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Using the volume averaging technique of Jackson (1997), we derive a set of two-fluid equations that describe the dynamics of a mono-disperse non-Brownian colloidal suspension in the semi-dilute regime. The equations are tensorial and can be applied in arbitrary geometries. Closure models are developed that represent the stress surrounding each particle as a sum of stresses due to fluid movement through a fixed bed of particles, and those due to interactions between particles. Emphasising pragmatism, the developed closure models are consistent with current knowledge of particle interactions in these systems but employ parameters that can be tuned to represent the microstructure of specific particle suspensions. Within the interaction model, a model for the particle distribution around each particle is used that depends on the strain rate field, allowing anisotropy of the microstructure (and hence normal suspension stresses) to develop within the suspension in response to arbitrary strain fields. Force moments acting on particles during particle interactions are calculating by summing hydrodynamic contributions between particle pairs, but adjusted to recognise that multi-particle interactions can increase the effective stress generated during each interaction. Finally, an order of magnitude analysis is performed on the derived momentum equations to determine what terms are significant, leaving only terms in the final equations that are required to predict behaviour during laminar flow within the targeted semi-dilute regime. In a companion paper (referred to as paper II, Noori et. al., 2024) we chose sets of microstructure parameters that predict experimentally measured bulk suspension behaviour, creating a link between particle properties (i.e. roughness), suspension microstructure and shear-induced particle migration in arbitrary flow fields.

The present study simulates shear-induced migration (SIM) in semi-dilute pressure-driven Stokes suspension flows using a multi-fluid (MF) model. Building on analysis from a companion paper (Harvie, 2024), the specific formulation uses volume-averaged phase stresses that are linked to the binary hydrodynamic interaction of spheres and suspension microstructure as represented by an anisotropic, piece-wise constant pair-distribution function (PDF). The form of the PDF is chosen to capture observations regarding the microstructure in sheared suspensions of rough particles, as reported in the literature. Specifically, a hydrodynamic roughness value is used to represent the width of the anisotropic region, and within this region the concentration of particles is higher in the compression zone than expansion zone. By numerically evaluating the hydrodynamic particle interactions and calculating the various shear and normal viscosities, the stress closure is incorporated into Harvie's volume-averaged MF framework, referred to as the MF-roughness model. Using multi-dimensional simulations the roughness and compression zone PDF concentration are then globally optimised to reproduce benchmark solid and velocity distributions reported in the literature for a variety of semi-dilute monodisperse suspension flows occurring within rectangular channels. For comparison, two different versions of the phenomenological stress closure by Morris and Boulay (1999) are additionally proposed as fully tensorial frame-invariant alternatives to the MF-roughness model. Referred to as MF-MB99-A and MF-MB99-B, these models use alternative assumptions for partitioning of the mixture normal stress between the solid and fluid phases. The optimised solid and velocity distributions from all three stress closures are similar and correlate well with the experimental data.

We establish a lower bound on the ADM mass of the time slices of $(n + 1)$-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. This extends the Minkowski inequality for static vacuum spacetimes, which relates the total mass of a time slice to the area and total mean curvature of its boundary. As a corollary, we obtain the Riemannian Penrose inequality in all dimensions and with rigidity for static manifolds under the TCC., Comment: 15 pages, no figures, comments welcome!

We prove that a proper weak solution $\{ \Omega_{t} \}_{0 \leq t < \infty}$ to inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n \leq 7$, is smooth and star-shaped by the time \begin{equation*} T= (n-1) \log \left( \frac{\text{sinh} \left( r_{+} \right)}{ \text{sinh} \left( r_{-} \right)} \right), \end{equation*} where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the initial domain $\Omega_{0}$. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in $\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on $\mathbb{H}^{n} \setminus \{ 0 \}$ in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains $\Omega_{0} \subset \mathbb{H}^{n}$ in dimensions $3 \leq n \leq 7$. From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of $\mathbb{H}^{n}$ in these dimensions., Comment: The lower bound on the support function from the original version was erroneous. The calculation in Section 3 and the resulting estimate have been corrected. Also added a result on proper weak IMCF in punctured hyperbolic space

