Employing shadow radius to constrain extra dimensions in black string space-time with dark matter halo

We investigate the shadow of five-dimensional black strings immersed in a dark matter environment. In particular, we demonstrate the influence of the momentum characteristic parameter $\widetilde{P_y}$ associated with the compact extra dimension on the shadow radius. Based on the observational data, we provide the constraint range of $0\le \widetilde{P_y}\lesssim 0.1712$ for the static Schwarzschild black string solution with a confidence level of $97.5\%$. Furthermore, we derive the valid range $2.16138\text{mm}\lesssim \ell <2.6\text{mm}$ for the length of the compact extra dimension $\ell$, and the valid range of $k\lesssim 0.07398$ for the compactness parameter $k$ in the vicinity of a black hole, with confidence levels of $84\%$ and $97.5\%$, respectively. Our results also reveal that the parameter $\widetilde{P_y}$ significantly influences the effective range of dark matter related parameters when the shadow radius is employed as a constraint tool.