Extended Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries

In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction-diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelled droplet.Comment: This document is the revised version of arXiv:0912.1288v

Smoothed Boundary Method for Solving Partial Differential Equations with General Boundary Conditions on Complex Boundaries

In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A function that has a prescribed value on the domain in which a differential equation is valid and smoothly but rapidly varying values on the boundary where boundary conditions are imposed is used to modify the original differential equations. The mathematical derivations are straight forward, and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions, (2) the diffusion equation with surface diffusion, (3) the mechanical equilibrium equation, and (4) the equation for phase transformation with additional boundaries. The solutions for a few of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelling droplet.Comment: A better smooth algorithm has been developed and tested, will soon replace Eq. 58 in page 16. We have also developed a level-set moving boundary SBM method, and it will replace the Navier-Stokes-Cahn-Hilliard type domain parameter tracking method in Section 5.

Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities

The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. A non-local lubrication equation is obtained; this non-local nature comes from the Laplace equation associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to moderately thick films for which it is shown that evaporation is diffusion limited and that the gas phase is saturated in vapour in the vicinity of the interface. The stability depends only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilising whereas capillarity and evaporation are stabilising processes. The results of the linear stability analysis are compared with the experiments of Poulard et al (2003) performed in a different geometry. In order to study the resulting patterns, the amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem unusually affects the amplitude equation

Phase separation during film growth

A diffusion equation describing phase separation during co‐deposition of a binary alloy is derived, and solved in the limit of dominant surface diffusion. Linear stability analysis yields results similar to bulk spinodal decomposition, except that long, and possibly all, wavelength are stabilized. Decomposition into two phases is investigated by solving the diffusion equation for lamellar and cylindrical symmetry. For the lamellar geometry, typically observed for near‐equal volume fractions, the diffusion equation does not yield wavelength selection criteria. These can be obtained if free energy minimization is assumed. For the cylindrical geometry, solutions for small volume fractions yield domain dimensions proportional to the deposition‐rate dependent surface diffusion length.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/71165/2/JAPIAU-72-2-442-1.pd

The Code DYN3DR for Steady-State and Transient Analyses of Light Water Reactor Cores with Rectangular Geometry

The code DYN3D/M2 was developed for steady-state and transient analysis of reactor cores with hexagonal fuel assemblies. The new version DYN3DR contains a method for neutron kinetics solving the two group neutron diffusion equation by a nodal method for cartesian geometry. The thermal-hydraulic model FLOCAL simulating the two phase flow of coolant and the fuel rod hehaviour is used in the two versions. The fundamentals of the neutron kinetics are described. The accuracy of the code is demonstated by comparisons with the results of rod ejection benchmarks for PWR with rectangular fuel assemblies. The developed algorithm of neutron kinetics are suitable for using parallel processing. The speedup of neutronic calculation is demonstrated for a steady state solution of diffusion equation

Thermal noise and dephasing due to electron interactions in non-trivial geometries

We study Johnson-Nyquist noise in macroscopically inhomogeneous disordered metals and give a microscopic derivation of the correlation function of the scalar electric potentials in real space. Starting from the interacting Hamiltonian for electrons in a metal and the random phase approximation, we find a relation between the correlation function of the electric potentials and the density fluctuations which is valid for arbitrary geometry and dimensionality. We show that the potential fluctuations are proportional to the solution of the diffusion equation, taken at zero frequency. As an example, we consider networks of quasi-1D disordered wires and give an explicit expression for the correlation function in a ring attached via arms to absorbing leads. We use this result in order to develop a theory of dephasing by electronic noise in multiply-connected systems.Comment: 9 pages, 6 figures (version submitted to PRB

