Temporal networks, defined as sequences of time-aggregated adjacency matrices, sample latent graph dynamics and trace trajectories in graph space. By interpreting each adjacency matrix as a different time snapshot of a scalar field, we show how fluid-mechanics methods can be applied to construct two distinct eigendecompositions of temporal networks. The first builds on the proper orthogonal decomposition (POD) of flowfields and decomposes the evolution of a network in terms of a basis of orthogonal network eigenmodes which are ordered in terms of their relative importance, hence enabling compression of temporal networks as well as their projection in low-dimensional embeddings. The second proposes a numerical approximation of the Koopman operator, a linear operator acting on a suitable observable of the graph space which provides the best linear approximation of the latent graph dynamics. Its eigendecomposition provides a data-driven spectral description of the temporal network dynamical stability, in terms of dynamic modes which grow, decay or oscillate over time. We illustrate and validate the application of both eigendecompositions in a suite of synthetic generative models of temporal networks with varying complexity.