This is not a problem that cannot be easily generalized. A normal mode analysis in the context of linear stability of a flow may give quantitatively different results depending on the approximation made. For example, the linear stability of plane Poiseuille flow subject to 2-D disturbances gives the correct stability result (Squires Theorem tells us that the most dangerous mode is 2-D not 3-D). But in other flows Squire's theorem may not hold, and a 3-D disturbance may be the most dangerous mode.
Flow in the annulus between two cylinders, in which the inner cylinder is rotating is unstable at a critical speed of rotation. GI Taylor analyzed this problem using linear stability theory to show that a new steady flow with vortices exists when the rotation speed exceeds critical value. This is a classic example of flow bifurcation, when the length of the cylinders is assumed to be infinite in extent. But when the cylinders are taken to be finite in extent such that end effects matter, one has what is known as an imperfect bifurcation, and the solution space changes dramatically. So in this case the 2-D flow and 3-D flow give quite different results.
Normal mode analysis can also be applied to steady flows. A nice example where one can compare say a 2-D base flow with a 1-D base flow is in the analysis of the shape of a thin liquid film downstream of a slot coater. The thickness of the liquid film approaches its asymptotic uniform thickness exponentially. A 2-D normal mode analysis predicts the rate of exponential decay which is good agreement with a 1-D lubrication model when capillary effects dominate.
See my paper on "Downstream Development of 2-D viscocapillary Flows" - see the discussion section in that paper
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