We prove that equality in the Minkowski inequality for asymptotically flat static manifolds is achieved only by slices of Schwarzschild space. To show this, we establish uniqueness of *quasi-spherical* static metrics: rotationally symmetric regions of Schwarzschild are the only static vacuum metrics which are quasi-spherical with vanishing shear vector. In addition, we observe that the static Minkowski inequality extends to all dimensions for a connected boundary and to asymptotically flat static manifolds of any positive decay order. Altogether, this yields a robust rigidity criterion for Schwarzschild space. Using this criterion, we recover Israel's static black hole uniqueness theorem under this mild decay assumption. Likewise, the uniqueness theorems for photon surfaces and static metric extensions from the prequel extend to all dimensions under these weaker asymptotics. Finally, as a notable by-product of our analysis, we establish regularity of weak inverse mean curvature flow in asymptotically flat manifolds -- that is, a weak IMCF is eventually smooth in an arbitrary asymptotically flat background., Comment: 57 pages. The main rigidity theorem (theorem 1.3) has been strengthened -- the mean-convex boundary assumption may be removed if the manifold is asymptotically flat of order \tau >0 (see Section 2), which yields the complete rigidity statement for these manifolds. Because of the upgraded rigidity statement, we also included a new proof of static black hole uniqueness in section 8

The self-attention mechanism utilizes large implicit weight matrices, programmed through dot product-based activations with very few trainable parameters, to enable long sequence modeling. In this paper, we investigate the possibility of discarding residual learning by employing large implicit kernels to achieve full context interaction at each layer of the network. To accomplish it, we introduce coordinate-based implicit MLPs as a slow network to generate hyper-kernels for another fast convolutional network. To get context-varying weights for fast dynamic encoding, we propose a $\mathrm{Hyper}\mathcal{Z{\cdot}Z{\cdot}W}$ operator that connects hyper-kernels ($\mathcal{W}$) and hidden activations ($\mathcal{Z}$) through simple elementwise multiplication, followed by convolution of $\mathcal{Z}$ using the context-dependent $\mathcal{W}$. Based on this design, we present a novel Terminator architecture that integrates hyper-kernels of different sizes to produce multi-branch hidden representations for enhancing the feature extraction capability of each layer. Additionally, a bottleneck layer is employed to compress the concatenated channels, allowing only valuable information to propagate to the subsequent layers. Notably, our model incorporates several innovative components and exhibits excellent properties, such as introducing local feedback error for updating the slow network, stable zero-mean features, faster training convergence, and fewer model parameters. Extensive experimental results on pixel-level 1D and 2D image classification benchmarks demonstrate the superior performance of our architecture., Comment: 10 pages, 6 figures, 5 tables

Hypothesis: Understanding contact angle hysteresis on rough surfaces is important as most industrially relevant and naturally occurring surfaces possess some form of random or structured roughness. We hypothesise that hysteresis originates from the energy dissipation during the $\textit{stick-slip}$ motion of the contact line and that this energy dissipation is key to developing a predictive equation for hysteresis. Experiments: We measured hysteresis on surfaces with randomly distributed and periodically arranged microscopic cylindrical pillars for a variety of different liquids in air. The inherent (flat surface) contact angles tested range from lyophilic ($\theta_{\rm{e}}=33.8^{\circ}$) to lyophobic ($\theta_{\rm{e}} = 112.0^{\circ}$). Findings: A new methodology for calculating the average advancing and receding contact angles on random surfaces is presented. Also, the correlations for roughness-induced energy dissipation were derived, and a predictive equation for the advancing and receding contact angles during homogeneous (Wenzel) wetting on random surfaces is presented. Significantly, equations that predict the onset of the alternate wetting conditions of hemiwicking, split-advancing, split-receding and heterogeneous (Cassie) wetting are also derived, thus defining the range of validity for the derived homogeneous wetting equation. A novel feature 'cluster' concept is introduced which explains the measurably higher hysteresis exhibited by structured surfaces compared to random surfaces observed experimentally., Comment: 20 pages, 11 figures in the main paper and 17 pages, 12 figures in the supplementary material. Submitted to the Journal of Colloid and Interface Science

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds $(M^{n}, g)$ with boundary and with dimension $ n < 8$ that was establishedby McCormick. First, we show that any asymptotically flat static $(M^{n},g)$ which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension $n < 8$ under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes., Comment: 37 pages, no figures. Correction made in the definition of Schwarzschild stability, asymptotic considerations clarified, and higher-dimensional result included for equipotetnial photon surfaces. Final version to appear in TAMS

