Viscosity bound for anisotropic superfluids with dark matter sector

The shear viscosity to the entropy density ratio $\eta /s$ of the anisotropic superfluid has been calculated by means of the gauge/gravity duality in the presence of the dark matter sector. The dark matter has been described by the Yang-Mills field analogous to the one describing the visible matter sector, and it is assumed to interact with the visible field with coupling constant $\alpha$. Close to the superfluid transition temperature ($T_c$), the analytical solution has been given up to the leading order in a symmetry breaking parameter and the ratio of the gravitational constant and Yang-Mils coupling. The tensor element of ratio $\eta /s$ remains unaffected by the dark matter for the viscosity tensor in the plane perpendicular to the symmetry breaking direction (here, $yz$). The temperature dependence and the linear correction in ($1-\alpha$) in the plane containing this direction (here, $xy$) was also revealed. The correction linearly vanishes for temperature tending to the critical one $T\to T_c$.