Projection method for droplet dynamics on groove-textured surface with merging and splitting

The geometric motion of small droplets placed on an impermeable textured substrate is mainly driven by the capillary effect, the competition among surface tensions of three phases at the moving contact lines, and the impermeable substrate obstacle. After introducing an infinite dimensional manifold with an admissible tangent space on the boundary of the manifold, by Onsager's principle for an obstacle problem, we derive the associated parabolic variational inequalities. These variational inequalities can be used to simulate the contact line dynamics with unavoidable merging and splitting of droplets due to the impermeable obstacle. To efficiently solve the parabolic variational inequality, we propose an unconditional stable explicit boundary updating scheme coupled with a projection method. The explicit boundary updating efficiently decouples the computation of the motion by mean curvature of the capillary surface and the moving contact lines. Meanwhile, the projection step efficiently splits the difficulties brought by the obstacle and the motion by mean curvature of the capillary surface. Furthermore, we prove the unconditional stability of the scheme and present an accuracy check. The convergence of the proposed scheme is also proved using a nonlinear Trotter-Kato's product formula under the pinning contact line assumption. After incorporating the phase transition information at splitting points, several challenging examples including splitting and merging of droplets are demonstrated.Comment: 26 page

A Multiscale Diffuse-Interface Model for Two-Phase Flow in Porous Media

In this paper we consider a multiscale phase-field model for capillarity-driven flows in porous media. The presented model constitutes a reduction of the conventional Navier-Stokes-Cahn-Hilliard phase-field model, valid in situations where interest is restricted to dynamical and equilibrium behavior in an aggregated sense, rather than a precise description of microscale flow phenomena. The model is based on averaging of the equation of motion, thereby yielding a significant reduction in the complexity of the underlying Navier-Stokes-Cahn-Hilliard equations, while retaining its macroscopic dynamical and equilibrium properties. Numerical results are presented for the representative 2-dimensional capillary-rise problem pertaining to two closely spaced vertical plates with both identical and disparate wetting properties. Comparison with analytical solutions for these test cases corroborates the accuracy of the presented multiscale model. In addition, we present results for a capillary-rise problem with a non-trivial geometry corresponding to a porous medium

Liquid-gas-solid flows with lattice Boltzmann: Simulation of floating bodies

This paper presents a model for the simulation of liquid-gas-solid flows by means of the lattice Boltzmann method. The approach is built upon previous works for the simulation of liquid-solid particle suspensions on the one hand, and on a liquid-gas free surface model on the other. We show how the two approaches can be unified by a novel set of dynamic cell conversion rules. For evaluation, we concentrate on the rotational stability of non-spherical rigid bodies floating on a plane water surface - a classical hydrostatic problem known from naval architecture. We show the consistency of our method in this kind of flows and obtain convergence towards the ideal solution for the measured heeling stability of a floating box.Comment: 22 pages, Preprint submitted to Computers and Mathematics with Applications Special Issue ICMMES 2011, Proceedings of the Eighth International Conference for Mesoscopic Methods in Engineering and Scienc

Role of particle conservation in self-propelled particle systems

Actively propelled particles undergoing dissipative collisions are known to develop a state of spatially distributed coherently moving clusters. For densities larger than a characteristic value, clusters grow in time and form a stationary well-ordered state of coherent macroscopic motion. In this work we address two questions. (i) What is the role of the particles’ aspect ratio in the context of cluster formation, and does the particle shape affect the system’s behavior on hydrodynamic scales? (ii) To what extent does particle conservation influence pattern formation? To answer these questions we suggest a simple kinetic model permitting us to depict some of the interaction properties between freely moving particles and particles integrated in clusters. To this end, we introduce two particle species: single and cluster particles. Specifically, we account for coalescence of clusters from single particles, assembly of single particles on existing clusters, collisions between clusters and cluster disassembly. Coarse graining our kinetic model, (i) we demonstrate that particle shape (i.e. aspect ratio) shifts the scale of the transition density, but does not impact the instabilities at the ordering threshold and (ii) we show that the validity of particle conservation determines the existence of a longitudinal instability, which tends to amplify density heterogeneities locally, and in turn triggers a wave pattern with wave vectors parallel to the axis of macroscopic order. If the system is in contact with a particle reservoir, this instability vanishes due to a compensation of density heterogeneities

GIZMO: A New Class of Accurate, Mesh-Free Hydrodynamic Simulation Methods

We present two new Lagrangian methods for hydrodynamics, in a systematic comparison with moving-mesh, SPH, and stationary (non-moving) grid methods. The new methods are designed to simultaneously capture advantages of both smoothed-particle hydrodynamics (SPH) and grid-based/adaptive mesh refinement (AMR) schemes. They are based on a kernel discretization of the volume coupled to a high-order matrix gradient estimator and a Riemann solver acting over the volume 'overlap.' We implement and test a parallel, second-order version of the method with self-gravity & cosmological integration, in the code GIZMO: this maintains exact mass, energy and momentum conservation; exhibits superior angular momentum conservation compared to all other methods we study; does not require 'artificial diffusion' terms; and allows the fluid elements to move with the flow so resolution is automatically adaptive. We consider a large suite of test problems, and find that on all problems the new methods appear competitive with moving-mesh schemes, with some advantages (particularly in angular momentum conservation), at the cost of enhanced noise. The new methods have many advantages vs. SPH: proper convergence, good capturing of fluid-mixing instabilities, dramatically reduced 'particle noise' & numerical viscosity, more accurate sub-sonic flow evolution, & sharp shock-capturing. Advantages vs. non-moving meshes include: automatic adaptivity, dramatically reduced advection errors & numerical overmixing, velocity-independent errors, accurate coupling to gravity, good angular momentum conservation and elimination of 'grid alignment' effects. We can, for example, follow hundreds of orbits of gaseous disks, while AMR and SPH methods break down in a few orbits. However, fixed meshes minimize 'grid noise.' These differences are important for a range of astrophysical problems.Comment: 57 pages, 33 figures. MNRAS. A public version of the GIZMO code, user's guide, test problem setups, and movies are available at http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.htm