Different bending models predict different dynamics of sedimenting elastic trumbbells

Abstract

The main goal of this paper is to examine theoretically and numerically the impact of a chosen bending model on the dynamics of elastic filaments settling in a viscous fluid under gravity at low-Reynolds-number. We use the bead-spring approximation of a filament and the Rotne-Prager mobility matrix to describe hydrodynamic interactions between the beads. We analyze the dynamics of trumbbells, for which bending angles are typically larger than for thin and long filaments. Each trumbbell is made of three beads connected by springs and it exhibits a bending resistance, described by the harmonic or - alternatively - by the 'cosine' (also called the Kratky-Porod) bending models, both often used in the literature. Using the harmonic bending potential, and coupling it to the spring potential by the Young's modulus, we find simple benchmark solutions: stable stationary configurations of a single elastic trumbbell and attraction of two elastic trumbbells towards a periodic long-lasting orbit. As the most significant result of this paper, we show that for very elastic trumbbells at the same initial conditions, the Kratky-Porod bending potential can lead to qualitatively and quantitatively different spurious dynamics, with artificially large bending angles and unrealistic shapes. We point out that for the bead models of an elastic filament, the range of applicability of the Kratky-Porod model might not go beyond bending angles smaller than π/2 for touching beads and beyond an even much lower value for beads well-separated from each other. The existence of stable stationary configurations of elastic trumbbells and a family of periodic oscillations of two elastic trumbbells are very important findings on their own.

Flexible fibers in shear flow approach attracting periodic solutions

“…Flexibility leads to complex dynamics with buckling [11], coil-stretch transitions [2,[12][13][14], migration across the streamlines of a flow [15][16][17][18][19][20][21], knotting [22][23][24][25] and a variety of deformed shapes [26][27][28][29][30][31]. Similar complexity of dynamics has been observed for flexible filaments in electro kinetic fields [32] or sedimenting due to gravity [33][34][35][36][37][38][39][40].…”

Sedimenting pairs of elastic microfilaments

“…In the vertical configuration the particles continue to rotate and next drift away from each other, tending towards a horizontal configuration again. Such tumbling dynamics was described before for quasi-2D periodic sedimentation of 2 pairs of point-particles, 48 2 rigid particles, 34,40,41 2 flexible trumbbells, 45 and for the non-periodic case of 2 elastic dumbbells. 49 Moreover, such vertical motions have been also generalized for 3D periodic solutions of many-particle systems.…”

“…In summary, the model of elastic filaments presented here is the same as in our previous work. 45 It is based on the harmonic elastic potential, similarly to work by Gruziel et al 46 Unlike in the latter reference, here the stiffness coefficients parameters k and A are coupled and very short range repulsion forces are absent. This elastic model is also similar to that in work by Schlagberger and Netz, 21 Gauger and Stark 47 and Marchetti et al 25 but with a harmonic bending potential instead of the 'cosine' one.…”

“…Our choice is motivated by possible spurious effects of the 'cosine' potential in some circumstances. 45…”

“…‡ Such a system may be considered as a single filament near a free surface located at the symmetry plane. 45 that the particles in such a configuration will neither approach each other, nor repel: their configuration remains constant, and their only motion is the rotation around the long rod axis combined with vertical settlement.…”