In this paper, we describe a novel approach to determine the flow behavior index of a power-law fluid by means of a microfluidic device. The concept of this method is based on a mathematical analysis by Aronsson and Janfalk [Eur. J. Appl. Math. 3, 343-366 (1992)] of Hele-Shaw flow of power-law liquids. We implement this approach by driving a non-Newtonian fluid through a glass microfluidic chip with a 100: 1 contraction. The flow in this chip satisfies the Hele-Shaw flow conditions in most of the device. Two conjugate p-Laplace equations describe the pressure and stream function in such flows. These equations depend on the flow behavior index, n. Therefore, by fitting the p-Laplace equation to the velocity field obtained from a micro particle image velocimetry measurement of the flow, the flow behavior index of the fluid in the chip can be determined. Because in practice, fluids rarely show perfectly inelastic power-law behavior, conditions under which the assumption of inelastic flow is valid were derived by analyzing Hele-Shaw flow of an Oldroyd-B fluid. The concept was tested using three different classes of model fluids, a Newtonian fluid, an inelastic power-law fluid, and a Boger fluid. In all three cases, satisfactory results were obtained, with values of n deviating at most 4% from values measured using conventional rheometry. The method presented here is expected to be potentially useful in online quality control in, for example, polymer or food processing. (C) 2013 The Society of Rheology.