The profiles of (a) radial transformation f(r), (b) velocity of background flow u(r), (c) mass density ρ, (d) speed of sound c(r), (e) pressure difference ΔP(r) = P − P₀ in comparison with the pressure P₀ at infinite, (f) constitutive relation between ρ and ΔP, (g) external potential U(r), (h) external mass source M(r), where the red, green and blue curves correspond to the transformations given in Eq. (12) with n = 2, n = 3, and n = 4 respectively. The quantity written in the square bracket beside each axis gives the unit of the corresponding coordinate axis.