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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 sta …

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 system …

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 equat …

[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 some lin …

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 Introductio …

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 irr …

$\bullet$ 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 bou …

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 …

I'll make an attempt at providing the steps you are seeking to go "from Euler equation to Bessel function". You start from the Euler equation, describing conservation of momentum, $$\rho\frac{\parti …

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 a …

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 Navie …

The equation in the OP is not correct, it should read $$\overline{u_iu_j} = \overline{(\bar{u_i}+u_i')(\bar{u_j} + u_j')} = \overline{\bar{u_i}\bar{u_j}+\bar{u_i}u_j'+u_i'\bar{u_j}+u_i'u_j'} = \bar{u_ …

Use that the velocity field is incompressible, $\nabla\cdot u=0$, to rewrite $$(u\cdot\nabla)u_j=\sum_{i} \nabla_i (u_iu_j).$$ You seek the curl of the curl of this expression, use that $$[\operatorna …

in the fluid (or continuum) limit, you should allow $k$ to vary continuously and consider instead of $\pi_k$ (the probability of a queue of exactly length $k$) the probability $\pi(k)dk$ that a queue …

for what it's worth, here is one rather recent & pessimistic assessment: Gian-Carlo Rota wrote that "The Reynolds operators are the potentials, in the language of probabilistic potential theory, …