Posts for the month of September 2009

Hydrodynamic limit with elevated interface diffusion

Increasing $\bar{M}_i$ certainly makes a big difference.

source:trunk/reactiveWetting/184/dropRadius.png

The image shows the drop radius against time. The inertial time scale scales time and initial radius of the drop scales the spreading distance $r$. There is an order of magnitude change in time between when the spreading occurs for the old hydrodynamic case and the new ones. I will need to plot the spreading rate curves to see how this knocks the solution away from Spelt. Also, there seems to be some difference between $\bar{M}_i=10^{-7}$ and $\bar{M}_i=10^{-5}$. Note that $t_{\text{inertial}}= 7 \times 10 ^{-8}$. The interface diffusion time scale is $t_{\text{ini}}=3\times 10^{-6}$ so we probably need to raise the interface diffusion coefficient by two orders from the fluid value to get it be less than the inertial time scale. If indeed that does matter. It certainly seems to.

  • Posted: 2009-09-25 18:50 (Updated: 2009-12-10 10:58)
  • Author: wd15
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Interface Diffusion

The following figure shows melting rates for various $\bar{M}_i$ with the sharp interpolation scheme.

source:trunk/reactiveWetting/146/box.png

Firstly, the melting rates agree well with analytical rates. Secondly, as pointed out in the previous post, $\bar{M}_i$ does not have much influence on the melting. It has some influence at early times, which is evidently related to hydrodynamics. By about $t = 10^{-5}$, the curves for $\bar{M}_f = 10^{-7}$ have almost all collapsed onto each other.

  • Posted: 2009-09-25 18:31 (Updated: 2009-09-25 18:55)
  • Author: wd15
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Interpolation Schemes for M

Carrying on from blog:SolidDiffusion1 it is clear that we have to modify the interpolation scheme. It showed that the time scale in the interface has to be fast enough to not influence the bulk diffusion properties. It is clear that for the purposes of diffusion that having the interface diffusion coefficient equal to the bulk diffusion is adequate. However, it may be necessary for hydrodynamic reasons to have faster diffusion in the interface.

source:trunk/reactiveWetting/146/interpolationScheme.png

In this figure $k$ refers to

$\bar{M} = \bar{M}_s^{\phi^k} \bar{M}_f^{1 - \phi^k} $

and $l$ refers to

$\bar{M} = \bar{M}_b + 2^{2l} \left( \bar{M}_i - \bar{M}_b^* / 2 \right) \phi^l \left(1 - \phi \right)^l $

where

$\bar{M}_{b} = \bar{M}_s \phi + \bar{M}_f \left( 1 - \phi \right) $

This red curve demonstrates that allowing the interface to have the fluid diffusion coefficient all the way through gets us back to the bulk limited value. It also demonstrates that the $l$ based interpolation scheme doesn't really work because it over influences the diffusion in the bulk and allows it to have much faster diffusion than it otherwise should do. This is due to the extent that $\phi$ extends out beyond the interface region. As shall be seen, a sharp cut off at $\phi=0.05$ ameliorates this issue.

As shall be seen in the subsequent post, the black line is actually agrees with the analytical solution and is bulk limited. It is also clear that as $l$ goes from 2 to 1 the melting is faster due to this issue.

  • Posted: 2009-09-25 18:05 (Updated: 2009-09-25 18:56)
  • Author: wd15
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Analytic Melting Solution

The following os an analytical melting solution from Bill and Geoff

$l = K \sqrt{4 D t}$

where $l$ is the distance the interface has moved. $K$ is given by solving

$K + S \frac{\exp \left( -K^2 \right) }{1 - \text{erf} \left( K \right) } \frac{1}{\sqrt{\pi}} = 0$

and $S$ is given by,

$S = \frac{X_1^l - X_1^{l, \text{equ}}}{X_1^s - X_1^l}$

In this case $X_1^s = X_1^{s, \text{equ}}$.

  • Posted: 2009-09-25 17:40 (Updated: 2009-09-25 17:51)
  • Author: wd15
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Influence of Solid Diffusion

The 1D simulations have demonstrated that the solid diffusion coefficient has a major impact on the rate of melting. The following image shows melting with $\bar{M}_s$ being varied.

source:trunk/reactiveWetting/146/solidDiffusion.png

It is clear that $\bar{M}_s$ has a considerable impact on the melting. This is due to the limiting nature of the interface diffusion. The interpolation scheme is

$\bar{M} = \bar{M}_s^{\phi} \bar{M}_f^{1 - \phi}$

which drives the value of the interface diffusion down a lot in the solid. The system is bulk limited as $\bar{M}_s$ approaches $10^{-4}$ as can be seen from the black dashed line. We though that the system was limited by the phase filed motion, but in fact it obviously isn't. The time scale for the phase field equation is given by,

$t_{\phi} = \frac{m T}{M_{\phi} p \left( \phi = 0.5 \right) \left( A_1 - A_2 \right) \rho \Delta X_1} \approx 8 \times 10^{-10}$

which is faster than the bulk diffusion time scale even when $\bar{M}_f = 10^{-2}$,

$t_{bulk} = \frac{l^2}{K^2 4 D} \approx 8 \times 10^{-9}$

where $l$ is the distance moved which I have chosen such that $l=X$. A reasonable choice in this case. Hardly surprising that this system is bulk limited.

