Observational constraints on viscous free-$γ$ fluid in $f(Q)$ gravity

We study the late-time cosmological dynamics of a spatially flat FLRW universe in the framework of $f(Q)$ gravity, where $Q$ denotes the nonmetricity scalar. The matter sector is modeled as a bulk viscous fluid with a free equation-of-state parameter $γ$, allowing for a generalized description of cosmic matter beyond the standard dust approximation. We derive the background evolution equations and analyze the resulting expansion history. The model parameters are constrained using a combination of observational datasets, including cosmic chronometers (CC), baryon acoustic oscillations from DESI DR2, and Type~Ia supernovae (GRBs and Union3). Using the best-fit parameters, we further employ the statefinder and $\mathrm{Om}(z)$ diagnostics to distinguish the viscous $f(Q)$ scenario from the standard $Λ$CDM model. In addition, we examine the evolution of the deceleration parameter, which exhibits a transition from an early decelerated phase to the current accelerated expansion, and analyze the effective equation of state behavior. Our results show that bulk viscosity within $f(Q)$ gravity provides a viable and observationally consistent description of late-time cosmic acceleration.