Fluid equations and different regimes of fluid flows

A fluid is a continuous medium whose state is characterised by its velocity field, u = u(x, t), pressure and density fields, p = p(x, t) and ρ = ρ(x, t) respectively, and possibly other relevant fields (e.g. temperature). Here, t is time and x ∈ Rd is the physical coordinate in the d-dimensional space, d = 1, 2 or 3. Respectively, u ∈ Rd, although in some special flows the dimensions of x and u may be different from each other.