Quintessence dark energy model in non-linear $f(Q)$ theory with bulk-viscosity

In this study, we investigate a locally rotationally symmetric (LRS) Bianchi type-I cosmological model in non-linear form of $f(Q)$ gravity with observational constraints. We solved the modified Einstein's field equations with a viscous fluid source and got a hyperbolic solution. First, we apply MCMC analysis to the cosmic chronometer (CC), Baryon Acoustic Oscillation (BAO) and Pantheon datasets to place observational constraints on the model parameters. Using constrained values of model parameters, we study the behavior of cosmological parameters, such as the Hubble parameter $H$, the deceleration parameter $q$, and the equation of state (EoS) parameter $\omega_{v}$ with the skewness parameter $\delta_{v}$ for the viscous fluid. In addition, we perform the Om diagnostics and statefinder analysis to categorize dark energy models. Also, we studied cosmographic series coefficients to explore the whole evolution of the derived universe model. We estimated the current age of the universe as $t_{0}\approx13.8$ Gyrs. We obtained a quintessential and ever-accelerating model with bulk viscosity fluid.