$h_g\equiv h/f_g$.

Perturbing equations (\[continuity-dust\])-(\[Euler-gas\]) (with $\Omega =
0$), still assuming incompressibility, gives the following system: $$(-i(\omega-\mathbf{k}\cdot\mathbf{V}_p)+\frac{1}{t_{\rm stop}})\mathbf{v}_p
-\frac{1}{t_{\mathrm{stop}}}\mathbf{v}_g=0$$ $$\begin{aligned}
-\frac{\epsilon}{t_{\mathrm{stop}}}\mathbf{v}_p
+(-i(\omega-\mathbf{k}\cdot\mathbf{V}_g)+\frac{\epsilon}{t_{\rm
stop}})\mathbf{v}_g  -\nonumber\\\frac{\epsilon}{t_{\rm
stop}}\delta_p(\mathbf{V}_p-\mathbf{V}_g)+ih_g\mathbf{k}=0\end{aligned}$$ $$i\mathbf{k}\cdot\mathbf{v}_p-i\omega\delta_p=0$$ $$i\mathbf{k}\cdot\mathbf{v}_g =0$$ where $\epsilon \equiv \rho_p/\rho_g$. identity matrix. Setting the determinant of the above 8$\times$8 system to zero, the following dispersion relation results: $$\begin{aligned}
\bigg(\omega-\mathbf{k}\cdot\mathbf{V}_p
+\frac{i}{t_{stop}}\bigg)\bigg(\omega^2+(\frac{i}{f_gt_{\mathrm{stop}}}-\mathbf{k}\cdot(\mathbf{V}_p+\mathbf{V}_g))\omega\nonumber\\+(\mathbf{k}\cdot\mathbf{V}_p)(\mathbf{k}\cdot\mathbf{V}_g)-i\frac{\mathbf{k}\cdot\mathbf{V}}{f_gt_{\mathrm{stop}}}\bigg)^2=0\end{aligned}$$ The first factor corresponds to a mode where the gas’ velocity field is unperturbed, with the gas compensating the disturbance of the dust’s with variation of pressure and density. The quadratic factor (squared) has roots: $$\begin{aligned}
\omega=\frac{1}{2}\bigg(\mathbf{k}\cdot(\mathbf{V}_p+\mathbf{V}_g)+\frac{i}{f_gt_{stop}}\bigg(-1\pm\nonumber\\\sqrt{(1+i(\epsilon-1)(f_gt_{stop})^2\mathbf{k}\cdot\mathbf{g}_e)^2
-4\epsilon(\mathbf{k}\cdot\mathbf{g}_e(f_gt_{stop})^2)^2}\bigg)\bigg),\end{aligned}$$ with $\mathbf{g}_e\equiv \nabla P/\rho$. Using the lemma $\mid\!\mathrm{Re}(\sqrt{(1+ia)^2-b^2})\!\mid \leq 1$ for $(a,b)\in\mathbb{R}^2$ (YG05), their imaginary part ($s$) is always negative. For $\mathbf{k}\cdot\mathbf{g}_e(f_gt_{stop})^2 \ll 1$ (as in the terminal velocity approximation limit), these roots may be expanded as $$\mathbf{k}\cdot(f_p\mathbf{V}_g+f_g\mathbf{V}_p)-i\epsilon(\mathbf{k}\cdot\mathbf{g}_e)^2(f_gt_{\mathrm{stop}})^3
\label{first root rotationless}$$ and $$\mathbf{k}\cdot\mathbf{V}-\frac{i}{f_gt_{\mathrm{stop}}}$$ the latter corresponding to a damping where dust and fluid converge in concert toward equilibrium.

There is thus no streaming instability in the absence of rotation (as was also clear from the YG05 numerical calculations). Rotation, as well as feedback, are both essential for the streaming instability, hence the difficulty in finding a simple toy model to explain it.

Toward an interpretation of the streaming instability
=====================================================

