#### Non-spherical particles in shear flow

Most solid particles we encounter in Nature are not spherical. Therefore it is necessary to derive and analyse models that account for non-spherical particle shapes, and that describe how these particles translate and rotate in the fluid.

Consider the tumbling of small rods in a steady channel flow. When fluid-inertia effects a negligible, the problem is degenerate: there are infinitely many different periodic solutions, the Jeffery orbits.

Using symmetry considerations, computer simulations, and experiments, we analysed how particle shape affects the angular dynamics (Figure 1) [3,4,6,10].

Any perturbation is expected to break the degeneracy of the angular dynamics, causing the particles to align in characteristic ways. We computed how the fluid inertia affects the angular dynamics of small disks and rods. Taking into account the symmetry of the problem from the outset allowed us to compute the stability exponents of the orbits, and to analyse their bifurcations [5,7,8,9]. This is important for the rheology of dilute suspensions of non-spherical particles.

**Figure 1. **Tumbling of a tiny triangular particle in a micro-channel flow. From Ref. [3]. See Ref. [4] for a theoretical analysis.

**Figure 2.** Bifurcation diagram for angular dynamics of small particles in a shear, as a function of Reynolds number (*y*-axis) and particle shape (*x*-axis). Our theoretical results are shown as solid lines, separating different dynamical regimes. Direct-numerical simulation results are shown as symbols. From Ref. [7], see this article for details.

[1] Time-dependent lift and drag on a rigid body in a viscous steady linear flow

F Candelier, B Mehlig, J Magnaudet, Journal of Fluid Mechanics 864 (2019) 554--595

[2] Intrinsic viscosity of a suspension of weakly Brownian ellipsoids in shear

G Almondo, J Einarsson, JR Angilella & B Mehlig, Physical Review Fluids 3 (2018) 064307

[3] Spinning and tumbling of micron-sized triangles in a micro-channel shear flow

J Fries, MV Kumar, BM Mihiretie, D Hanstorp & B Mehlig, Physics of Fluids 30 (2018), 033304

[4] Angular dynamics of small crystals in viscous flow

J Fries, J Einarsson & B Mehlig, Physical Review Fluids 2 (2017) 014302

[5] Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number

J Meibohm, F Candelier, T Rosen, J Einarsson, F Lundell & B Mehlig, Physical Review Fluids 1 (2016) 084203

[6] Tumbling of asymmetric microrods in a microchannel flow

J Einarsson, BM Mihiretie, A Laas, S Ankardal, JR Angilella, D Hanstorp & B Mehlig, Physics of Fluids 28 (2016), 013302

[7] Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers

T Rosén, J Einarsson, A Nordmark, CK Aidun, F Lundell & B Mehlig, Physical Review E 92 (2015) 063022

[8] Rotation of a spheroid in a simple shear at small Reynolds number

J Einarsson, F Candelier, F Lundell, JR Angilella & B Mehlig, Physics of Fluids 27 (2015), 063301

[9] Effect of weak fluid inertia upon Jeffery orbits

J Einarsson, F Candelier, F Lundell, JR Angilella & B Mehlig, Physical Review E 91 (2015) 041002

[10] Periodic and aperiodic tumbling of microrods advected in a microchannel flow

J Einarsson, A Johansson, SK Mahato, YN Mishra, JR Angilella, D Hanstorp & B Mehlig, Acta Mechanica 224 (2013) 2281-2289