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104 votes
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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 ...
Carlo Beenakker's user avatar
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: 2051–2054 where they use gauge theory to describe micro-swimming. Because the ...
Richard Montgomery's user avatar
13 votes
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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 ...
Carlo Beenakker's user avatar
10 votes
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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....
Denis Serre's user avatar
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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.
Denis Serre's user avatar
  • 51.9k
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 ...
Carlo Beenakker's user avatar
6 votes
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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}...
Willie Wong's user avatar
  • 38.7k
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: ...
Tobias Diez's user avatar
  • 5,632
6 votes
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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 ...
Willie Wong's user avatar
  • 38.7k
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 ...
Willie Wong's user avatar
  • 38.7k
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 ...
Carlo Beenakker's user avatar
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 ...
Carlo Beenakker's user avatar
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 ...
Carlo Beenakker's user avatar
5 votes
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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 ...
Hollis Williams's user avatar
4 votes
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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 ...
Carlo Beenakker's user avatar
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 ...
Carlo Beenakker's user avatar
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 ...
Sudharsan Madhavan's user avatar
4 votes

Book about fluid dynamics

Perhaps this CFD crash course and also A crash course in fluid mechanics could help a bit.
just-learning's user avatar
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{...
Bazin's user avatar
  • 15.7k
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.
user378654's user avatar
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)\...
Denis Denisov's user avatar
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 $H^{n-1}(M)=0$ ...
Igor Khavkine's user avatar
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
Paul's user avatar
  • 76
3 votes
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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 ...
Carlo Beenakker's user avatar
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 Navier–Stokes equation, including the nonlinear drift term, has been proven by Menaldi and Sritharan, Stochastic 2-D ...
Carlo Beenakker's user avatar
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 ...
Jean Duchon's user avatar
  • 3,065
2 votes
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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}...
Marko Rajkovic's user avatar
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.
Michael Renardy's user avatar
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 ...
Denis Serre's user avatar
  • 51.9k
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^...
Jean Duchon's user avatar
  • 3,065

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