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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 …

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 …

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 …

These are two different integrals. To see they are different, you could for example take $\textbf{F}(\textbf{r})=\textbf{r}$. Then the curl of $\textbf{F}$ vanishes, so the integral on the right-hand- …

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_ …

Two-dimensional plane-periodic Couette flow is non-turbulent for long times at any Reynolds number. The mechanism by which laminar flow is recovered is analyzed in Enhanced dissipation and inviscid da …

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 …

[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 …

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 …

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 …

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 …

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 …

If you take the curl of your equation for $v$ (after the correction $iv\mapsto\partial v/\partial t$), you find $$\nabla\times(\partial v/\partial t-s^{-2}\nabla^2 v)=0.$$ This partial differential …

$\bullet$ 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 …

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 …