still produces the streaming instability.

Fundamental two-fluid equations and local stability analysis
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Let us model the dust as a collection of identical, indestructible spheres, of mass density $\rho_p$ and velocity $\mathbf{V}_p$. The gas density and velocity, on the other hand, we denote by $\rho_g$ and $\mathbf{V}_g$, respectively. Their evolution equations are (see Appendix A for justification of the dust equations): $$\frac{\partial \rho_p}{\partial t} + \nabla\cdot(\rho_p\mathbf{V}_p)=0
\label{continuity-dust}$$ $$\frac{\partial \rho_g}{\partial t} + \nabla\cdot(\rho_g\mathbf{V}_g)=0
\label{continuity-gas}$$ $$\mathcal{D}_p\mathbf{V}_p=-\Omega^2 \mathbf{R} -
\frac{\mathbf{V}_p-\mathbf{V}_g}{t_{\mathrm{stop}}}
\label{Euler-dust}$$ $$\mathcal{D}_g\mathbf{V}_g=-\Omega^2 \mathbf{R}
+\frac{\rho_p}{\rho_g}\frac{\mathbf{V}_p-\mathbf{V}_g}{t_{\mathrm{stop}}}-\frac{\nabla
P}{\rho_g},
\label{Euler-gas}$$ where $\Omega$ is the Keplerian angular velocity, $\mathcal{D}_{p,g}\equiv\partial /\partial t+\mathbf{V}_{p,g}\cdot\nabla$ the particle/gas Lagrangian derivatives, and $\mathbf{R}$ the cylindrical vector radius. We ignore the vertical component of the solar gravity, as well as self-gravity.

The particle stopping time $t_{\rm stop}$ depends on the size and velocity regime [e.g. @Weidenschilling1977]. For example, for particles that are both small compared to the gas mean-free-path and drifting subsonically relative to the gas (as appropriate, e.g., for chondrule-sized bodies at a few AUs in a Minimum Mass Solar Nebula), Epstein’s law [@Epstein1924] applies: $$t_{\mathrm{stop}}=\frac{\rho_s a}{\rho_g v_T},$$ with $v_T=\sqrt{8k_BT/\pi m}$ (with $m$ the molecular mass and $T$ the temperature), roughly the sound speed, $\rho_s$ the *internal* density of the grains (not to be 