Bulk-viscous limit of a material-density-damped condensate (${\partial }_t{\mathbf{v}}_s+\frac{1}{2}\nabla |{\mathbf{v}}_s{|}^2=-\nabla \rho +\zeta \nabla \nabla \cdot [\rho ({\mathbf{v}}_s-{\mathbf{v}}_n)]+O({ϵ}^3)$)

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 $O(ϵ)$: the quantum potential force is $O({ϵ}^3)$, while the bulk term is $O({ϵ}^2)$.