 NSFPAR ID:
 10437368
 Date Published:
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 952
 ISSN:
 00221120
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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This paper presents a theory to obtain the force per unit length acting on a slender filament with a noncircular crosssection moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has $O(1/\ln (2A))+O(\unicode[STIX]{x1D6FC}/\ln ^{2}(2A))$ contributions driven by the relative motion of the particle and the local fluid velocity and an $O(\unicode[STIX]{x1D6FC}/(\ln (2A)A))$ contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ( $A=l/a_{0}$ ) is defined as the ratio of the particle size ( $l$ ) to the crosssectional dimension ( $a_{0}$ ) and $\unicode[STIX]{x1D6FC}$ is the amplitude of the noncircular perturbation. Using thought experiments, we show that twolobed and threelobed crosssections affect the response to relative motion and velocity gradients, respectively. A twodimensional Stokes flow calculation is used to extend the perturbation analysis to crosssections that deviate significantly from a circle (i.e. $\unicode[STIX]{x1D6FC}\sim O(1)$ ). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasiperiodically if they have twolobed crosssection, and rotate chaotically and translate diffusively if they have a combination of two and threelobed crosssections. Finally, we show the remarkable ability of our theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios ( ${\sim}10$ ) and crosssections that deviate significantly from a circle, thereby making our theory a computationally inexpensive alternative to other Stokes flow solvers.more » « less

Dynamic organization of the cytoskeletal filaments and rodlike proteins in the cell membrane and other biological interfaces occurs in many cellular processes. Previous modeling studies have considered the dynamics of a single rod on fluid planar membranes. We extend these studies to the more physiologically relevant case of a single filament moving in a spherical membrane. Specifically, we use a slenderbody formulation to compute the translational and rotational resistance of a single filament of length L moving in a membrane of radius R and 2D viscosity ηm, and surrounded on its interior and exterior with Newtonian fluids of viscosities η− and η+. We first discuss the case where the filament's curvature is at its minimum κ=1/R. We show that the boundedness of spherical geometry gives rise to flow confinement effects that increase in strength with increasing the ratio of filament's length to membrane radius L/R. These confinement flows only result in a mild increase in filament's resistance along its axis, ξ∥, and its rotational resistance, ξΩ. As a result, our predictions of ξ∥ and ξΩ can be quantitatively mapped to the results on a planar membrane. In contrast, we find that the drag in perpendicular direction, ξ⊥, increases superlinearly with the filament's length, when L/R>1 and ultimately ξ⊥→∞ as L/R→π. Next, we consider the effect of the filament's curvature, κ, on its parallel motion, while fixing the membrane's radius. We show that the flow around the filament becomes increasingly more asymmetric with increasing its curvature. These flow asymmetries induce a net torque on the filament, coupling its parallel and rotational dynamics. This coupling becomes stronger with increasing L/R and κ.more » « less

Semiflexible slender filaments are ubiquitous in nature and cell biology, including in the cytoskeleton, where reorganization of actin filaments allows the cell to move and divide. Most methods for simulating semiflexible inextensible fibers/polymers are based on discrete (beadlink or bloblink) models, which become prohibitively expensive in the slender limit when hydrodynamics is accounted for. In this paper, we develop a novel coarsegrained approach for simulating fluctuating slender filaments with hydrodynamic interactions. Our approach is tailored to relatively stiff fibers whose persistence length is comparable to or larger than their length and is based on three major contributions. First, we discretize the filament centerline using a coarse nonuniform Chebyshev grid, on which we formulate a discrete constrained Gibbs–Boltzmann (GB) equilibrium distribution and overdamped Langevin equation for the evolution of unitlength tangent vectors. Second, we define the hydrodynamic mobility at each point on the filament as an integral of the Rotne–Prager–Yamakawa kernel along the centerline and apply a spectrally accurate “slenderbody” quadrature to accurately resolve the hydrodynamics. Third, we propose a novel midpoint temporal integrator, which can correctly capture the Ito drift terms that arise in the overdamped Langevin equation. For two separate examples, we verify that the equilibrium distribution for the Chebyshev grid is a good approximation of the bloblink one and that our temporal integrator for overdamped Langevin dynamics samples the equilibrium GB distribution for sufficiently small time step sizes. We also study the dynamics of relaxation of an initially straight filament and find that as few as 12 Chebyshev nodes provide a good approximation to the dynamics while allowing a time step size two orders of magnitude larger than a resolved bloblink simulation. We conclude by applying our approach to a suspension of crosslinked semiflexible fibers (neglecting hydrodynamic interactions between fibers), where we study how semiflexible fluctuations affect bundling dynamics. We find that semiflexible filaments bundle faster than rigid filaments even when the persistence length is large, but show that semiflexible bending fluctuations only further accelerate agglomeration when the persistence length and fiber length are of the same order.more » « less

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