Physics of Fluids
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Demand is growing for a larger catalogue of experimental techniques to measure flow rates through micro-/nanoscale systems for both fundamental research and device development. Flow emerging from a hole in a plane wall is a common system of interest in such work for its relevance to membrane separation. In this paper, we consider the possibility of measuring volume flow rates through small scale orifice plates from images of dye dispersions downstream. Based on approximate analytical solutions to the advection–diffusion equation, we show that, at low Reynolds numbers, the concentration in the nearly hemispherical plume that forms increases linearly with inverse distance from the pore and that the slope is proportional to volume flow rate. From micrographs of fluorescent dye plumes taken downstream of micropores of three different diameters, we demonstrate that, at Reynolds numbers below 15, the volume flow rate can be determined by extracting this slope from fluorescence intensity images. At higher Reynolds numbers, laminar jets form. In this regime, we derive an approximate similarity solution for the concentration field and show agreement of imaged dye dispersion shapes with both analytical expressions for the streamlines and isoconcentration contours at Reynolds numbers above 25. The results validate a scalable method for flow rate measurements applicable to small micropores of any geometry in plane walls and to small areas of porous materials relevant to membrane systems.