The wetting behaviour of surfaces is important for various applications like super-hydrophobic surfaces, enhanced oil recovery, mining of metal ores and anti-icing surfaces etc. For rough surfaces, which are the rule rather than the exception, designing textured surfaces that have wetting properties tailored to suit these applications generally involves either bio-mimicry or trial and error. Existing wetting theories such as the well-known Wenzel (1936) and Cassie & Baxter (1944) models do not predict wetting regimes and importantly, only give a single equilibrium angle rather than a range of stable contact angles (hysteresis) as observed in reality. In this paper, we use a roughness-scale mechanical energy balance (derived in part I) combined with simulations of micro-scale interface dynamics based on open-source software (Surface Evolver (Brakke 1992)) to calculate the energy dissipation during the motion of an interface over a chemically homogeneous rough surface. This dissipation is used to predict contact angle hysteresis (CAH) from knowledge of just the surface roughness topography and equilibrium contact angle. We simulate interface dynamics over a surface decorated with a periodic array of round-edge square pillars and show that the energy dissipated varies approximately as ${\phi}ln{\phi}$ with the area fraction (${\phi}$), and becomes zero as ${\phi} \to 0$. The CAH predicted by our method is in good agreement with the experimental results of Forsberg et al. (2010), especially at low area fractions. We also compute CAH for an interface moving at 45${\deg}$ to the surface periodicity direction, showing that the experimental measurements are bracketed by the 0${\deg}$ and 45${\deg}$ advance direction results., Comment: To be published in Journal of Fluid Mechanics, 55 pages, 39 figures

Using as a starting point conservation of momentum, a multiphase mechanical energy balance equation is derived that accounts for multiple material phases and interfaces present within a moving control volume. This balance is applied to a control volume that is anchored to a three phase contact line as it advances over the surface of a rough and chemically homogeneous solid. Using semi-quantitative models for the material behaviour occurring within the control volume, an order-of-magnitude analysis is performed to find what terms within the balance are significant, producing an equation that can be used to predict contact angle hysteresis from a knowledge of interface dynamics occurring around the three phase contact line. In addition to this equation, the theory also answers several questions that have been discussed within the wetting literature: Namely that (static) contact angle hysteresis is a function of conditions around the three phase contact line, as opposed to the surrounding flow system; That contact angle hysteresis results from interface `jumps' that dissipate energy, rather than directly from contact line deformation; That interfacial dynamics is required to predict contact angle hysteresis, but that these dynamics should be interpreted via energy conservation, and; That dynamic contact angles depend on kinetic energy transport around the three phase contact line, as well as local energy dissipation. The framework has been derived using assumptions of incompressible Newtonian fluids, reversible interface formation and zero-strain solids -- future work could relax these assumptions to make the theory more generally applicable.

The Open Lock-In Amplifier (OLIA) is a microcontroller-based digital lock-in amplifier built from a small number of inexpensive and easily sourced electronic components. Despite its small credit card-sized form-factor and low build-cost of around US$35, OLIA is a capable instrument that offers many features associated with far costlier commercial devices. Key features include dual-phase lock-in detection at multiple harmonic frequencies up to 50 kHz, internal and external reference modes, adjustable levels of input gain, a choice between low-pass filtering and synchronous filtering, noise estimation, and a comprehensive programming interface for remote software control. OLIA comes with an optional optical breakout board that allows noise-tolerant optical detection down to the 40 pW level. OLIA and its breakout board are released here as open hardware, with technical diagrams, full parts-lists, and source-code for the firmware provided as Supporting Information.

We give a simple proof to the computation of ADM mass of the static extensions of small spheres in Wiygul \cite{W1, W2}. It makes use of the mass formula $m = \frac{1}{4\pi} \int_{\partial M} \frac{\partial V}{\partial \nu}$ for an asymptotically flat static manifold with boundary.

For an $H>0$ rotationally symmetric embedded torus $N_{0} \subset \mathbb{R}^{3}$, evolved by Inverse Mean Curvature Flow, we show that the total curvature $|A|$ remains bounded up to the singular time $T_{\max}$. We then show convergence of the $N_{t}$ to a $C^{1}$ rotationally symmetric embedded torus $N_{T_{\max}}$ as $t \rightarrow T_{\max}$ without rescaling. Later, we observe a scale-invariant $L^{2}$ energy estimate on any embedded solution of the flow in $\mathbb{R}^{3}$ that may be useful in ruling out curvature blowup near singularities in general., Comment: 15 pages, 2 figures, some minor typos corrected. Final version to appear in Proceedings of the American Mathematical Society

We prove that the Inverse Mean Curvature Flow of a non-star-shaped, mean-convex embedded sphere in $\mathbb{R}^{n+1}$ with symmetry about an axis and sufficiently long, thick necks exists for all time and homothetically converges to a round sphere as $t \rightarrow \infty$. Our approach is based on a localized version of the parabolic maximum principle. We also present two applications of this result. The first is an extension of the Minkowski inequality to the corresponding non-star-shaped, mean-convex domains in $\mathbb{R}^{n+1}$. The second is a connection between IMCF and minimal surface theory. Based on previous work by Meeks and Yau and using foliations by IMCF, we establish embeddedness of the solution to Plateau's problem and a finiteness property of stable immersed minimal disks for certain Jordan curves in $\mathbb{R}^{3}$., Comment: 40 pages, 7 figures. This version includes more exposition on rotationally symmetric immersions, some additional applications, and a simpler example of a non-star-shaped admissible initial surface