So why does the solid diffusion eventually slow things down? Maybe we could say that the interface has to have reached some sort of equilibration before the motion occurs. This would lead to a time scale on the order of,

$t_{int} = \frac{X^2}{D} \approx 3 \times 10^{-11} $

This is probably too fast, it probably also has a prefactor like the $K$ in the bulk. However, it should be faster than the bulk kinetics. This seems obvious now for some reason. Anyway, it makes sense that we need to reduce the solid diffusion by three orders of magnitude before the interface kinetics start to become slower than the bulk kinetics.

  • Posted: 2009-09-25 16:40 (Updated: 2009-09-25 18:56)
  • Author: wd15
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Time scales asccoiated with diffusion

Having done a large number of 1D simulations it has become apparent that the comments in blog:SolidDiffusion entry seem to hit the nail on the head and are probably correct. The following image shows the liquid fraction from a number of 1D simulations that start with an system that has a 1/3 of each phase, but with a low concentration is the liquid. The liquid should expand to 1.12 of its original volume.

source:trunk/reactiveWetting/146/solidDiffusion.png

Clearly, the solid diffusion coefficient has no effect on the melting as long as it is above a given threshold. This agrees with our earlier time scale analysis in the previous blog entry. In the following image the interpolation scheme is varied. It is clear that if the fluid diffusion is used in the interface the solid diffusion coefficient does not effect the melting.

source:trunk/reactiveWetting/146/interpolationScheme.png

Given the sharp interpolation scheme, is it still possible to have interface effects. Well, maybe, if $t_{int} > t_{diff}$ we could be in for some problems. To avoid this:

$t_{int} < t_{diff}$

$\frac{X^4 R}{X_1 X_2 \epsilon_i m \rho L^2} < 1$

For the numbers that we are using, the expression on the right hand side is approximately 0.01. That means we are in good shape for the 2D simulations from the perspective of not being interface limited for melting.

However, the absolute value for $t_{int}$ is $6.7 \times 10^{-6}$, which is a lot slower than many of the other time scales in the system. The inertial time scale for example is two orders of magnitude faster. This means that we are in extremely bad shape since interface effects are still way slower than the spreading. Ideally for a physical system, interface effects should be much faster than the spreading. This would indicate that we could be interface limited for spreading, which is bad.

  • Posted: 2009-09-21 11:56 (Updated: 2009-09-21 12:03)
  • Author: wd15
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Melting dependence on solid diffusion coefficient

It is becoming apparent that the rate of melting is not independent of the solid diffusion coefficient. The following link is to a movie where the system has the same viscosity and diffusion coefficient (2E-3 and 2E-2) in all phases.

trunk/reactiveWetting/147/movie/movie.gif

The dynamics is quite fast and the pressure and chemical potential equilibration occur at similar time scales. There is a nice concentration gradient and the phase filed relaxes back as the liquid melts. Note how the chemical potentials equilibrate nicely through the interface and large gradients are quickly removed.

The following is a simulation where the solid diffusion coefficient is 2E-11 (down from 2E-2).

trunk/reactiveWetting/148/movie/movie.gif

From a pure diffusion argument the solid diffusion coefficient should have no influence on the melting rate of the liquid. Clearly in this example it is having an effect. If you look at frames from similar times in both the above simulations, the rate of melting is clearly different and the concentration profile in the liquid has no gradients in the second movie. Also, note that the chemical potential has not equilibrated through the solid liquid interface. This is very interesting.

This does seem somewhat disturbing. However, if we look at the diffusion equation, there are two time scales, the first being the diffusion time scale through the liquid,

$ t_{\text{diff}} = \frac{m \rho L^2}{R \bar{M}_f} $

where $L$ is the length of the liquid phase (it can't be shorter than this because the diffusion equation creates a profile across the entire liquid domain!). The second time scale is and interface time scale associated with the fourth order term,

$ t_{\text{int}} = \frac{X^4}{\bar{M}_s X_1 X_2 \epsilon_i} $

where $X$ is the depth of the interface. $t_{\text{diff}}$ is very fast about 9.27E-9 equivalent to the other time scales in the system such as the phase field and inertial flow scales. Since the chemical potential needs to equilibrate through the entire interface I am going to postulate that I can use \bar{M} in the solid to evaluate $ t_{\text{int}}$ giving a time scale of 2.35E-4. This is very slow. This represents a worse case since $\bar{M}$ in the interface is interpolated between the liquid and solid, but there are still regions of the interface that require a time scale close to this for equilibration.

