= Bulk-viscous limit of a material-density-damped condensate
{title2=$\partial_t\mathbf v_s+\tfrac12\nabla|\mathbf v_s|^2=-\nabla\rho+\zeta\nabla\nabla\cdot[\rho(\mathbf v_s-\mathbf v_n)]+O(\epsilon^3)$}
Write $\psi=\sqrt\rho e^{i\theta}$ away from zeros, and $\mathbf v_s=\nabla\theta$. Its <Madelung equations> give $\rho_t+\nabla\cdot(\rho\mathbf v_s)=0$ and $\theta_t+|\mathbf v_s|^2/2=1-\rho+\nabla^2\sqrt\rho/(2\sqrt\rho)-\zeta D_n\rho$. For an incompressible <normal fluid>, $D_n\rho=-\nabla\cdot[\rho(\mathbf v_s-\mathbf v_n)]$. Taking a gradient proves the displayed equation when each space-time derivative of the slowly varying amplitudes is $O(\epsilon)$: the <quantum potential> force is $O(\epsilon^3)$, while the bulk term is $O(\epsilon^2)$.
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