More Like this

1. Nonlinear electrophoretic velocity of a spherical colloidal particle

   https://doi.org/10.1017/jfm.2023.537  

   Cobos, Richard; Khair, Aditya S. (August 2023, Journal of Fluid Mechanics)

   Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength$$\beta =a^*e^*E_\infty ^*/k_B^*T^*$$, defined as the ratio of the product of the applied electric field magnitude$$E_\infty ^*$$and particle radius$$a^*$$, to the thermal voltage$$k_B^*T^*/e^*$$, where$$k_B^*$$is Boltzmann's constant,$$T^*$$is the absolute temperature, and$$e^*$$is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density$$\sigma$$over a wide range of$$\beta$$. Here,$$\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$$, where$$\sigma ^*$$is the dimensional surface charge density, and$$\epsilon ^*$$is the permittivity of the electrolyte. For moderately charged particles ($$\sigma ={O}(1)$$), the electrophoretic velocity is linear in$$\beta$$when$$\beta \ll 1$$, and its dependence on the ratio of the Debye length ($$1/\kappa ^*$$) to particle radius (denoted by$$\delta =1/(\kappa ^*a^*)$$) agrees with Henry's formula. As$$\beta$$increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is$$\delta$$-dependent. For$$\beta \gg 1$$, the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all$$\delta$$. For highly charged particles ($$\sigma \gg 1$$) in the thin-Debye-layer limit ($$\delta \ll 1$$), our computations are in good agreement with recent experimental and asymptotic results. 

   more » « less
2. Stable laws for random dynamical systems

   https://doi.org/10.1017/etds.2024.5  

   AIMINO, ROMAIN; NICOL, MATTHEW; TÖRÖK, ANDREW (February 2024, Ergodic Theory and Dynamical Systems)

   Abstract In this paper, we consider random dynamical systems formed by concatenating maps acting on the unit interval$$[0,1]$$in an independent and identically distributed (i.i.d.) fashion. Considered as a stationary Markov process, the random dynamical system possesses a unique stationary measure$$\nu $$. We consider a class of non-square-integrable observables$$\phi $$, mostly of form$$\phi (x)=d(x,x_0)^{-{1}/{\alpha }}$$, where$$x_0$$is a non-recurrent point (in particular a non-periodic point) satisfying some other genericity conditions and, more generally, regularly varying observables with index$$\alpha \in (0,2)$$. The two types of maps we concatenate are a class of piecewise$$C^2$$expanding maps and a class of intermittent maps possessing an indifferent fixed point at the origin. Under conditions on the dynamics and$$\alpha $$, we establish Poisson limit laws, convergence of scaled Birkhoff sums to a stable limit law, and functional stable limit laws in both the annealed and quenched case. The scaling constants for the limit laws for almost every quenched realization are the same as those of the annealed case and determined by$$\nu $$. This is in contrast to the scalings in quenched central limit theorems where the centering constants depend in a critical way upon the realization and are not the same for almost every realization. 

   more » « less
3. On the dynamics of aligned inertial particles settling in a quiescent, stratified two-layer medium

   https://doi.org/10.1017/jfm.2024.462  

   Kang, Soohyeon; Best, James L; Chamorro, Leonardo P (June 2024, Journal of Fluid Mechanics)

