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

We study the shadow of five-dimensional black strings immersed in dark matter environment. We show the influence of the momentum characteristic parameter $\widetilde{P_{y}}$ of the extra compact dimension on the shadow radius, and we provide the constraint range $0~\leq~\widetilde{P_{y}}~\lesssim~0.171215$ in the static Schwarzschild black string solution based on the observed data. Furthermore, we give the effective range $2.16138~\text{mm}~\lesssim~\ell~<~2.6~\text{mm}$ for the extra length $\ell$ and $\mathrm{k}~\lesssim~0.07398$ for the compactness parameter $\mathrm{k}$ in the environment near a black hole. We also find that the momentum characteristic parameter $\widetilde{P_{y}}$ of extra dimension will affect the effective range of dark matter related parameters when using shadow radius as a constraint tool.