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Title: Rotation of a fibre in simple shear flow of a dilute polymer solution

The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio,$\kappa$. A regular perturbation expansion in the polymer concentration,$c$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the$O(c)$correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon$c\, De$($De$is the imposed shear rate times the polymer relaxation time) and$\kappa$and quantitatively on$c$. At a small but finite$c\, De$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing$\kappa$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate$c\, De$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller$c\, De$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing$c\, De$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing$c\, De$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.

Chan, William; Jackson, Stephen; Trang, Nam(
, Forum of Mathematics, Sigma)

Abstract

This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.

The following summarizes the main results proved under suitable partition hypotheses.

If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.

If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.

If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.

Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.

Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).

Cobos, Richard; Khair, Aditya S.(
, 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.

Joshi, P.; Anderson, W.(
, Journal of Fluid Mechanics)

Large-eddy simulation was used to model turbulent atmospheric surface layer (ASL) flow over canopies composed of streamwise-aligned rows of synthetic trees of height,$h$, and systematically arranged to quantify the response to variable streamwise spacing,$\delta _1$, and spanwise spacing,$\delta _2$, between adjacent trees. The response to spanwise and streamwise heterogeneity has, indeed, been the topic of a sustained research effort: the former resulting in formation of Reynolds-averaged counter-rotating secondary cells, the latter associated with the$k$- and$d$-type response. No study has addressed the confluence of both, and results herein show secondary flow polarity reversal across ‘critical’ values of$\delta _1$and$\delta _2$. For$\delta _2/\delta \lesssim 1$and$\gtrsim 2$, where$\delta$is the flow depth, the counter-rotating secondary cells are aligned such that upwelling and downwelling, respectively, occurs above the elements. The streamwise spacing$\delta _1$regulates this transition, with secondary cell reversal occurring first for the largest$k$-type cases, as elevated turbulence production within the canopy necessitates entrainment of fluid from aloft. The results are interpreted through the lens of a benchmark prognostic closure for effective aerodynamic roughness,$z_{0,{Eff.}} = \alpha \sigma _h$, where$\alpha$is a proportionality constant and$\sigma _h$is height root mean square. We report$\alpha \approx 10^{-1}$, the value reported over many decades for a broad range of rough surfaces, for$k$-type cases at small$\delta _2$, whereas the transition to$d$-type arrangements necessitates larger$\delta _2$. Though preliminary, results highlight the non-trivial response to variation of streamwise and spanwise spacing.

We study the spaces of twisted conformal blocks attached to a$\Gamma$-curve$\Sigma$with marked$\Gamma$-orbits and an action of$\Gamma$on a simple Lie algebra$\mathfrak {g}$, where$\Gamma$is a finite group. We prove that if$\Gamma$stabilizes a Borel subalgebra of$\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$\mathscr {G}$be the parahoric Bruhat–Tits group scheme on the quotient curve$\Sigma /\Gamma$obtained via the$\Gamma$-invariance of Weil restriction associated to$\Sigma$and the simply connected simple algebraic group$G$with Lie algebra$\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$\mathscr {G}$-torsors on$\Sigma /\Gamma$when the level$c$is divisible by$|\Gamma |$(establishing a conjecture due to Pappas and Rapoport).

Bal, Deepak; DeBiasio, Louis(
, 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.

Sharma, Arjun, and Koch, Donald L. Rotation of a fibre in simple shear flow of a dilute polymer solution. Retrieved from https://par.nsf.gov/biblio/10513120. Journal of Fluid Mechanics 976. Web. doi:10.1017/jfm.2023.823.

Sharma, Arjun, & Koch, Donald L. Rotation of a fibre in simple shear flow of a dilute polymer solution. Journal of Fluid Mechanics, 976 (). Retrieved from https://par.nsf.gov/biblio/10513120. https://doi.org/10.1017/jfm.2023.823

@article{osti_10513120,
place = {Country unknown/Code not available},
title = {Rotation of a fibre in simple shear flow of a dilute polymer solution},
url = {https://par.nsf.gov/biblio/10513120},
DOI = {10.1017/jfm.2023.823},
abstractNote = {The motion of a freely rotating prolate spheroid in a simple shear flow of a dilute polymeric solution is examined in the limit of large particle aspect ratio,$\kappa$. A regular perturbation expansion in the polymer concentration,$c$, a generalized reciprocal theorem, and slender body theory to represent the velocity field of a Newtonian fluid around the spheroid are used to obtain the$O(c)$correction to the particle's orientational dynamics. The resulting dynamical system predicts a range of orientational behaviours qualitatively dependent upon$c\, De$($De$is the imposed shear rate times the polymer relaxation time) and$\kappa$and quantitatively on$c$. At a small but finite$c\, De$, the particle spirals towards a limit cycle near the vorticity axis for all initial conditions. Upon increasing$\kappa$, the limit cycle becomes smaller. Thus, ultimately the particle undergoes a periodic motion around and at a small angle from the vorticity axis. At moderate$c\, De$, a particle starting near the flow–gradient plane departs it monotonically instead of spirally, as this plane (a limit cycle at smaller$c\, De$) obtains a saddle and an unstable node. The former is close to the flow direction. Upon further increasing$c\, De$, the saddle node changes to a stable node. Therefore, depending upon the initial condition, a particle may either approach a periodic orbit near the vorticity axis or obtain a stable orientation near the flow direction. Upon further increasing$c\, De$, the limit cycle near the vorticity axis vanishes, and the particle aligns with the flow direction for all starting orientations.},
journal = {Journal of Fluid Mechanics},
volume = {976},
publisher = {Cambridge University Press},
author = {Sharma, Arjun and Koch, Donald L},
}

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