This argument suggests that $\bar{M}_s$ needs to be large enough or $\bar{M}_l$ needs to be small enough such that $ t_{\text{int}} < t_{\text{diff}}$ and only at this point is melting independent of $\bar{M}_s$. This cross over point for our numbers is roughly when $\bar{M}_s$ is three or four orders of magnitude less than $\bar{M}_f$.

The other thing to note is that for a physical system $X$ is much smaller so for a realistic system $ t_{\text{int}} \ll t_{\text{diff}} $ in general.

Another hand wavy argument may be to say that $ t_{\text{int}} $ represents the time scale for the initial formation of the interface and if this is really slow all bets are off.

I am beginning to realize why it is important to have a more realistic interface depth. The above argument suggests the diffusion coefficient should be large enough such that the time scale associated with the fourth order term in the diffusion equation is always faster than other relevant time scales. All the spreading dynamics will be screwed up otherwise.

  • Posted: 2009-09-17 18:19 (Updated: 2009-09-21 11:22)
  • Author: wd15
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MPI command

mpirun -np 12 -machinefile machinefile ./input.py >> out 2>&1 &

  • Posted: 2009-09-10 18:21 (Updated: 2009-09-10 18:23)
  • Author: wd15
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Parallel on the Cluster

Results for parallel simulations on the cluster

  • 1 processor, r005
  • sweep times: 29.3520491123 23.3660550117 23.3256959915 23.1911690235 23.1299791336 22.9121041298 22.8525710106 23.0343871117 22.8907659054 23.039219141 23.1625990868 23.013864994
  • 2 processors, r006 r005
  • sweep times: 15.5924270153 12.1713759899 12.2448968887 12.2202451229 12.2246148586 12.2457139492 12.3183469772 12.3248221874 12.215831995 12.2335240841 12.2320721149
  • 4 processors, r006 r007 r006 r005
  • sweep times: 21.6786100864 6.6802740097 6.63790607452 6.84371995926 6.65304708481 7.14012789726 6.46965503693 6.99537992477 6.83790421486 6.62651705742 6.71980905533 6.77136397362 6.41707897186
  • 8 processors, r007 r008 r006 r007 r006 r008 r007 r005
  • sweep times: 17.380297184 17.7295508385 17.6523389816 17.4138000011 31.6395330429 17.4052929878 24.2372221947 31.4160881042 3.41191792488 3.63104009628 3.52340602875 10.1916849613 17.3036520481 10.5115458965 3.95448207855 3.49768590927
  • 16 processors, r007 r012 r008 r006 r007 r006 r022 r013 r012 r012 r008 r007 r011 r020 r010 r005
  • sweep times: 30.4704580307 15.7536921501 16.4713590145 17.762444973 3.03759098053 3.31603288651 15.6013848782 9.57383298874 43.7733981609 3.11927390099 3.11927390099 15.9374248981 2.89413809776 2.70106816292 2.70635294914 16.7859950066

Results demonstrate:

  • good scaling from 1 to 4 processors
  • something weird happening at 8 processors
  • fastest times for 8 processors does scale correctly, might indicate network traffic
  • similarly, for 16 processors there is a 2.7s sweep
  • Posted: 2009-09-10 14:17
  • Author: wd15
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Melting Simulations

source:trunk/reactiveWetting/128/dropRadiusMelt.png

  • 125 is the base simulation
  • 126 is with 50% reduction of $X_1$.
  • 127 is with $M_{\phi}$ increased by an order of magnitude
  • Not a great deal of variation will take $M_{\phi}$ higher.
  • 126 is has the slowest spreading. makes sense since a reduction in concentration reduces spreading rate (slightly more melting maybe).
  • 127 is faster as $M_{\phi}$ is increased
  • Posted: 2009-09-10 12:14
  • Author: wd15
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Base Simulation with Higher Viscosity

source:trunk/reactiveWetting/128/dropRadiusBase.png

  • 119 is the original base simulation with $\mu_s = 2 \times 10^4$
  • 125 has a higher solid viscosity $\mu_s = 2 \times 10^4$ and a deeper solid region
  • 128 is the same as 125, but with a tolerance of $R = 1 \times 10^{-2}$ instead of $R = 1 \times 10^{-1}$.
  • Figure demonstrates that 128 and 125 map fairly well indicating that $R = 1 \times 10^{-1}$ is adequate
  • 119 has slower spreading dynamics than 125, which is counter intuitive, this may have something to do with the deeper solid region
  • It is clear from the movies for source:trunk/reactiveWetting/119/movie/119.gif and source:trunk/reactiveWetting/125/movie/125.gif, that the solid wave are substantially decreased with the higher solid viscosity.

  • Posted: 2009-09-10 11:45 (Updated: 2009-09-10 11:48)
  • Author: wd15
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