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Markovian fluid flow models are used extensively in performance analysis of communication networks. They are also instances of Markov reward models that find applications in several areas like storage theory, insurance risk and financial models, and inventory control. This paper deals with the transient (time dependent) analysis of such models. Given a Markovian fluid flow, we construct on the same probability space a sequence of queues that are stochastically coupled to the fluid flow in the sense that at certain selected random epochs, the distribution of the fluid level and the phase (the state of the modulating Markov chain) is identical to that of the work in the queue and the phase. The fluid flow is realized as a stochastic process limit of the processes of work in the system for the queues, and the latter are analyzed using the matrix-geometric method. These in turn provide the needed characterization of transient results for the fluid model