975} application of Langevin equation to fluid suspensions; includes fluid inertia and interaction with suspension.\\