The lift and drag forces acting on a small spherical particle moving with a finite slip in single-wall-bounded flows are investigated via direct numerical simulations. The effect of slip velocity on the particle force is analysed as a function of separation distance for low slip and shear Reynolds numbers ($10^{-3} \leq Re_{\gamma}, Re_{\text{slip}} \leq 10^{-1}$) in both quiescent and linear shear flows. A generalised lift model valid for arbitrary particle-wall separation distances and $Re_{\gamma}, Re_{\text{slip}} \leq 10^{-1}$ is developed based on the results of the simulations. The proposed model can now predict the lift forces in linear shear flows in the presence or absence of slip,and in quiescent flows when slip is present. Existing drag models are also compared with numerical results for both quiescent and linear shear flows to determine which models capture near wall slip velocities most accurately for low particle Reynolds numbers. Finally, we compare the results of the proposed lift model to previous experimental results of buoyant particles and to numerical results of neutrally-buoyant (force-free) particles moving near a wall in quiescent and linear shear flows. The generalised lift model presented can be used to predict the behaviour of particle suspensions in biological and industrial flows where the particle Reynolds numbers based on slip and shear are $\mathcal{O}(10^{-1})$ and below.

We experimentally study the coherence time of a below-threshold Raman laser in which the gain medium is a gas of magneto-optically trapped atoms. The second-order optical coherence exhibits photon bunching with a correlation time which is varied by two orders of magnitude by controlling the gain. Results are in good agreement with a simple analytic model which suggests the effect is dominated by gain, rather than dispersion, in this system. Cavity ring-down measurements show the photon lifetime, related to the first-order coherence time, is also increased., Comment: 4 pages, 3 figures; minor changes in response to referee review; version accepted for publication in Optics Letters

The lift and drag forces acting on a small, neutrally-buoyant spherical particle in a single-wall-bounded linear shear flow are examined via numerical computation. The effects of shear rate are isolated from those of slip by setting the particle velocity equal to the local fluid velocity (zero slip), and examining the resulting hydrodynamic forces as a function of separation distance. In contrast to much of the previous numerical literature, low shear Reynolds numbers are considered ($10^{-3} \lesssim Re_{\gamma} \lesssim 10^{-1}$). This shear rate range is relevant when dealing with particulate flows within small channels, for example particle migration in microfluidic devices being used or developed for the biotech industry. We demonstrate a strong dependence of both the lift and drag forces on shear rate. Building on previous theoretical $Re_{\gamma} \ll 1$ studies, a wall-shear based lift correlation is proposed that is applicable when the wall lies both within the inner and outer regions of the disturbed flow. Similarly, we validate an improved drag correlation that includes higher order terms in wall separation distance that more accurately captures the drag force when the particle is close to, but not touching, the wall. Application of the new correlations shows that the examined shear based lift force is as important as the previously examined slip based lift force, highlighting the need to account for shear when predicting the near-wall movement of neutrally-buoyant particles., Comment: Currently submitted to JFM for review

Shear induced particle pressure has been measured and modelled for concentrated suspensions of particles. Importantly, the significance of the shear induced particle pressure has not been recognized in polymer rheology. The shear induced particle pressure results in an inward pressure on the polymer chains resulting in a shear dependent compressive force. The analytical form of the force balance equations that incorporate the effect of shear induced particle pressure predict a reduced polymer blob size and reducing viscosity with increasing shear rate as has been observed experimentally. Power law behavior is found for the viscosity in accord with the general behavior observed for the rheology of concentrated polymer solutions and melts. Differing powers are found for the behavior depending on the concentration regime., Comment: 11 pages, 7 equations, 2 Figures, 1 table and 58 references. arXiv admin note: text overlap with arXiv:1806.00963

We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection within a prescribed time interval depending only on initial data. Finally, we establish the existence of either finite-time singularities or intersections for certain topological spheres under IMCF., Comment: 22 pages, 2 figures. Final version to appear in Mathematical Research Letters. Added clarification in Section 3 on how one relates the global variational solution of IMCF to the variational solution induced by the classical flow. The most recent update was to fix some typos and update contact info in the document, but no further edits were made

We study optical gain in a gas of cold 39K atoms. The gain is observed during operation of a conventional magneto-optical trap without the need for additional fields. Measurements of transmission spectra from a weak probe show that the gain is due to stimulated Raman scattering between hyperfine ground states. The experimental results are reproduced by a simplified six-level model, which also helps explain why such gain is not observed in similar experiments with rubidium or cesium., Comment: 7 pages, 7 figures; submitted for publication; v2: minor changes in response to Referee comments; v3: version accepted for publication in PRA