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1. Near-contact approach of two permeable spheres

   https://doi.org/10.1017/jfm.2021.588  

   Reboucas, Rodrigo B; Loewenberg, Michael (October 2021, Journal of Fluid Mechanics)

   An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit$$K=k/a^{2} \ll 1$$is considered, where$$k$$is the arithmetic mean permeability, and$$a^{-1}=a^{-1}_{1}+a^{-1}_{2}$$is the reduced radius, and$$a_1$$and$$a_2$$are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact$$F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$$, where$$\mu$$is the fluid viscosity,$$W$$is the relative velocity and$$\tilde {f}_c$$depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions,$$\tilde {f}_c=0.7507$$; for a permeable particle in near contact with a spherical drop,$$\tilde {f}_c$$is reduced by a factor of$$2^{-6/5}$$. 

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2. Higher uniformity of arithmetic functions in short intervals I. All intervals

   https://doi.org/10.1017/fmp.2023.28  

   Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni (January 2023, Forum of Mathematics, Pi)

   Abstract We study higher uniformity properties of the Möbius function$$\mu $$, the von Mangoldt function$$\Lambda $$, and the divisor functions$$d_k$$on short intervals$$(X,X+H]$$with$$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$$for a fixed constant$$0 \leq \theta < 1$$and any$$\varepsilon>0$$. More precisely, letting$$\Lambda ^\sharp $$and$$d_k^\sharp $$be suitable approximants of$$\Lambda $$and$$d_k$$and$$\mu ^\sharp = 0$$, we show for instance that, for any nilsequence$$F(g(n)\Gamma )$$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$ when$$\theta = 5/8$$and$$f \in \{\Lambda , \mu , d_k\}$$or$$\theta = 1/3$$and$$f = d_2$$. As a consequence, we show that the short interval Gowers norms$$\|f-f^\sharp \|_{U^s(X,X+H]}$$are also asymptotically small for any fixedsfor these choices of$$f,\theta $$. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$$L^2$$. Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$$II$$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$$I_2$$sums. 

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3. Large monochromatic components in expansive hypergraphs

   https://doi.org/10.1017/S096354832400004X  

   Bal, Deepak; DeBiasio, Louis (March 2024, Combinatorics, Probability and Computing)

   Abstract A result of Gyárfás [12] exactly determines the size of a largest monochromatic component in an arbitrary$$r$$-colouring of the complete$$k$$-uniform hypergraph$$K_n^k$$when$$k\geq 2$$and$$k\in \{r-1,r\}$$. We prove a result which says that if one replaces$$K_n^k$$in Gyárfás’ theorem by any ‘expansive’$$k$$-uniform hypergraph on$$n$$vertices (that is, a$$k$$-uniform hypergraph$$G$$on$$n$$vertices in which$$e(V_1, \ldots, V_k)\gt 0$$for all disjoint sets$$V_1, \ldots, V_k\subseteq V(G)$$with$$|V_i|\gt \alpha$$for all$$i\in [k]$$), then one gets a largest monochromatic component of essentially the same size (within a small error term depending on$$r$$and$$\alpha$$). As corollaries we recover a number of known results about large monochromatic components in random hypergraphs and random Steiner triple systems, often with drastically improved bounds on the error terms. Gyárfás’ result is equivalent to the dual problem of determining the smallest possible maximum degree of an arbitrary$$r$$-partite$$r$$-uniform hypergraph$$H$$with$$n$$edges in which every set of$$k$$edges has a common intersection. In this language, our result says that if one replaces the condition that every set of$$k$$edges has a common intersection with the condition that for every collection of$$k$$disjoint sets$$E_1, \ldots, E_k\subseteq E(H)$$with$$|E_i|\gt \alpha$$, there exists$$(e_1, \ldots, e_k)\in E_1\times \cdots \times E_k$$such that$$e_1\cap \cdots \cap e_k\neq \emptyset$$, then the smallest possible maximum degree of$$H$$is essentially the same (within a small error term depending on$$r$$and$$\alpha$$). We prove our results in this dual setting. 

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4. 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$$f_p$$. 

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5. 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. 

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