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We propose a new model for the viscosity of cosmic matters, which can be applied to different epochs of the universe. Using this model, we include the bulk viscosities as practical corrections to the perfect fluid models of the baryonic and dark matters since the material fluids in the real world may have viscosities due to thermodynamics. Such inclusion is put to the test within the framework of $f(T)$ gravity that is proved to be successful in describing the cosmic acceleration, where $T$ denotes the torsion scalar. We perform an observational fit to our model and constrain the cosmological and model parameters by using various latest cosmological datasets. Based on the fitting result, we discuss several cosmological implications including the dissipation of matters, the evolutionary history of the universe, $f(T)$ modification as an effective dark energy, and the Hubble tension problem. The corresponding findings are (i) The late time dissipation will make the density parameters of the matters vanish in the finite future. Moreover, the density ratio between the baryonic and dark matters will change over time. (ii) The radiation dominating era, matter dominating era and the accelerating era can be recovered and the model can successfully describe the known history of the universe. (iii) The $f(T)$ modification is the main drive of the acceleration expansion and currently mimics a phantom-like dark energy. But the universe will eventually enter a de Sitter expansion phase. (iv) The Hubble tension between local and global observations can be significantly alleviated in our model.