104
votes

Accepted

### Should water at the scale of a cell feel more like tar?

There is a beautiful article (a write-up of a talk, actually), by E.M. Purcell, Life at low Reynolds number, that explains how bacteria swim.
Low Reynolds number is the technical way to phrase the ...

18
votes

### Should water at the scale of a cell feel more like tar?

You may be interested in Shapere, A., and F. Wilczek. 1987. Self-propulsion at low Reynolds number. Phys. Rev. Lett. 58: 2051â€“2054 where they use gauge theory to describe micro-swimming. Because the ...

13
votes

Accepted

### Riemann, fluid dynamics, and critical lines

Q: Does anyone know of a reference which discusses more thoroughly the critical line appearing in Riemann's hydrodynamics problem?
A: A recent reference is Elliptical instability in hot Jupiter ...

10
votes

Accepted

### Explanation for why an ideal fluid doesn't have increasing entropy?

This is a very important issue, to which an answer must be made in mathematical terms, rather than by waving hands.
Yes, the Euler system (conservation of mass, momentum and energy) is time-reversible....

7
votes

### Compressible Ebin-Marsden?

I have worked for decades on compressible gas dynamics, and I am not aware of such a followup by Ebin & Marsden. But I think you will find something in the book of Arnol'd & Khesin.

6
votes

### Navier-Stokes fluid dynamics, Einstein gravity and holography

The first point to make is that the fluid/gravity correspondence relates the general theory of relativity to relativistic fluid dynamics. I don't see how the usual non-relativistic Navier-Stokes ...

6
votes

Accepted

### A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global?

$\dot{H}^{1/2}$ is critical with respect to scaling.
Let $\tilde{u}(t,x) = \lambda u(\lambda^2 t, \lambda x)$. Then $\tilde{u}$ solves the Navier-Stokes equation up to $\tilde{T}^* = T^* \lambda^{-2}...

6
votes

### Compressible Ebin-Marsden?

The compressible case uses semidirect products of groups (group of diffeomorphisms times functions).
To my knowledge, the first paper that discusses this in detail is
Marsden, Ratiu, Weinstein: ...

6
votes

Accepted

### Quasilinear wave equations without (weak) null conditions and conjectures

What you've found is basically "survivor bias", so it helps for me to describe a bit where the null conditions came about.
Assertion 1: Quasilinear partial differential equations, in ...

5
votes

### Showing nonlinearity of PDEs arising from physics by mathematical argument alone

The "very few" include at least
Einstein's equation in general relativity, derivable from a Lagrangian formulation by looking for the critical point of the Einstein-Hilbert functional.
Any place ...

5
votes

### Explanation for why an ideal fluid doesn't have increasing entropy?

Q: Explanation for why an ideal fluid doesn't have increasing entropy?
A: The entropy will in fact increase for the most probable initial conditions.
The question in the OP refers to the socalled ...

5
votes

### Why are solenoidal fields called solenoidal?

[To expand on Wojowu's comment.]
Q: "Why the description of a divergence-free field as solenoidal? I expect that this name had historical origins but its unlikely that it was so named without ...

5
votes

### Textbook suggestions for rigorous fluid dynamics

An older, classic text is Mathematical Theory of Compressible Fluid Flow by Richard von Mises.
More recent text books include
Introduction to Mathematical Fluid Dynamics by R.E. Meyer.
An ...

5
votes

Accepted

### Textbook suggestions for rigorous fluid dynamics

A few possibilities:
JC Robinson, JL Rodrigo, & W Sadowski (2016) Classical theory of the three-dimensional Navier-Stokes equations.
OA Ladyzhenskaya (1963) The mathematical theory of viscous ...

4
votes

Accepted

### Incompressible Navier-Stokes equation with heat conduction

There is an extensive literature, this could be helpful entry point:
Solving Navier-Stokes equations coupled with a heat transfer equation (2015)
In this paper, the dynamics of an incompressible ...

4
votes

### References on thin film equation: derivation and properties

$\bullet$ Physical model: There is no physical model that gives this equation for arbitrary $m$; the values $m=1,2,3$ appear in viscous flow, as summarized in "Viscous Thin Films": For the no-slip ...

4
votes

### Book about fluid dynamics

Since this is for a job interview, I would focus on the more applied side of computational fluid dynamics (CFD). I really liked CFD Python: 12 steps to Navier-Stokes
A brief description of this course ...

4
votes

### Book about fluid dynamics

Perhaps this CFD crash course and also A crash course in fluid mechanics could help a bit.

4
votes

Accepted

### Vorticity equation for incompressible 2D fluid dynamics

The vorticity equation for the Euler equation in 3D is, with $\omega=\text{curl } v$,
$$
\dot\omega + (v\cdot\nabla)\omega-(\omega\cdot\nabla)v=0,
$$
so that if $v$ is two-dimensional, i.e.
$
v=\begin{...

4
votes

### Textbook suggestions for rigorous fluid dynamics

Constantin, Foias, Navier-Stokes Equations (1988).
This classic is not a textbook exactly, but gives a lot of detail and should be pretty readable.

3
votes

### A question about intuition of fluid limit in queuing system

For fixed $t<1/\lambda$ it seems that intuition B is correct as $N \to \infty $. In this case, using the large deviations, one can see that it is highly unlikely that more than $(N+\varepsilon)\...

3
votes

Accepted

### Compactly supported transverse traceless tensors

The answer is Yes, at least under the reasonable conditions that (i) the number of conformal Killing vectors locally admitted by $(M,g)$ is constant and that (ii) the de Rham cohomology $H^{n-1}(M)=0$ ...

3
votes

Accepted

### Generalising results on superfluid Kubo formulas

Not quite sure what you are asking, but Shukla and Kovtun have provided all Kubo formulas for non-dissipative transport coefficients here

3
votes

Accepted

### Fluid dynamics textbook discussing Hele-Shaw flow

A mathematics-oriented text book is Conformal and Potential Analysis in Hele-Shaw Cells, by Gustafsson and Vasil'ev (2006).
This monograph aims at giving a presentation of recent and new ideas
that ...

2
votes

### Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation

For $d=2$ the existence and uniqueness of strong solutions for the stochastic Navierâ€“Stokes equation, including the nonlinear drift term, has been proven by Menaldi and Sritharan, Stochastic 2-D ...

2
votes

### Reformulation of the classical Navier-Stokes equation as a semilinear evolution equation and corresponding mild solutions

You are right, $(4)$ has to be understood in the special sense that the semigroup $S(t)$ (and the (Leray?) $L^2$-projection operator to $H$, as well) extend to a wider space, that of "distributional ...

2
votes

Accepted

### Hadamard-Rybczynski problem

Yes, I partially missunderstood Batchelor, leading myself by some other literature. Spherical form of drop is due to conitnuity of normal stresses but in form $-p_{abs}+2\mu \sigma_{rr}$ where $p_{abs}...

2
votes

### Steady Euler flows with compact support?

According to the following paper, it is an open problem whether such solutions exist:
N. Nadirashvili, Liouville theorem for Beltrami flow, Geometric and Functional Analysis 24 (2014), 916-921.

2
votes

### Navier-Stokes fluid dynamics, Einstein gravity and holography

This is a bit too long for being a comment.
An important remark is that the NS equation is parabolic and therefore the velocity at which information propagates is unbounded. For instance, if the ...

2
votes

### Stationary Navier-Stokes solutions

Tai-Peng Tsai's book Lectures on Navier-Stokes Equations (2018) cites as Theorem 8.3 (p.149) a theorem of V. Sverak (2011) that excludes the existence of minus one homogeneous solutions on $\mathbb R^...

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