Possible dark states could be induced after derivations of the entrainment of matter induced by a surface wave propagating along the flexible vacuum-matter boundary by considering the nonlinear coupling between the interface and the rarefaction effect. The nonrelativistic limit of the relativistic Navier-Stokes equations was considered and analytically solved by a perturbation approach. The critical reflux values associated with the product of the second-order body forcing and the Reynolds number (representing the viscous dissipations) decrease as the Knudsen number (representing the rarefaction measure) increases from zero to 0.1. We obtained the critical bounds for possible dark states corresponding to specific Reynolds numbers (ratio of wave inertia and viscous dissipation effects) and wave numbers which might be linked to the dissipative evolution of certain large-scale structure during the relativistic heavy-ion collisions.
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