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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.

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

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 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.

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

Electrokinetic transport behavior in nanochannels is different to that in larger sized channels. Specifically, molecular dynamics (MD) simulations in nanochannels have demonstrated two little understood phenomena which are not observed in microchannels, being: (i) the decrease of average electroosmotic mobility at high surface charge density, and (ii) the decrease of channel conductance at high salt concentrations, as the surface charge is increased. However, current electric double layer models do not capture these results. In this study we provide evidence that this inconsistency primarily arises from the neglect of the viscoelectric effect (being the increase of local viscosity near charged surfaces due to water molecule orientation) in conventional continuum models. It is shown that predictions of electroosmotic mobility in a slit nanochannel, derived from a viscoelectric-modified continuum model, are in quantitative agreement with previous MD simulation results. Furthermore, viscoelectric effects are found to dominate over ion steric and dielectric saturation effects in both electroosmotic and ion transport processes. Finally, we indicate that mechanisms of the previous MD-observed phenomena can be well-explained by the viscoelectric theory., Comment: 6 pages, 5 figures