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Brownian microscopy is an intriguing technique that enables in situ determination of the local fluid properties and velocity fields in microfluidic devices. We report application of Brownian microscopy to steady laminar microchannel flows and demonstrate accurate measurements of the viscosity of glycerin/water mixtures, as well as the velocity profiles. A theoretical model is developed to evaluate statistical errors in the measured Brownian diffusivity while accounting for complications associated with the electronic noise and finite exposure time of an imaging system. The model is validated using both Monte Carlo simulations and experiments.