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I'm probably missing something stupid. In the paper mention above (see link, pg. 4) the hydrodynamical instability of the disk is reviewed with Navier-Stokes equation.

Having the unperturbed velocity field in cylindrical coordinates $\mathbf{u_0}=u_0(R)\mathbf{\hat{\phi}}$, and the small perturbation $\mathbf{u}$ in $\mathbf{\hat{R}}$ and $\mathbf{\hat{\phi}}$ directions, one can write the linearized Navier-Stokes equation (neglecting the perturbations of the density and gravitational potential) as: $$ \frac{\partial \mathbf{u}}{\partial t}+\left(\mathbf{u_0}\cdot\nabla\right)\mathbf{u}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u_0}. $$ It's good so far. Now breaking this equation into components, one can write:

$\mathbf{\hat{R}}$ direction

$$ \frac{\partial u_R}{\partial t}+\frac{u_0}{R}\partial_\phi u_R=0 $$

$\mathbf{\hat{\phi}}$ direction $$ \frac{\partial u_\phi}{\partial t}+\frac{u_0}{R}\partial_\phi u_\phi+u_ru_0'=0 $$

The problem is, that in the original paper the authors get extra terms:

$\mathbf{\hat{R}}$ direction

$$ \frac{\partial u_R}{\partial t}+\frac{u_0}{R}\partial_\phi u_R\mathbin{\color{red}{-2\frac{u_0}{R}u_\phi}}=0 $$

$\mathbf{\hat{\phi}}$ direction $$ \frac{\partial u_\phi}{\partial t}+\frac{u_0}{R}\partial_\phi u_\phi+u_ru_0'\mathbin{\color{red}{+\frac{u_0}{R}u_R}}=0 $$

I should be missing something. Can you, please, point me on that?

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Use the expression for $(\vec{A}\cdot\vec{\nabla})\vec{B}$ in cylindrical coordinates as given here - https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates with $\rho\equiv R$. In case of $R$-component, $-2u_0u_\phi/R$ will come from the last term $A_\phi B_\phi/\rho$ as given in the above link.

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  • $\begingroup$ Thanks, I missed this row in Wiki. This term most likely comes out nontrivially from the coordinate transformation. $\endgroup$ – Hayk Hakobyan Sep 1 '16 at 19:18

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