Viscoelastic fingering during the sedimentation of a sphere

The motion of particles in complex fluids is ubiquitous in biological and physical processes. The most popular toy model for understanding the physics of such systems is the settling of a solid sphere in a viscoelastic fluid. There is a general agreement that an elastic wake develops downstream of the sphere, causing the breakage of fore-and-aft symmetry, while the flow remains axisymmetric, independent of fluid viscoelasticity and flow conditions. Using a continuum mechanics model, we reveal that axisymmetry holds only for weak viscoelastic flows. Beyond a critical value of the settling velocity, steady, nonaxisymmetric disturbances develop peripherally of the sphere rear pole, giving rise to a 4-lobed fingering instability. The transition to nonaxisymmetric flow is characterized by a regular bifurcation and solely depends on the interplay between shear and extensional properties of the viscoelastic fluid under different flow regimes. At higher settling velocities, each lobe tip is split into two new lobes, resembling fractal fingering in interfacial flows. For the first time, we capture a fingering instability under steady-state conditions and provide the missing information for understanding and predicting such instabilities in the response of soft media.