Abstract In this work, we study a flat Friedmann–Robertson–Walker universe filled with dark matter and viscous new holographic dark energy. We present four possible solutions of the model depending on the choice of the viscous term. We obtain the evolution of the cosmological quantities such as scale factor, deceleration parameter and transition redshift to observe the effect of viscosity in the evolution. We also emphasis upon the two independent geometrical diagnostics for our model, namely the statefinder and the Om diagnostics. In the first case we study new holographic dark energy model without viscous and obtain power-law expansion of the universe which gives constant deceleration parameter and statefinder parameters. In the limit of the parameter, the model approaches to $$Lambda CDM$$ ΛCDM model. In new holographic dark energy model with viscous, the bulk viscous coefficient is assumed as $$zeta =zeta _{0}+zeta _{1}H$$ ζ=ζ0+ζ1H , where $$zeta _{0}$$ ζ0 and $$zeta _{1}$$ ζ1 are constants, and H is the Hubble parameter. In this model, we obtain all possible solutions with viscous term and analyze the expansion history of the universe. We draw the evolution graphs of the scale factor and deceleration parameter. It is observed that the universe transits from deceleration to acceleration for small values of $$zeta $$ ζ in late time. However, it accelerates very fast from the beginning for large values of $$zeta $$ ζ . By illustrating the evolutionary trajectories in $$r-s$$ r-s and $$r-q$$ r-q planes, we find that our model behaves as an quintessence like for small values of viscous coefficient and a Chaplygin gas like for large values of bulk viscous coefficient at early stage. However, model has close resemblance to that of the $$Lambda CDM$$ ΛCDM cosmology in late time. The $$ Om$$ Om has positive and negative curvatures for phantom and quintessence models, respectively depending on $$zeta $$ ζ . Our study shows that the bulk viscosity plays very important role in the expansion history of the universe.
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