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1. Rheology of dense suspensions of ideally conductive particles in an electric field

   https://doi.org/10.1017/jfm.2023.980  

   Mirfendereski, Siamak; Park, Jae Sung (December 2023, Journal of Fluid Mechanics)

   The rheological behaviour of dense suspensions of ideally conductive particles in the presence of both electric field and shear flow is studied using large-scale numerical simulations. Under the action of an electric field, these particles are known to undergo dipolophoresis (DIP), which is the combination of two nonlinear electrokinetic phenomena: induced-charge electrophoresis (ICEP) and dielectrophoresis (DEP). For ideally conductive particles, ICEP is predominant over DEP, resulting in transient pairing dynamics. The shear viscosity and first and second normal stress differences$$N_1$$and$$N_2$$of such suspensions are examined over a range of volume fractions$$15\,\% \leq \phi \leq 50\,\%$$as a function of Mason number$$Mn$$, which measures the relative importance of viscous shear stress over electrokinetic-driven stress. For$$Mn < 1$$or low shear rates, the DIP is shown to dominate the dynamics, resulting in a relatively low-viscosity state. The positive$$N_1$$and negative$$N_2$$are observed at$$\phi < 30\,\%$$, which is similar to Brownian suspensions, while their signs are reversed at$$\phi \ge 30\,\%$$. For$$Mn \ge 1$$, the shear thickening starts to arise at$$\phi \ge 30\,\%$$, and an almost five-fold increase in viscosity occurs at$$\phi = 50\,\%$$. Both$$N_1$$and$$N_2$$are negative for$$Mn \gg 1$$at all volume fractions considered. We illuminate the transition in rheological behaviours from DIP to shear dominance around$$Mn = 1$$in connection to suspension microstructure and dynamics. Lastly, our findings reveal the potential use of nonlinear electrokinetics as a means of active rheology control for such suspensions. 

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2. The exact consistency strength of the generic absoluteness for the universally Baire sets

   https://doi.org/10.1017/fms.2023.127  

   Sargsyan, Grigor; Trang, Nam (January 2024, Forum of Mathematics, Sigma)

   Abstract A set of reals isuniversally Baireif all of its continuous preimages in topological spaces have the Baire property.$$\mathsf {Sealing}$$is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by forcing. The$$\mathsf {Largest\ Suslin\ Axiom}$$($$\mathsf {LSA}$$) is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable bijections. Let$$\mathsf {LSA-over-uB}$$be the statement that in all (set) generic extensions there is a model of$$\mathsf {LSA}$$whose Suslin, co-Suslin sets are the universally Baire sets. We show that over some mild large cardinal theory,$$\mathsf {Sealing}$$is equiconsistent with$$\mathsf {LSA-over-uB}$$. In fact, we isolate an exact large cardinal theory that is equiconsistent with both (see Definition 2.7). As a consequence, we obtain that$$\mathsf {Sealing}$$is weaker than the theory ‘$$\mathsf {ZFC} +$$there is a Woodin cardinal which is a limit of Woodin cardinals’. A variation of$$\mathsf {Sealing}$$, called$$\mathsf {Tower\ Sealing}$$, is also shown to be equiconsistent with$$\mathsf {Sealing}$$over the same large cardinal theory. The result is proven via Woodin’s$$\mathsf {Core\ Model\ Induction}$$technique and is essentially the ultimate equiconsistency that can be proven via the current interpretation of$$\mathsf {CMI}$$as explained in the paper. 

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3. DIVERGENT MODELS WITH THE FAILURE OF THE CONTINUUM HYPOTHESIS

   https://doi.org/10.1017/jsl.2023.92  

   TRANG, NAM (December 2023, The Journal of Symbolic Logic)

   Abstract We construct divergent models of$$\mathsf {AD}^+$$along with the failure of the Continuum Hypothesis ($$\mathsf {CH}$$) under various assumptions. Divergent models of$$\mathsf {AD}^+$$play an important role in descriptive inner model theory; all known analyses of HOD in$$\mathsf {AD}^+$$models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of$$\mathsf {AD}^+$$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings of the reals and principles implying$$\neg \mathsf {CH}$$, like$$\mathsf {MM}$$, that divergent models shed light on, see Question 5.1. 

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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. 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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