Finite-time future singularities models in $f(T)$ gravity and the effects of viscosity

We investigate models of future finite-time singularities in $f(T)$ theory, where $T$ is the torsion scalar. The algebraic function $f(T)$ is put as the teleparallel term $T$ plus an arbitrary function $g(T)$. A suitable expression of the Hubble parameter is assumed and constraints are imposed in order to provide an expanding universe. Two parameters $\beta$ and $H_s$ that appear in the Hubble parameter are relevant in specifying the types of singularities. Differential equations of $g(T)$ are established and solved, leading to the algebraic $f(T)$ models for each type of future finite time singularity. Moreover, we take into account the viscosity in the fluid and discuss three interesting cases: constant viscosity, viscosity proportional to $\sqrt{-T}$ and the general one where the viscosity is proportional to $(-T)^{n/2}$, where $n$ is a natural number. We see that for the first and second cases, in general, the singularities are robust against the viscous fluid, while for the general case, the Big Rip and the Big Freeze can be avoided from the effects of the viscosity for some values of $n$.