11 problems

Let the periodic channel be ${\mathbb T}\times{\mathbb R}$, and consider the linearised Euler--Boussinesq system near the constantly stratified Couette flow … Here…

Let $alpha$ and $c$ denote the parameters in the dispersion relation for the Orr–Sommerfeld equation, and suppose that both are of order $nu^{1/4}$. It turns out that this is an ar…

Let $s$ denote the steepness of a Stokes wave. A modulational instability branch is a spectral instability branch associated with the Benjamin–Feir instability, while a localized i…

Let $s$ denote the steepness of a Stokes wave, let $u$ be the Floquet parameter, and consider the localized instability branch. The modes with $u=0$ and $u=1/2$ are respectively…

Let $\alpha$ and $c$ be parameters in the dispersion relation for the Orr–Sommerfeld equation, and suppose that both are of order $\nu^{1/4}$. Most unstable instabilities conjectur…

Let $\gamma>0$ denote the characteristic length scale of the shear flow, and consider the transient growth rate described by Gill for a shear flow of the form $\big(b(y),0\big)$. G…

Energy-maximization conjecture. The maximum $m$ is attained among the functions of ${\cal S}_0$.

Trefethen et al.'s conjecture. Transition to turbulence of eigenvalue-stable shear flows should be driven by an essentially linear destabilizing mechanism, with amplitude threshold

Kelvin's conjecture. The Poiseuille flow is stable, but its stability threshold decreases as $uto0$.

Generic shear-flow instability conjecture. Generically, shear flows for the Navier–Stokes equations are linearly unstable, and the Prandtl expansion is not valid in Sobolev spaces.

A three-dimensional shear flow has the form $(U(y),0,W(y))$, where $U$ and $W$ are the streamwise and spanwise velocity profiles. A type I 3D shear is a shear of the type classifie…