In typical, three-dimensional context, classical fluid flow is described by 3D velocity vector, fluid density and pressure function. This is all required information.

These functions are obtained from conservation laws, in some sense. Conservation of mass gives rise to continuity equation, from which, knowing the velocity vector, we can, in principle, determine the density. Euler's equation, being simply second Newton's law, hence "preservation of momentum", allows us to derive Bernoulli's law (preservation of energy), therefore finding the pressure function. All above is done assuming that we already know velocity.

That's it for three dimensions, however analogous question in higher-dimensional-framework seems to me at least puzzling, since everywhere I looked for an answer, I've found only familiar three-dimensional setting.

My core question is that in three dimensions I need five functions in order to fully describe fluid, so in n (arbitrary) dimensions I need n+2 quantites(scalar fdunctions)? Or is this a simplistic view and this number n+2 becomes more ccomplicated function Z(n)?