Posts for the month of July 2008

Reactive Wetting Equations in Cylindrical Coordinates

$\partial_j = \left( \partial_r, \partial_z \right) $

Note

We need to remove the radial distance $r$ and embed it in the term's coefficient. For a general equation in cylindrical coordinates that are axisymmetric the general equation can be written,

$ \frac{\partial \phi}{\partial t} + \frac{1}{r} \partial_r \left( r u_r \phi \right) + \partial_z \left( u_z \phi \right) = \frac{1}{r} \partial_r \left( r \Gamma \partial_r \phi \right) + \partial_z \left( \Gamma \partial_z \phi \right)$

Multiply through by $r$ to get,

$ \frac{\partial \left( r \phi \right)}{\partial t} + \partial_r \left( r u_r \phi \right) + \partial_z \left( r u_z \phi \right) = \partial_r \left( r \Gamma \partial_r \phi \right) + \partial_z \left( r \Gamma \partial_z \phi \right)$

which is,

$ \frac{\partial \left( r \phi \right) }{\partial t} + \partial_j \left( r u_j \phi \right) = \partial_j \left( r \Gamma \partial_j \phi \right) $

Thus, in general, it is very easy to recast equations in cylindrical axi-symmetric coordinates with fipy. The fourth order term in the concentration equations may be an issue and the correction equation also has a fourth order term.

Fourth Order Term

$\frac{1}{r} \partial_r \left( r \Gamma_1 \partial_r \left[ \frac{1}{r} \partial_r \left( r \Gamma_2 \partial_r \phi \right) + \partial_z \left( \Gamma_1 \partial_z \phi \right) \right] \right) + \partial_z \left( \Gamma_1 \partial_z \left[ \frac{1}{r} \partial_r \left( r \Gamma_2 \partial_r \phi \right) + \partial_z \left( \Gamma_1 \partial_z \phi \right) \right] \right) $

This can be rewritten as

$\frac{1}{r} \partial_r \left( r \Gamma_1 \partial_r \left[ \frac{1}{r} \partial_r \left( r \Gamma_2 \partial_r \phi \right) + \frac{1}{r} \partial_z \left( r \Gamma_1 \partial_z \phi \right) \right] \right) + \frac{1}{r} \partial_z \left( r \Gamma_1 \partial_z \left[ \frac{1}{r} \partial_r \left( r \Gamma_2 \partial_r \phi \right) + \frac{1}{r} \partial_z \left( r \Gamma_1 \partial_z \phi \right) \right] \right) $

There is an awkward $\frac{1}{r}$ that is not next to a $\Gamma$ and thus can not just multiply a regular coefficient. The outer $\frac{1}{r}$ gets dealt with because we multiply all the terms in the equation by $r$. The above can be rewritten as,

$\frac{1}{r} \partial_j \left( \Gamma_1 \left[\partial_j - \frac{\hat{r}_j}{r} \right] \partial_k \left[ r \Gamma_2 \partial_k \phi \right] \right)$

Unfortunately, The last term needs to be added as a source term. If we had a third order convection term then it could be added implicitly.

Continuity

$ \frac{\partial \left(r \rho \right)}{\partial t} + \partial_j \left( r u_j \rho \right) = 0 $

Velocity

$ \frac{\partial \left(r \rho u_i \right)}{\partial t} + \partial_j \left( r u_j u_i \rho \right) = \partial_j \left( r \mu \left[ \partial_j u_i + \partial_i u_j \right] \right) - r \rho_k \partial_i \mu_k + \epsilon_k T r \rho_k \partial_i \left( \frac{1}{r} \partial_j \left(r \partial_j \rho_k \right) \right) $

Concentration Equation

$ \frac{\partial \left(r \rho X_2 \right)}{\partial t} + \partial_j \left( r u_j X_2 \rho \right) = \partial_j \left( \frac{ r \bar{M}}{T} X_1 X_2 \left[ \left(\frac{ \left( A_2 - A_1 \right) }{m} - \frac{ \left( e_2 - e_1 \right) }{m^2} \rho \right) \partial_j p + \left( 1 - p \right) \frac{ \left( e_2 - e_1 \right)}{m^2} \partial_j \rho \right] \right) $

$ + \partial_j \left( \frac{ r \bar{M} R }{m} \partial_j X_2 \right) - \partial_j \left( \bar{M} X_1 X_2 \left[ \epsilon_1 + \epsilon_2 \right] \left[\partial_j - \frac{\hat{r}_j}{r} \right] \partial_k \left[ r \rho \partial_k X_2 \right] \right) $

$+ \partial_j \left( r \bar{M} X_1 X_2 \partial_j \left[ \frac{1}{r} \partial_k \left( r \left[ \epsilon_1 X_1 - \epsilon_2 X_2 \right] \partial_k \rho \right) \right] \right)$

The last term can remain in its original form as it is a convection term.

Alternate Method

The formulation above could lead to some coding issues and maybe other difficulties concerning the fourth order source terms. A more expedient and relatively trivial method that is probably no less accurate is to define a CylindricalGrid2D object that has its mesh volumes and face areas modified accordingly. I just realized that the discretiztion is exactly equivalent in the finite volume method when using a cylindrically modified grid and rewriting the equations.

Reactive Wetting Paper

Outline

  1. Introduction
    1. Boiler plate - motivation
    2. Summary of physical parameters coinciding with typical applications (water-ice-salt, solder alloys)
    3. Thermodynamics
  2. Model
    1. Transport
    2. Thermodynamics
    3. Surface Tensions
  3. Parameters Choices - Numerics
    1. Governing equations summary.
    2. Numerical Approach
      1. Solution algorithm
      2. Parasitic Currents
    3. One dimensional liquid vapor analysis
      1. Dimensionless form of the pure-liquid vapor
      2. Convergence of a one-dimensional pure liquid-vapor system
        1. $\delta$,
        2. $Re$
        3. $M$,
        4. $Ca$
        5. $\kappa$ liquid
      3. Convergence of a one-dimensional binary liquid vapor system
        1. $Pe$, $Pe \rightarrow \infty$, What is the meaning?
        2. $Re > 1$ issues.
    4. Two dimensional Binary Solid-Liquid-Vapor
      1. Obtaining Plausible contact angles
      2. $\kappa$ solid
      3. Approximation of a solid with $\mu \rightarrow \infty$, density trapping.
      4. Comparison with analytical solution in equilibrium
  4. Results
    1. Extracting interface locations and junctions with phase field methods
      1. Ridge detection
      2. Wheeler method
      3. Villanueva method
    2. $\theta_{\text{contact}}$ versus $t$.
    3. Cox comparisons - look at later cox work with surfactants.
    4. $R\left(t\right)$ for different initial conditions.
    5. Pure Hydrodynamics with and without reactive wetting ($\theta_{\text{liquid}}$ vs. $\vec{v}$)
    6. Pretty pictures
    7. Compare with Gustav
      1. Steady state versus transient
      2. Compare streamlines
    8. $\theta$ versus diffusion
      1. $\bar{M}\ll 1$ versus $\bar{M}=0$
      2. $\bar{M}_{\phi}=0$
    9. Compare with S. Troian
    10. Spherical cap deviation
    11. Best measures of angle and shape
    12. Compare with Jim and Bill's right angle assumption

Copies of notes

Warren's notes

Wheeler's notes