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We study varying forms of viscous dark matter and try to address theintriguing tensions of the standard model of cosmology with recent cosmologicaldata, including the Hubble and $S_8$ tensions. We note that by assuming thedark matter viscosity depends on the Hubble parameter, dark matter density, orboth, one can improve the statistics. Although the models tend to aggravate theHubble tension a bit, they tend to reduce the $S_8$ tension, even in comparisonwith the constant viscosity case. Since similar to viscosity massive neutrinossuppress the power spectrum of matter on small length scales, considering themalong with the viscous dark matter, we find that the neutrino mass range istightened.

Dark Matter Cosmology with Varying Viscosity: a Possible Resolution to the $S_8$ Tension