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 $0\le \widetilde{P_y}\lesssim 0.171215$ in the static Schwarzschild black string solution based on the observed data. Furthermore, we give the effective range $2.16138\text{ mm }\lesssim \mathrm{\ell }<2.6\text{ mm}$ for the extra length $\mathrm{\ell }$ 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.