Allen-Cahn Equation

Applicability of two kinds of computational-fluid-dynamics method adopting Cahn-Hilliard (CH) and Allen-Cahn (AC)-type diffuse-interface advection equations based on a phase-field model (PFM) is examined to simulation of motions of microscopic incompressible two-phase fluid on solid surface. A capillarity-driven gas-liquid motion in rectangular channel is simulated by use of a PFM method for solving Navier-Stokes (NS) equations and a CH equation, whereas an immiscible liquid-liquid flow in a microchannel with T-junction and square cross section is simulated by use of another PFM method proposed in this study, which adopts a lattice-Boltzmann method based on fictitious particles kinematics as numerical scheme for solving NS equations and an AC equation that is modified to improve volume-of-fluid conservation. The major findings are as follows: (1) effect of capillary force on the dynamic two-phase fluid system with a high density ratio is well predicted for cross-sectional aspect ratio of the channel = 1 and 2; (2) mono-dispersed slug flow pattern transition is reproduced in good agreement with experimental observations in terms of variation in length and interval of droplets as increasing their volumetric flow rates at a constant flow rate ratio = 1. These results prove that the PFM methods can be used for analyzing two-phase fluid motions in various microfluidic devices and micro fabrication processes.

A preliminary numerical simulation of the microscopic two-phase fluid motion on a solid surface was conducted using an interface-tracking method based on the phase-field model (PFM). Two variations of the lattice Boltzmann method (LBM) based on fictitious particle kinematics are proposed for solving diffuse-interface advection equations which were revised to improve volume-of-fluid conservation in the PFM simulations. The major findings are as follows: (1) the interface-tracking method accurately predicted the capillary force effect on dynamic two-phase fluid systems with a high density ratio between parallel plates; (2) the initial shape and volume of the two-phase fluid were retained adequately in linear translation with the use of the LBMs. These results proved that the PFM-based method and the LBM-based advection schemes can be used for simulating two-phase fluid motions in various macro- and microfluidics problems for devices, machineries and higher-throughput microdevice fabrication processes.

We consider minimal surfaces $M$ which are complete, embedded and have finite total curvature in $\R^3$, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation $\Delta u + f(u) = 0 \hbox{in} \R^3 $. Here $f=-W'$ with $W$ bistable and balanced, for instance $W(u) =\frac 14 (1-u^2)^2$. We assume that $M$ has $m\ge 2$ ends, and additionally

Cox and Matthews [3] developed a class of Exponential Time Differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic equations; Kassam and Trefethen [12] have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order (ETDRK4) scheme. They use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour for larger problems. Neither treatment addresses problems with nonsmooth data, where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully. Such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications, including computational chemistry and financial engineering. We introduce a new version of the fourth order Cox-Matthews, Kassam-Trefethen ETDRK4 scheme designed to...