We consider the problem of reducing the dimensionality of labeled data for classification. Unfortunately, the optimal approach of finding the low-dimensional projection with minimal Bayes classification error is intractable, so most standard algorithms optimize a tractable heuristic function in the projected subspace. Here, we investigate a physics-based model where we consider the labeled data as interacting fluid distributions. We derive the forces arising in the fluids from information theoretic potential functions, and consider appropriate low rank constraints on the resulting acceleration and velocity flow fields. We show how to apply the Gauss principle of least constraint in fluids to obtain tractable solutions for low rank projections. Our fluid dynamic approach is demonstrated to better approximate the Bayes optimal solution on Gaussian systems, including infinite dimensional Gaussian processes.