Newtonian hydrodynamics in the comoving gauge

Einstein's gravity has the exact Newtonian limit as we take the infinite-speed-of-light limit in the zero-shear gauge and the uniform-expansion gauge. Although lacking proper Newtonian gravitational potential, here we show that Newtonian hydrodynamics for density and velocity is also recovered in the comoving gauge using the weak gravity and negligible pressure limits but without using the slow-motion and subhorizon conditions. This curious correspondence, however, is available only for the irrotational fluid.