OK, I have encountered this theory again, after many years. The idea is to describe separation between two phases as the minimum of a free energy with respect to an order parameter :
When a becomes negative, the minimum of g changes from 0 to other values, . The function then has the celebrated double minimum feature, which features prominently in many symmetry-breaking theories, including of course the appearance of particle mass mass and, you know, the Big Bang and the Universe.
But here we are just considering phase separation in materials. The interface between two coexisting phases must have some associated cost, and the simplest (lowest order) way to include it is by introducing a total free energy functional
This is also called a London-Linzburg-Landau free energy, also appearing in their theory of superconductors.
Now, parameters a, b, and c are not easy to measure (or, at least, estimate) experimentally, but they are related to: the surface tension, the width of the interface, and the magnitude of the bulk equilibrium order parameter (i.e. ). Here I show how to obtain it in a slightly more general setting, since I was not able to find it on the internet (it can be found e.g, in the book by Rowlinson & Widom).
The Taylor-Green vortex sheet is a solution to the 2D Navier-Stokes equations for an incompressible Newtonian fluid:
where is the velocity field, is the pressure, is the kinematic viscosity, and is the fixed density of the fluid. The time derivative is a total derivative:
It is common to choose parameters that simplify the equations, but that can obscure the role of the different parameters. In the following, I provide expressions with all relevant parameters included, with their physical dimensions. I later pass to dimensionless, or reduced, units, in terms of the Reynolds and Courant numbers.
The Rayleigh-Taylor instability is a well-known benchmarck for CFD codes. The idea is to start with two phases, on on top of the other, the lighter one being underneath. The interface is slightly perturbed, and this plume appears. I describe a quick and dirty way of getting this instability.
Just a simple derivation of the role of attenuation in the standard sound wave equation. Original work: Stokes, 1845.
Starting with the Navier-Stokes momentum equation
where is a Lamé viscosity coefficient. The bulk viscosity coeficient is defined as . The last term is often neglected, even in compressible flow, but sound attenuation is one of the few cases where it may have some influence. All viscosities are assumed to be constant, but in this case this is a safe assumption, since we are going to assume small departures about equilibrium values.
Back to scientific computing. Lately, I have been using the Eigen libraries for linear algebra. They were recommended by the CGAL project, and they indeed share a common philosophy. Thanks to the rather high C++ level they can accomplish this sort of magic:
int n = 100;
VectorXd x(n), b(n);
// solve Ax = b
x = solver.solveWithGuess(b,x0);
Notice that A is a sparse matrix! I am next describing how to use this in order to solve the 1D Poisson equation.
A quick cheatsheet for OpenFOAM. In italics, things that are useful but not part of OpenFOAM proper. Interesting read: The OpenFOAM Technology Primer
Shortcuts to directories
(type “alias” to reveal these)
- run (go to own’s running directory)
- foam3rdParty (hit <tab> for these longish commands!)
- echo $FOAM_ <tab> (directories)
- echo $WM_ <tab> (building, aka compiling, settings)
- Import / export
- fluentMeshToFoam, etc …
- foamHelp (e.g. foamHelp boudary -field U)
- many, many others
- foamNew source App …
- doxygen doxyfile
- aliases for settings: wm32, wm64, wmSP, wmDP, wmSET, wmUNSET
We ended up the article about weighted residuals pointing out three possible approaches to the problem. We expand them here.