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In this paper, we study the $f(T,\mathcal{T})$ gravity model in the presenceof the bulk viscosity by the flat-FRW metric. The field equation is obtained byteleparallel gravity with tetrad field. The universe components are consideredas matter and dark energy which the dark energy component associates from theviscous $f(T,\mathcal{T})$ gravity. After calculating the Friedmann equations,we obtain the energy density, the pressure and the EoS of dark energy in termsof the redshift parameter. Afterward, we plot the corresponding cosmologicalparameters versus the redshift parameter and examine the accelerated expansionof the universe. In the end, we explore the system stability by a functioncalled the speed sound parameter.

Observational constraints and stability in viscous $f(T,\mathcal{T})$ gravity