   We explored the settling dynamics of vertically aligned particles in a quiescent, stratified two-layer fluid using particle tracking velocimetry. Glass spheres of$$d=4\,{\rm mm}$$diameter were released at frequencies of 4, 6 and 8 Hz near the free surface, traversing through an upper ethanol layer ($$H_1$$), whereHis height or layer thickess, varying from$$10d$$to$$40d$$and a lower oil layer. Results reveal pronounced lateral particle motion in the ethanol layer, attributed to a higher Galileo number ($$Ga = 976$$, ratio of buoyancy–gravity to viscous effects), compared with the less active behaviour in the oil layer ($$Ga = 16$$). The ensemble vertical velocity of particles exhibited a minimum just past the density interface, becoming more pronounced with increasing$$H_1$$, and suggesting that enhanced entrainment from ethanol to oil resulted in an additional buoyancy force. This produced distinct patterns of particle acceleration near the density interface, which were marked by significant deceleration, indicating substantial resistance to particle motion. An increased drag coefficient occurred for$$H_1/d = 40$$compared with a single particle settling in oil; drag reduced as the particle-release frequency ($$\,f_p$$) increased, likely due to enhanced particle interactions at closer proximity. Particle pair dispersions, lateral ($$R^2_L$$) and vertical ($$R^2_z$$), were modulated by$$H_1$$, initial separation$$r_0$$and$$f_p$$. The$$R^2_L$$dispersion displayed ballistic scaling initially, Taylor scaling for$$r_0 < H_1$$and Richardson scaling for$$r_0 > H_1$$. In contrast,$$R^2_z$$followed a$$R^2_z \sim t^{5.5}$$scaling under$$r_0 < H_1$$. Both$$R^2_L$$and$$R^2_z$$plateaued at a distance from the interface, depending on$$H_1$$and$$f_p$$. 

   more » « less
4. On stability of rainbow matchings

   https://doi.org/10.1017/S096354832510031X  

   Lu, Hongliang; Wang, Yan; Yu, Xingxing (March 2026, Combinatorics, Probability and Computing)

   Abstract We show that for any integer$$k\ge 1$$there exists an integer$$t_0(k)$$such that, for integers$$t, k_1, \ldots , k_{t+1}, n$$with$$t\gt t_0(k)$$,$$\max \{k_1, \ldots , k_{t+1}\}\le k$$, and$$n \gt 2k(t+1)$$, the following holds: If$$F_i$$is a$$k_i$$-uniform hypergraph with vertex set$$[n]$$and more than$$ \binom{n}{k_i}-\binom{n-t}{k_i} - \binom{n-t-k}{k_i-1} + 1$$edges for all$$i \in [t+1]$$, then either$$\{F_1,\ldots , F_{t+1}\}$$admits a rainbow matching of size$$t+1$$or there exists$$W\in \binom{[n]}{t}$$such that$$W$$intersects$$F_i$$for all$$i\in [t+1]$$. This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every$$t$$and$$n \gt 2k^3t$$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings. 

   more » « less
5. Large deformation of elastic capsules under uniaxial extensional flow

   https://doi.org/10.1017/jfm.2025.10216  

   Yariv, Ehud; Howell, Peter D; Stone, Howard A (June 2025, Journal of Fluid Mechanics)

   A spherical capsule (radius$$R$$) is suspended in a viscous liquid (viscosity$$\mu$$) and exposed to a uniaxial extensional flow of strain rate$$E$$. The elasticity of the membrane surrounding the capsule is described by the Skalak constitutive law, expressed in terms of a surface shear modulus$$G$$and an area dilatation modulus$$K$$. Dimensional arguments imply that the slenderness$$\epsilon$$of the deformed capsule depends only upon$$K/G$$and the elastic capillary number$${Ca}=\mu R E/G$$. We address the coupled flow–deformation problem in the limit of strong flow,$${Ca}\gg 1$$, where large deformation allows for the use of approximation methods in the limit$$\epsilon \ll 1$$. The key conceptual challenge, encountered at the very formulation of the problem, is in describing the Lagrangian mapping from the spherical reference state in a manner compatible with hydrodynamic slender-body formulation. Scaling analysis reveals that$$\epsilon$$is proportional to$${Ca}^{-2/3}$$, with the hydrodynamic problem introducing a dependence of the proportionality prefactor upon$$\ln \epsilon$$. Going beyond scaling arguments, we employ asymptotic methods to obtain a reduced formulation, consisting of a differential equation governing a mapping field and an integral equation governing the axial tension distribution. The leading-order deformation is independent of the ratio$$K/G$$; in particular, we find the approximation$$\epsilon ^{2/3} {Ca}\approx 0.2753\ln (2/\epsilon ^2)$$for the relation between$$\epsilon$$and$$Ca$$. A scaling analysis for the neo-Hookean constitutive law reveals the impossibility of a steady slender shape, in agreement with existing numerical simulations. More generally, the present asymptotic paradigm allows us to rigorously discriminate between strain-softening and strain-hardening models. 

   more » « less