The Navier-Stokes equation conceals the paradox of instantaneous propagation in the same way as the classical heat equation. Hence, it is no longer valid for modelling phenomena of ultra-fast or start-up fluid flow. In this paper, the paradox of instantaneous propagation in the Navier-Stokes equation is addressed and a modified version of the equation to resolve the paradox is proposed. It follows that, in addition to viscous dissipation, the existence of momentum waves and apparent acoustic sources is essential for resolving the paradox of instantaneous propagation in fluid flow.