The evolution of fluids is known (eg via experiments and numerical studies) to occur on multiple timescales. We discuss how this can be analyzed rigorously in two fundamental models of fluids: the 1D Burgers equation and the 2D Navier-Stokes equation. For Burgers equation, we provide a complete geometric explanation involving invariant manifolds in the phase space of the evolution. For Navier-Stokes on the 2D torus, we discuss two complementary approaches. The first involves the theory of hypocoercive operators, and the second involves invariant manifolds and geometric singular perturbation theory in the Fourier phase space.