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1. Plunging breakers. Part 2. Droplet generation

   https://doi.org/10.1017/jfm.2023.378  

   Erinin, M.A. ; Liu, C. ; Wang, S.D. ; Liu, X. ; Duncan, J.H. ( July 2023 , Journal of Fluid Mechanics)

   An experimental study of the dynamics and droplet production in three mechanically generated plunging breaking waves is presented in this two-part paper. In the present paper (Part 2), in-line cinematic holography is used to measure the positions, diameters ($d\geq 100\ \mathrm {\mu }{\rm m}$), times and velocities of droplets generated by the three plunging breaking waves studied in Part 1 (Erininet al.,J. Fluid Mech., vol. 967, 2023, A35) as the droplets move up across a horizontal measurement plane located just above the wave crests. It is found that there are four major mechanisms for droplet production: closure of the indentation between the top surface of the plunging jet and the splash that it creates, the bursting of large bubbles that were entrapped under the plunging jet at impact, splashing and bubble bursting in the turbulent zone on the front face of the wave and the bursting of small bubbles that reach the water surface at the crest of the non-breaking wave following the breaker. The droplet diameter distributions for the entire droplet set for each breaker are fitted with power-law functions in separate small- and large-diameter regions. The droplet diameter where these power-law functions cross increases monotonically from 820 to 1480$\mathrm {\mu }{\rm m}$from the weak to the strong breaker, respectively. The droplet diameter and velocity characteristics and the number of the droplets generated by the four mechanisms are found to vary significantly and the processes that create these differences are discussed.

    

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2. Almost Everywhere Behavior of Functions According to Partition Measures

   https://doi.org/10.1017/fms.2023.130  

   Chan, William ; Jackson, Stephen ; Trang, Nam ( January 2024 , Forum of Mathematics, Sigma)

   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}}$).

    

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3. Coherent motions in a turbulent wake of an axisymmetric bluff body

   https://doi.org/10.1017/jfm.2023.231  

   Zhang, Fengrui ; Peet, Yulia T. ( May 2023 , Journal of Fluid Mechanics)

   The wake flow past an axisymmetric body of revolution at a diameter-based Reynolds number$Re=u_{\infty }D/\nu =5000$is investigated via a direct numerical simulation. The study is focused on identification of coherent vortical motions and the dominant frequencies in this flow. Three dominant coherent motions are identified in the wake: the vortex shedding motion with the frequency of$St=fD/u_{\infty }=0.27$, the bubble pumping motion with$St=0.02$, and the very-low-frequency (VLF) motion originated in the very near wake of the body with the frequency$St=0.002$–$0.005$. The vortex shedding pattern is demonstrated to follow a reflectional symmetry breaking mode, whereas the vortex loops are shed alternatingly from each side of the vortex shedding plane, but are subsequently twisted and tangled, giving the resulting wake structure a helical spiraling pattern. The bubble pumping motion is confined to the recirculation region and is a result of a Görtler instability. The VLF motion is related to a stochastic destabilisation of a steady symmetric mode in the near wake and manifests itself as a slow, precessional motion of the wake barycentre. The VLF mode with$St=0.005$is also detectable in the intermediate wake and may be associated with a low-frequency radial flapping of the shear layer.

    

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4. 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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5. The Matsuno–Gill model on the sphere

   https://doi.org/10.1017/jfm.2023.369  

   Shamir, Ofer ; Garfinkel, Chaim I. ; Gerber, Edwin P. ; Paldor, Nathan ( June 2023 , Journal of Fluid Mechanics)

   We extend the Matsuno–Gill model, originally developed on the equatorial$\beta$-plane, to the surface of the sphere. While on the$\beta$-plane the non-dimensional model contains a single parameter, the damping rate$\gamma$, on a sphere the model contains a second parameter, the rotation rate$\epsilon ^{1/2}$(Lamb number). By considering the different combinations of damping and rotation, we are able to characterize the solutions over the$(\gamma, \epsilon ^{1/2})$plane. We find that the$\beta$-plane approximation is accurate only for fast rotation rates, where gravity waves traverse a fraction of the sphere's diameter in one rotation period. The particular solutions studied by Matsuno and Gill are accurate only for fast rotation and moderate damping rates, where the relaxation time is comparable to the time on which gravity waves traverse the sphere's diameter. Other regions of the parameter space can be described by different approximations, including radiative relaxation, geostrophic, weak temperature gradient and non-rotating approximations. The effect of the additional parameter introduced by the sphere is to alter the eigenmodes of the free system. Thus, unlike the solutions obtained by Matsuno and Gill, where the long-term response to a symmetric forcing consists solely of Kelvin and Rossby waves, the response on the sphere includes other waves as well, depending on the combination of$\gamma$and$\epsilon ^{1/2}$. The particular solutions studied by Matsuno and Gill apply to Earth's oceans, while the more general$\beta$-plane solutions are only somewhat relevant to Earth's troposphere. In Earth's stratosphere, Venus and Titan, only the spherical solutions apply.

    

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