Abstract:
In this work, closure of the Boltzmann--BGK moment hierarchy is accomplished via projection of the distribution function $f$ onto a space $\mathbb{H}^{N}$ spanned by $N$-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of $f$, the presented procedure produces a hierarchy of (single) $N$-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ($N$-order) lattice Boltzmann--BGK (LBGK) simulation. Numerical analysis is performed with LBGK models and direct simulation Monte Carlo (DSMC) for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number $Wi=\tau\nu k^2$ (i.e. Knudsen number $Kn=\lambda k=\sqrt{Wi}$); $k$ is the wavenumber, $\tau$ the relaxation time of the system, $\lambda\simeq\tau c_s$ the mean-free path, and $c_s$ the speed of sound. The present results elucidate the applicability of LBGK simulation under general non-equilibrium conditions.

Abstract:
The ability to control wettability is important for a wide range of technological applications in which precise microfluidic handling is required. It is known that predesigned roughness at a micro- or nano- scale enhances the wetting properties of solid materials giving rise to super-hydrophobic or super-hydrophilic behavior. In this work, we study the dependence of the apparent wettability of a stripe-patterned solid surface on the stripe geometry, utilizing systems level analysis and mesoscopic Lattice-Boltzmann (LB) simulations. Through the computation of both stable and unstable states we are able to determine the energy barriers separating distinct metastable wetting states that correspond to the well-known Cassie and Wenzel states. This way the energy cost for inducing certain wetting transitions is computed and its dependence on geometric features of the surface pattern is explored.

Abstract:
Nanoparticles with different surface morphologies that straddle the interface between two immiscible liquids are studied via molecular dynamics simulations. The methodology employed allows us to compute the interfacial free energy at different angular orientations of the nanoparticle. Due to their atomistic nature, the studied nanoparticles present both microscale and macroscale geometrical features and cannot be accurately modeled as a perfectly smooth body (e.g., spheres, cylinders). Under certain physical conditions, microscale features can produce free energy barriers that are much larger than the thermal energy of the surrounding media. The presence of these energy barriers can effectively "lock" the particle at specific angular orientations with respect to the liquid-liquid interface. This work provides new insights on the rotational dynamics of Brownian particles at liquid interfaces and suggests possible strategies to exploit the effects of microscale features with given geometric characteristics.

Abstract:
Plane Poiseuille flow past a nanoscale cylinder that is arbitrarily confined (i.e., symmetrically or asymmetrically confined) in a slit channel is studied via hydrodynamic lubrication theory and molecular dynamics simulations, considering cases where the cylinder remains static or undergoes thermal motion. Lubrication theory predictions for the drag force and volumetric flow rate are in close agreement with molecular dynamics simulations of flows having molecularly thin lubrication gaps, despite the presence of significant structural forces induced by the crystalline structure of the modeled solid. While the maximum drag force is observed in symmetric confinement, i.e., when the cylinder is equidistant from both channel walls, the drag decays significantly as the cylinder moves away from the channel centerline and approaches a wall. Hence, significant reductions in the mean drag force on the cylinder and hydraulic resistance of the channel can be observed when thermal motion induces random off-center displacements. Analytical expressions and numerical results in this work provide useful insights into the hydrodynamics of colloidal solids and macromolecules in confinement.

Abstract:
The adsorption dynamics of a colloidal particle at a fluid interface is studied theoretically and numerically, documenting distinctly different relaxation regimes. The adsorption of a perfectly smooth particle is characterized by a fast exponential relaxation to thermodynamic equilibrium where the interfacial free energy has a minimum. The short relaxation time is given by the ratio of viscous damping to capillary forces. Physical and/or chemical heterogeneities in a colloidal system, however, can result in multiple minima of the free energy giving rise to metastability. In the presence of metastable states we observe a crossover to a slow logarithmic relaxation reminiscent of physical aging in glassy systems. The long relaxation time is determined by the thermally-activated escape rate from metastable states. Analytical expressions derived in this work yield quantitative agreement with molecular dynamics simulations and recent experimental observations. This work provides new insights on the adsorption dynamics of colloidal particles at fluid interfaces.

