  In this paper, we study the $f(T,\mathcal{T})$ gravity model in the presence
of the bulk viscosity by the flat-FRW metric. The field equation is obtained by
teleparallel gravity with tetrad field. The universe components are considered
as matter and dark energy which the dark energy component associates from the
viscous $f(T,\mathcal{T})$ gravity. After calculating the Friedmann equations,
we obtain the energy density, the pressure and the EoS of dark energy in terms
of the redshift parameter. Afterward, we plot the corresponding cosmological
parameters versus the redshift parameter and examine the accelerated expansion
of the universe. In the end, we explore the system stability by a function
called the speed sound parameter.