RG flow of transport quantities

The RG flow equation of various transport quantities are studied in arbitrary spacetime dimensions, in the fixed as well as fluctuating background geometry both for the Maxwellian and DBI type of actions. The regularity condition on the flow equation of the conductivity at the horizon for the DBI action reproduces naturally the leading order result of {\it Hartnoll et al.}, [{\it JHEP}, {\bf 04}, 120 (2010)]. Motivated by the result of {\it van der Marel et al.}, [{\it science}, {\bf 425}, 271 (2003], we studied, analytically, the conductivity versus frequency plane by dividing it into three distinct parts: $\omega T$ and $\omega >> T$. In order to compare, we choose 3+1 dimensional bulk spacetime for the computation of the conductivity. In the $\omega <T$ range, the conductivity does not show up the Drude like form in any spacetime dimensions. In the $\omega > T$ range and staying away from the horizon, for the DBI action with unit dynamical exponent, non-zero magnetic field and charge density, the conductivity goes as $\omega^{-2/3}$, whereas the phase of the conductivity, goes as, $ArcTan(Im\sigma^{xx}/Re\sigma^{xx})=\pi/6$ and $ArcTan(Im\sigma^{xy}/Re\sigma^{xy})=-\pi/3$. There exists a universal quantity at the horizon that is the phase angle of conductivity, which either vanishes or an integral multiple of $\pi$. Furthermore, we calculate the temperature dependence to the thermoelectric and the thermal conductivity at the horizon. The charge diffusion constant for the DBI action is studied.Comment: 1+68 pages, 12 figures and 4 appendices; V2: The charge diffusion constant is calculated for arbitrary spacetime dimensions and related references added; v3: Connection with the RG flow of 1010.4036 is made; v4: Several corrections, typos fixed and a ref. adde

Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations

By considering a lattice model of extended phase space, and using techniques of noncommutative differential geometry, we are led to: (a) the conception of vector fields as generators of motion and transition probability distributions on the lattice; (b) the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure; (c) the general prescription for the observables' evolution in analogy with classical dynamics. We show that, in the limit of a continuous description, these results lead to the time evolution of observables in terms of (the adjoint of) generalized Fokker-Planck equations having: (1) a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion; (2) a drift term given by the microscopic average of the dynamical equations in the present context. These results are applied to 1D and 2D problems. Specifically, we derive: (I) The equations of diffusion, Smoluchowski and Fokker-Planck in velocity space, thus indicating the way random walk models are incorporated in the present context; (II) Kramers' equation, by further assuming that, motion is deterministic in coordinate spaceComment: LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfi

In-vitro cartilage growth: macroscopic mass transport modelling in a three-phase system

Transplantation of engineered tissues is of major interest as an alternative to autogenic alogenic or exogenic grafts. In this study, in vitro cartilage cell culture on a fibrous biodegradable polymer scaffold is under concern. The scaffold is first seeded with cells which adhere to the fibres and the system is then grown in a bioreactor. As reported in the literature, hydrodynamics and transport of nutrients and metabolic products during this growth process is of considerable importance, motivating our analysis. A one-equation macroscopic model was first developed in order to describe macroscopic mass transport during in vitro tissue growth using the volume averaging method. This model takes into account a three phase system composed of solid fibres, cell phase and fluid phase and allows determination of the macroscopic quantities as a function of microscopic properties and geometry at any stage of growth. In a second step, numerical tools for the computation of the effective properties were developed and validated. This validation is carried out using results available in the literature for some sub-classes of our model (namely, diffusion, diffusion/reaction and diffusion/advection problems in 2D systems). The behaviour of the macroscopic dispersion tensor for the complete model (diffusion/reaction/advection) in a three phase configuration is studied and the influence of different parameters such as the volume fractions of the phases, Peclet and Kinetic numbers is discussed

An active particle diffusion theory of flame quenching for laminar flames / Dorothy M. Simon and Frank E. Belles

An equation for quenching distance based on the destruction of chain carriers by the surface is derived. The equation expresses the quenching distance in terms of the diffusion coefficients and partial pressures of the chain carriers and gas phase molecules, the efficiency of the surface as a chain breaker, the total pressure of the mixture, and a constant which depends on the geometry of the quenching surface. Quenching distances measured by flashback for propane-air flames are shown to be consistent with the mechanism. The derived equation is used with the lean inflammability limit and a rate constant calculated from burning velocity data to estimate quenching distances for propane-air (hydrocarbon lean) flames satisfactorily