Abstract:
We study numerically the hydrodynamics of dip coating from a suspension and report a mechanism for colloidal assembly and pattern formation on smooth and uniform substrates. Below a critical withdrawal speed of the substrate, capillary forces required to deform the meniscus prevent colloidal particles from entering the coating film. Capillary forces are overcome by hydrodynamic drag only after a minimum number of particles organize in a close-packed formation within the meniscus. Once within the film, the formed assembly moves at nearly the withdrawal speed and rapidly separates from the next assembly. The interplay between hydrodynamic and capillary forces can thus produce periodic and regular structures within the curved meniscus that extends below the withdrawn film. The hydrodynamically-driven assembly documented here is consistent with stripe pattern formations observed experimentally in the so-called thin-film entrainment regime.

Abstract:
Theoretical analysis and fully atomistic molecular dynamics simulations reveal a Brownian ratchet mechanism by which thermal fluctuations drive the net displacement of immiscible liquids confined in channels or pores with micro- or nanoscale dimensions. The thermally-driven displacement is induced by surface nanostructures with directional asymmetry and can occur against the direction of action of wetting or capillary forces. Mean displacement rates in molecular dynamics simulations are predicted via analytical solution of a Smoluchowski diffusion equation for the position probability density. The proposed physical mechanisms and derived analytical expressions can be applied to engineer surface nanostructures for controlling the dynamics of diverse wetting processes such as capillary filling, wicking, and imbibition in micro- or nanoscale systems.

Abstract:
We present a model based on the lattice Boltzmann equation that is suitable for the simulation of dynamic wetting. The model is capable of exhibiting fundamental interfacial phenomena such as weak adsorption of fluid on the solid substrate and the presence of a thin surface film within which a disjoining pressure acts. Dynamics in this surface film, tightly coupled with hydrodynamics in the fluid bulk, determine macroscopic properties of primary interest: the hydrodynamic slip; the equilibrium contact angle; and the static and dynamic hysteresis of the contact angles. The pseudo- potentials employed for fluid-solid interactions are composed of a repulsive core and an attractive tail that can be independently adjusted. This enables effective modification of the functional form of the disjoining pressure so that one can vary the static and dynamic hysteresis on surfaces that exhibit the same equilibrium contact angle. The modeled solid-fluid interface is diffuse, represented by a wall probability function which ultimately controls the momentum exchange between solid and fluid phases. This approach allows us to effectively vary the slip length for a given wettability (i.e. the static contact angle) of the solid substrate.

Abstract:
In this work, we employ a kinetic theory based approach to predict the hydrodynamic forces on electromechanical resonators operating in gaseous media. Using the Boltzmann-BGK equation, we investigate the influence of the resonator geometry on the fluid resistance in the entire range of nondimensional frequency variation $0\le\tau\omega\le\infty$; here the fluid relaxation time $\tau=\mu/p$ is determined by the gas viscosity $\mu$ and pressure $p$ at thermodynamic equilibrium, and $\omega$ is the (angular) oscillation frequency. Our results support the experimentally observed transition from viscous to viscoelastic flow in simple gases at $\tau\omega\approx1$. They are also in remarkable agreement with the measured geometric effects in resonators in a broad linear dimension, frequency, and pressure range.

Abstract:
Using kinetic equation in the relaxation approximation (RTA), we investigate a flow generated by an infinite plate oscillating with frequency $\omega$. Geometrical simplicity of the problem allows a solution in the entire range of dimensionless frequency variation $0\leq \omega \tau\leq \infty$, where $\tau$ is a properly defined relaxation time. A transition from viscoelastic behavior of Newtonian fluid ($\omega\tau\to 0$) to purely elastic dynamics in the limit $\omega\tau\to \infty$ is discovered. The relation of the derived solutions to microfluidics (high-frequency micro-resonators) is demonstrated on an example of a "plane oscillator .