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The rheology of suspensions of rings (tori) rotating in an unbounded low Reynolds number simple shear flow is calculated using numerical simulations at dilute particle number densities ( n ≪ 1 ). Suspensions of nonBrownian rings are studied by computing pair interactions that include hydrodynamic interactions modeled using slender body theory and particle collisions modeled using a shortrange repulsive force. Particle contact and hydrodynamic interactions were found to have comparable influences on the steadystate Jeffery orbit distribution. The average tilt of the ring away from the flowvorticity plane increased during pairwise interactions compared to the tilt associated with Jeffery rotation and the steadystate orbit distribution. Particle stresses associated with the increased tilt during the interaction were found to be comparable to the stresses induced directly by particle contact forces and the hydrodynamic velocity disturbances of other particles. The hydrodynamic diffusivity coefficients in the gradient and vorticity directions were also obtained and were found to be two orders of magnitude larger than the corresponding values in fiber suspensions at the same particle concentrations. Rotary Brownian dynamics simulations of isolated Brownian rings were used to understand the shear rate dependence of suspension rheology. The orbit distribution observed in the regime of weak Brownian motion, P e ≫ ϕ T − 3, was surprisingly similar to that obtained from pairwise interaction calculations of nonBrownian rings. Here, the Peclet number P e is the ratio of the shear rate and the rotary diffusivity of the particle and ϕ T is the effective inverseaspect ratio of the particle (approximately equal to 2 π times the inverse of its nondimensional Jeffery time period). Thus, the rheology results obtained from pairwise interactions should retain accuracy even for weakly Brownian rings ( n ≪ 1 and ϕ T − 3 ≪ P e ).more » « lessFree, publiclyaccessible full text available April 14, 2024

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Free, publiclyaccessible full text available January 1, 2024

null (Ed.)The collisions in a dilute polydisperse suspension of subKolmogorov spheres with negligible inertia settling in a turbulent flow and interacting through hydrodynamics including continuum breakdown on close approach are studied. A statistically significant decrease in ideal collision rate without gravity is resolved via a Lagrangian stochastic velocitygradient model at Taylor microscale Reynolds number larger than those accessible by current direct numerical simulation capabilities. This arises from the difference between the mean inward velocity and the rootmeansquare particle relative velocity. Differential sedimentation, comparable to the turbulent shear relative velocity, but minimally influencing the sampling of the velocity gradient, diminishes the Reynolds number dependence and enhances the ideal collision rate i.e. the rate without interactions. The collision rate is retarded by hydrodynamic interactions between sphere pairs and is governed by noncontinuum lubrication as well as full continuum hydrodynamic interactions at larger separations. The collision efficiency (ratio of actual to ideal collision rate) depends on the relative strength of differential sedimentation and turbulent shear, the size ratio of the interacting spheres and the Knudsen number (defined as the ratio of the meanfree path of the gas to the mean radius of the interacting spheres). We develop an analytical approximation to concisely report computed results across the parameter space. This accurate closed form expression could be a critical component in computing the evolution of the size distribution in applications such as water droplets in clouds or commercially valuable products in industrial aggregators.more » « less

null (Ed.)Collisions in a dilute polydisperse suspension of spheres of negligible inertia interacting through noncontinuum hydrodynamics and settling in a slow uniaxial compressional flow are studied. The ideal collision rate is evaluated as a function of the relative strength of gravity and uniaxial compressional flow and it deviates significantly from a linear superposition of these driving terms. This nontrivial behaviour is exacerbated by interparticle interactions based on uniformly valid noncontinuum hydrodynamics, that capture noncontinuum lubrication at small separations and full continuum hydrodynamic interactions at larger separations, retarding collisions driven purely by sedimentation significantly more than those driven purely by the linear flow. While the ideal collision rate is weakly dependent on the orientation of gravity with the axis of compression, the rate including hydrodynamic interactions varies by more than $100\,\%$ with orientation. This dramatic shift can be attributed to complex trajectories driven by interparticle interactions that prevent particle pairs from colliding or enable a circuitous path to collision. These and other important features of the collision process are studied in detail using trajectory analysis at near unity and significantly smaller than unity size ratios of the interacting spheres. For each case analysis is carried for a large range of relative strengths and orientations of gravity to the uniaxial compressional flow, and Knudsen numbers (ratio of mean free path of the media to mean radius).more » « less

We analyse the electrophoresis of a weakly charged particle with a thin double layer in a dilute polymer solution. The particle velocity in polymer solutions modelled with different constitutive equations is calculated using a regular perturbation in the polymer concentration and the generalized reciprocal theorem. The analysis shows that the polymer is strongly stretched in two regions, the birefringent strand and the highshear region inside the double layer. The electrophoretic velocity of the particle always decreases with the addition of polymers due to both increased viscosity and fluid elasticity. At a small Weissenberg number ( $Wi$ ), which is the product of the polymer relaxation time and the shear rate, the polymers inside the double layer contribute to most of the velocity reduction by increasing the fluid viscosity. With increasing $Wi$ , viscoelasticity decreases and shear thinning increases the particle velocity. Polymer elasticity alters the fluid velocity disturbance outside the double layer from that of a neutral squirmer to a pullertype squirmer. At high $Wi$ , the strong extensional stress inside the birefringent strand downstream of the particle dominates the velocity reduction. The scaling of the birefringent strand is used to estimate the particle velocity.more » « less

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

The plasma membranes of cells are thin viscous sheets in which some transmembrane proteins have twodimensional mobility and some are immobilized. Previous studies have shown that immobile proteins retard the shorttime diffusivity of mobile particles through hydrodynamic interactions and that steric effects of immobile proteins reduce the longtime diffusivity in a model that neglects hydrodynamic interactions. We present a rigorous derivation of the longtime diffusivity of a single mobile protein interacting hydrodynamically and thermodynamically with an array of immobile proteins subject to periodic boundary conditions. This method is based on a finite element method (FEM) solution of the probability density of the mobile protein diffusing with a positiondependent mobility determined through a multipole solution of Stokes equations. The simulated longtime diffusivity in square arrays decreases as the spacing in the array approaches the particle size in a manner consistent with a lubrication analysis. In random arrays, steric effects lead to a percolation threshold volume fraction above which longtime diffusion is arrested. The FEM/multipole approach is used to compute the longtime diffusivity far away from this threshold. An approximate analysis of mobile protein diffusion through a network of pores connected by bonds with resistances determined by the FEM/multipole calculations is then used to explore higher immobile area fractions and to evaluate the finite simulation cell size scaling behaviour of diffusion near the percolation threshold. Surprisingly, the ratio of the longtime diffusivity to the spatially averaged shorttime diffusivity in these twodimensional fixed arrays is higher in the presence of hydrodynamic interactions than in their absence. Finally, the implications of this work are discussed, including the possibility of using the methods developed here to investigate more complex diffusive phenomena observed in cell membranes.more » « less