Geometry of Hele-Shaw flow, the in-plane dimensions are much larger than the depth of the flow, i.e., .
Streamlines, i.e., lines of constant ψ, of flow in a 90° salient corner, following from an exact solution to the 2D p-Laplace equations, Eq. (5) , units on the axes are arbitrary. The positive x- and y-axes represent (semi-infinite) solid walls, the flow is from top left to bottom right. Left: n = 0.2 (shear-thinning), middle: n = 1 (Newtonian), right: n = 1.8 (shear-thickening).
Left: Dimensions of the flow cell with 100:1 contraction (not to scale, dimensions in mm). Right: Schematic drawing of measurement setup.
Streamlines resulting from PIV measurements, superimposed on an image of the Hele-Shaw cell (flow is from right to left). Top: Glycerol, 20 μl/min, middle: Xanthan gum solution, 20 μl/min, bottom: PEG-PEO Boger fluid, 5 μl/min. The part of the flow where the streamlines are shown is also the part that was used for the fitting procedure. The axes represent the PIV interrogation window number.
Rheological behavior of the xanthan gum and PEG-PEO solutions used as test fluid. Top: Loss and storage modulus in an oscillatory frequency sweep; bottom: viscosity in steady shear.
Contraction flow of a liquid crystalline solution of 10 wt. % of poly-2,2′-disulfonylbenzidine-terephtalamide (PBDT) in water (sample obtained from J. Gao, TU Delft). The same flow geometry was used as in the other experiments in this paper. It was placed between two oppositely oriented circular polarizers to improve the visibility of the structure in the flow. No tracer particles are needed here to measure the velocity field and/or the streamlines.
Results of streamline fitting, n fit is the power law index resulting from the fitting procedure, n rheom is the value resulting from conventional rheometry on the samples, S is a measure of the signal-to-noise ratio of the correlation peak, with “av” indicating the average value and “crit” the percentage of the measurement points with a signal-to-noise ratio larger than the critical value of 5.6 (see Sec. III B ).
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