# Using deformable particles for single-particle measurements of velocity gradient tensors

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### Abstract

We measure the deformation of particles made of several slender arms in a two-dimensional (2D) linear shear and a three-dimensional (3D) turbulent flow. We show how these measurements of arm deformations along with the rotation rate of the particle allow us to extract the velocity gradient tensor of the flow. The particles used in the experiments have three symmetric arms in a plane (triads) and are fabricated using 3D printing of a flexible polymeric material. Deformation measurements of a particle free to rotate about a fixed axis in a 2D simple shear flow are used to validate our model relating particle deformations to the fluid strain. We then examine deformable particles in a 3D turbulent flow created by a jet array in a vertical water tunnel. Particle orientations and deformations are measured with high precision using four high-speed cameras and have an uncertainty on the order of \(10^{-4}\) radians. Measured deformations in 3D turbulence are small and only slightly larger than our orientation measurement uncertainty. Simulation results for triads in turbulence show deformations similar to the experimental observations. Deformable particles offer a promising method for measuring the full local velocity gradient tensor from measurements of a single particle where traditionally a high concentration of tracer particles would be required.

### Graphic abstract

## Notes

### Acknowledgements

This work was supported by NSF Grant DMR-1508575, and Army Research Office Grants W911NF-15-1-0205 and W911NF-17-1-0176. We benefited from discussions with Stefan Kramel and Lee Walsh on software development and data analysis.

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