Flow birefringence measurement in a radial Hele-Shaw cell considering three-dimensional effects

Abstract

Flow birefringence measurement is an emerging technique for visualizing stress fields in fluid flows. This study investigates flow birefringence in the steady radial Hele-Shaw flow. In the radial Hele-Shaw flow, stress is dominant along the gap direction, challenging the applicability of the conventional stress-optic law (SOL) with measurement from the gap direction. To overcome this problem, we used two types of flow birefringence measurement using radial Hele-Shaw cell and rheometer. We conduct flow birefringence measurements at various flow rates and compare the results with theoretical predictions. The observed phase retardation cannot be quantitatively explained using the conventional SOL, but is successfully described using the second-order SOL, which accounts for stress along the optical direction. The stress-optic coefficient in the second-order SOL was obtained by rheo-optical measurements. This study demonstrates that the combination of the second-order SOL and rheo-optical measurements is essential for an accurate interpretation of flow birefringence in Hele-Shaw flow, providing a noninvasive approach for stress field analysis in high-aspect-ratio geometries.

Figures (9)

• Figure 1: Configuration and coordinates in radial Hele-Shaw flow
• Figure 2: (a)Experimental setup using a Hele-Shaw cell. (b) Example of raw image. The white dotted circle has $r$ = 35 mm, the radius at the furthest interface from the center of inlet is defined as $r_\mathrm{max}$ and the radius at the nearest interface from the center of inlet is defined as $r_\mathrm{min}$. The white bar indicates a scale of 10 mm.
• Figure 3: Shear viscosity measured by rheometer at ${20}^\circ$C; measurements were repeated three times for each condition. Blue circles: No sonication; Yellow triangles: Sonication for 200 s; Red squares: Sonication for 400 s. Red solid line is the weighted average value for 10$<\dot{\gamma}<$700.
• Figure 4: (a) Experimental setup for rheo-optical measurements, consisting of high-speed polarization camera and a rheometer with a glass parallel plate and stage. $T$ is torque, $\omega$ is angular velocity. Plate diameter is $2R_o$ = 43.016 mm. (b) Example of raw image and (c) diagram of a quarter area of phase retardation distribution for $\omega$ = 11.2 rad/s. Region of interest (ROI) is shown. The phase retardation in the ROI was used to calculate $C_2$ (Eq. \ref{['eq:C2_calc']}).
• Figure 5: (a) Phase retardation measured using rheo-optical setup (Fig. \ref{['fig:rheometer']}) for various shear rates. (b) Phase retardation $\Delta$ and $C_2$ as a function of shear rate $\dot{\gamma}$. Blue and purple plots are measured $\Delta$ and yellow and red plots are $C_2$ calculated using Eq. \ref{['eq:C2_calc']}. Different plot colors indicate results with different shear rate settings in the rheometer. Black solid line is the fitting curve given by Eq. \ref{['eq:C2_fit']}.