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Abstract:
Three-dimensional dynamics of flexible fibers in shear flow are studied numerically, with a qualitative comparison to experiments. Initially, the fibers are straight, with different orientations with respect to the flow. By changing the rotation speed of a shear rheometer, we change the ratio A of bending to shear forces. We observe fibers in the flow-vorticity plane, which gives insight into the motion out of the shear plane. The numerical simulations of moderately flexible fibers show that they rotate along effective Jeffery orbits, and therefore the fiber orientation rapidly becomes very close to the flow-vorticity plane, on average close to the flow direction, and the fiber remains in an almost straight configuration for a long time. This ``ordering'' of fibers is temporary since they alternately bend and straighten out while tumbling. We observe numerically and experimentally that if the fibers are initially in the compressional region of the shear flow, they can undergo a compressional buckling, with a pronounced deformation of shape along their whole length during a short time, which is in contrast to the typical local bending that originates over a long time from the fiber ends. We identify differences between local and compressional bending and discuss their competition, which depends on the initial orientation of the fiber and the bending stiffness ratio A. There are two main finding. First, the compressional buckling is limited to a certain small range of the initial orientations, excluding those from the flow-vorticity plane. Second, since fibers straighten out in the flow-vorticity plane while tumbling, the compressional buckling is transient - it does not appear for times longer than 1/4 of the Jeffery period. For larger times, bending of fibers is always driven by their ends.

Keywords:
Stokes flow, flexible fibers, bending, buckling, orientational order

Abstract:
We present a numerical study of the dynamics of an elastic fibre in a shear flow at low Reynolds number, and seek to understand several aspects of the fibre's motion using the equations for slender-body theory coupled to the elastica. The numerical simulations are performed in the bead-spring framework including hydrodynamic interactions in two theoretical schemes: the generalized Rotne-Prager-Yamakawa model and a multipole expansion corrected for lubrication forces. In general, the two schemes yield similar results, including for the dominant scaling features of the shape that we identify. In particular, we focus on the evolution of an initially straight fibre oriented in the flow direction and show that the time scales of fibre bending, curling and rotation, which depend on its length and stiffness, determine the overall motion and evolution of the shapes. We document several characteristic time scales and curvatures representative of the shape that vary as power laws of the bending stiffness and fibre length. The numerical results are further supported by an interpretation using an elastica model.

Keywords:
Stokesian dynamics, particle/fluid flow, slender-body theory

Abstract:
The charging of electrical double layers inside a cylindrical pore has applications to supercapacitors, batteries, desalination and biosensors. The charging dynamics in the limit of thin double layers, i.e., when the double layer thickness is much smaller than the pore radius, is commonly described using an effective RC transmission line circuit. Here, we perform direct numerical simulations (DNS) of the Poisson-Nernst-Planck equations to study the double layer charging for the scenario of overlapping double layers, i.e., when the double layer thickness is comparable to the pore radius. We develop an analytical model that accurately predicts the results of DNS. Also, we construct a modified effective circuit for the overlapping double layer limit, and find that the modified circuit is identical to the RC transmission line but with different values and physical interpretation of the capacitive and resistive elements. In particular, the effective surface potential is reduced, the capacitor represents a volumetric current source, and the charging timescale is weakly dependent on the ratio of the pore radius and the double layer thickness.

Abstract:
Two oppositely charged surfaces separated by a dielectric medium attract each other. In contrast we observe a strong repulsion between two plates of a capacitor that is filled with an aqueous electrolyte upon application of an alternating potential difference between the plates. This long-range force increases with the ratio of diffusion coefficients of the ions in the medium and reaches a steady state after a few minutes, which is much larger than the millisecond timescale of diffusion across the narrow gap. The repulsive force, an order of magnitude stronger than the electrostatic attraction observed in the same setup in air, results from the increase in osmotic pressure as a consequence of the field-induced excess of cations and anions due to lateral transport from adjacent reservoirs.

Abstract:
The three-dimensional dynamics of a single non-Brownian flexible fiber in shear flow is evaluated numerically, in the absence of inertia. A wide range of ratios A of bending to hydrodynamic forces and hundreds of initial configurations are considered. We demonstrate that flexible fibers in shear flow exhibit much more complicated evolution patterns than in the case of extensional flow, where transitions to higher-order modes of characteristic shapes are observed when A exceeds consecutive threshold values. In shear flow, we identify the existence of an attracting steady configuration and different attracting periodic motions that are approached by long-lasting rolling, tumbling, and meandering dynamical modes, respectively. We demonstrate that the final stages of the rolling and tumbling modes are effective Jeffery orbits, with the constant parameter C replaced by an exponential function that either decays or increases in time, respectively, corresponding to a systematic drift of the trajectories. In the limit of C→0, the fiber aligns with the vorticity direction and in the limit of C→∞, the fiber periodically tumbles within the shear plane. For moderate values of A, a three-dimensional meandering periodic motion exists, which corresponds to intermediate values of C. Transient, close to periodic oscillations are also detected in the early stages of the modes.

Abstract:
We use numerical simulations of a bead–spring model chain to investigate the evolution of the conformations of long and flexible elastic fibers in a steady shear flow. In particular, for rather open initial configurations, and by varying a dimensionless elastic parameter, we identify two distinct conformational modes with different final size, shape, and orientation. Through further analysis we identify slipknots in the chain. Finally, we provide examples of initial configurations of an 'open' trefoil knot that the flow unknots and then knots again, sometimes repeating several times.

Keywords:
knots, low Reynolds number flows, multipole method

Keywords:
Stokes equations, Euler-Bernoulli beam, elastica, elastic filament, shear flow, buckling

Keywords:
fluid dynamics, Stokes flows, particulate flows, hydrodynamic interactions, bending

Abstract:
Dynamics of a single non-Brownian flexible fiber in shear flow at low Reynolds number is investigated numerically. Initially, the fiber is straight and at the equilibrium. For different initial orientations and values of bending stiffness, three generic scenarios are observed: the fiber tends to: align along the vorticity direction, tumble within the plane perpendicular to vorticity, or perform a periodic motion superposed with translation along the flow.

Keywords:
Stokes equations, flexible fibers, shear flow