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
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We extend the Matsuno–Gill model, originally developed on the equatorial
 Award ID(s):
 2004572
 NSFPAR ID:
 10468677
 Publisher / Repository:
 Cambridge University Press
 Date Published:
 Journal Name:
 Journal of Fluid Mechanics
 Volume:
 964
 ISSN:
 00221120
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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, defined as the ratio of the product of the applied electric field magnitude$\beta =a^*e^*E_\infty ^*/k_B^*T^*$ and particle radius$E_\infty ^*$ , to the thermal voltage$a^*$ , where$k_B^*T^*/e^*$ is Boltzmann's constant,$k_B^*$ is the absolute temperature, and$T^*$ 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$e^*$ over a wide range of$\sigma$ . Here,$\beta$ , where$\sigma =(e^*a^*/\epsilon ^*k_B^*T^*)\sigma ^*$ is the dimensional surface charge density, and$\sigma ^*$ is the permittivity of the electrolyte. For moderately charged particles ($\epsilon ^*$ ), the electrophoretic velocity is linear in$\sigma ={O}(1)$ when$\beta$ , and its dependence on the ratio of the Debye length ($\beta \ll 1$ ) to particle radius (denoted by$1/\kappa ^*$ ) agrees with Henry's formula. As$\delta =1/(\kappa ^*a^*)$ increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is$\beta$ dependent. For$\delta$ , the electrophoretic velocity again becomes linear in field strength, approaching the Hückel limit of electrophoresis in a dielectric medium, for all$\beta \gg 1$ . For highly charged particles ($\delta$ ) in the thinDebyelayer limit ($\sigma \gg 1$ ), our computations are in good agreement with recent experimental and asymptotic results.$\delta \ll 1$ 
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
is a cardinal,$\kappa $ ,$\epsilon < \kappa $ ,${\mathrm {cof}}(\epsilon ) = \omega $ and$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the almost everywhere short length continuity property: There is a club$\Phi $ and a$C \subseteq \kappa $ so that for all$\delta < \epsilon $ , if$f,g \in [C]^\epsilon _*$ and$f \upharpoonright \delta = g \upharpoonright \delta $ , then$\sup (f) = \sup (g)$ .$\Phi (f) = \Phi (g)$ If
is a cardinal,$\kappa $ is countable,$\epsilon $ holds and$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the strong almost everywhere short length continuity property: There is a club$\Phi $ and finitely many ordinals$C \subseteq \kappa $ so that for all$\delta _0, ..., \delta _k \leq \epsilon $ , if for all$f,g \in [C]^\epsilon _*$ ,$0 \leq i \leq k$ , then$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$ .$\Phi (f) = \Phi (g)$ If
satisfies$\kappa $ ,$\kappa \rightarrow _* (\kappa )^\kappa _2$ and$\epsilon \leq \kappa $ , then$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$ satisfies the almost everywhere monotonicity property: There is a club$\Phi $ so that for all$C \subseteq \kappa $ , if for all$f,g \in [C]^\epsilon _*$ ,$\alpha < \epsilon $ , then$f(\alpha ) \leq g(\alpha )$ .$\Phi (f) \leq \Phi (g)$ Suppose dependent choice (
),$\mathsf {DC}$ and the almost everywhere short length club uniformization principle for${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$ hold. Then every function${\omega _1}$ satisfies a finite continuity property with respect to closure points: Let$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$ be the club of$\mathfrak {C}_f$ so that$\alpha < {\omega _1}$ . There is a club$\sup (f \upharpoonright \alpha ) = \alpha $ and finitely many functions$C \subseteq {\omega _1}$ so that for all$\Upsilon _0, ..., \Upsilon _{n  1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$ , for all$f \in [C]^{\omega _1}_*$ , if$g \in [C]^{\omega _1}_*$ and for all$\mathfrak {C}_g = \mathfrak {C}_f$ ,$i < n$ , then$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$ .$\Phi (g) = \Phi (f)$ Suppose
satisfies$\kappa $ for all$\kappa \rightarrow _* (\kappa )^\epsilon _2$ . For all$\epsilon < \kappa $ ,$\chi < \kappa $ does not inject into$[\kappa ]^{<\kappa }$ , the class of${}^\chi \mathrm {ON}$ length sequences of ordinals, and therefore,$\chi $ . As a consequence, under the axiom of determinacy$[\kappa ]^\chi  < [\kappa ]^{<\kappa }$ , these two cardinality results hold when$(\mathsf {AD})$ is one of the following weak or strong partition cardinals of determinacy:$\kappa $ ,${\omega _1}$ ,$\omega _2$ (for all$\boldsymbol {\delta }_n^1$ ) and$1 \leq n < \omega $ (assuming in addition$\boldsymbol {\delta }^2_1$ ).$\mathsf {DC}_{\mathbb {R}}$ 
Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and nonselfadjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the nonselfadjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter
, where$q=2{\rm i}k\,{\textit {Pe}}$ is the nondimensional wavenumber of the tracer in the streamwise direction, and$k$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scaledependent scalar decay of tracer variance, and characterize the distinct selfsimilar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.${\textit {Pe}}$ 
Abstract The wellstudied moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient
, as a Baily–Borel compactification of a ball quotient${\mathcal {M}}^{\operatorname {GIT}}$ , and as a compactified${(\mathcal {B}_4/\Gamma )^*}$ K moduli space. From all three perspectives, there is a unique boundary point corresponding to nonstable surfaces. From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup , whereas from the ball quotient point of view, it is natural to consider the toroidal compactification${\mathcal {M}}^{\operatorname {K}}\rightarrow {\mathcal {M}}^{\operatorname {GIT}}$ . The spaces${\overline {\mathcal {B}_4/\Gamma }}\rightarrow {(\mathcal {B}_4/\Gamma )^*}$ and${\mathcal {M}}^{\operatorname {K}}$ have the same cohomology, and it is therefore natural to ask whether they are isomorphic. Here, we show that this is in fact${\overline {\mathcal {B}_4/\Gamma }}$ not the case. Indeed, we show the more refined statement that and${\mathcal {M}}^{\operatorname {K}}$ are equivalent in the Grothendieck ring, but not${\overline {\mathcal {B}_4/\Gamma }}$ K equivalent. Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients. 
Abstract Given a set
defined over a field and an infinite sequence$S=\{x^2+c_1,\dots,x^2+c_s\}$ of elements of$\gamma$ S , one can associate an arboreal representation to , generalising the case of iterating a single polynomial. We study the probability that a random sequence$\gamma$ produces a “largeimage” representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets$\gamma$ S defined over , and we conjecture a similar positiveprobability result for suitable sets over$\mathbb{Z}[t]$ . As an application of largeimage representations, we prove a densityzero result for the set of prime divisors of some associated quadratic sequences. We also consider the stronger condition of the representation being finiteindex, and we classify all$\mathbb{Q}$ S possessing a particular kind of obstruction that generalises the postcritically finite case in singlepolynomial iteration.