# sandbox/ecipriano/run/epsteinplesset.c

# Epstein Plesset

This test case was borrowed from sandbox/farsoiya and it was adapted to my phase change model. It describes the consumption of a liquid droplet in an isothermal environment, where the evaporation is driven by a gradient of chemical species.

This test case considers mass transfer in pure diffusive conditions. Therefore, the Stefan flow is not considered in the theretical solution, obtained assuming quasi-static conditions. In this context, the chemical species concentration profile is assumed to be equal to the steady-state concentration profile, at any simulation time instants. This approximation is justified when the diffusion is much faster than the interface regression velocity. Using this approximation, the theretical solution describing the evolution of the droplet radius in time was obtained by Epstein and Plesset in 1950:

\displaystyle \dfrac{dR}{dt} = -MW\dfrac{\mathcal{D} (\hat{c} - c_{bulk})}{\rho_g} \left(\dfrac{1}{R} + \dfrac{1}{\sqrt{\pi \mathcal{D} t}}\right)

Even if evaporation.h is specifically conceived for evaporation problems including the Stefan convection, its formulation is general and it works also for diffusive conditions. The initial setup of this simulation allows the expansion term in the projection step to be null due to the density ratio equal to 1, and the interface mass fraction is chosen to be sufficiently small, so that the total vaporization rate tends to the pure diffusive conditions:

## Simulation Setup

We use the centered solver with the divergence source term, and the extended velocity can be obtained using the centered-doubled procedure. Note that, for the conditions analyzed in this test case, the extended velocity calculation can be skipped, in order to save computational time, since *uf* and *ufext* will have exactly the same value. The evaporation model is used in combination with the species-gradient mechanism, which manages the calculation of the vaporization rate for the pure droplet, and the solution of the chemical species mass fraction field.

```
#include "axi.h"
#include "navier-stokes/centered-evaporation.h"
#define ufext uf
#include "two-phase.h"
#include "tension.h"
#include "evaporation.h"
#include "species-gradient.h"
#include "view.h"
```

### Boundary Consitions

We let symmetry boundary conditions everywhere. The top and right mass fractions are set to the bulk value.

```
YG[top] = dirichlet (0.);
YG[right] = dirichlet (0.);
```

The original simulation setup is expressed in terms of chemical species concentration, while the species-gradient.h model solves the mass fraction fields. Therefore, we declare a concentration field used just for post-processing.

`scalar c[];`

### Problem Data

We declare the variables required by the species-gradient model, and other useful quantities.

```
int maxlevel, minlevel = 5;
double Dmix1, Dmix2, YIntVal, MW;
double YL0, YG0, R0, effective_radius0;
int main (void) {
```

We set the material properties of the fluids, the initial mass fractions, and the interface mass fraction.

```
rho1 = 1., rho2 = 1.;
mu2 = 1./20., mu1 = mu2/20.;
Dmix1 = 0., Dmix2 = 1.;
MW = 0.001;
YG0 = 0., YL0 = 1.*MW/rho1, YIntVal = YL0*0.8;
```

We change the dimension of the domain, as a function of the initial droplet radius.

` R0 = 1., L0 = 10.*R0;`

We change the surface tension coefficient of the droplet. The surface tension of the original test case is set to 0. A non-null surface tension is used here in order to reduce the time step, for the stability of the diffusion step, which includes an explicit source.

` f.sigma = 0.1;`

We increase the grid size to speed up the simulation on the Basilisk wiki server.

```
for (maxlevel=8; maxlevel<=8; maxlevel++) {
init_grid (1 << maxlevel);
run();
}
}
#define circle(x,y,R) (sq(R) - sq(x) - sq(y))
```

In the initialization event, the volume fraction is initialized at the lower-left corner of the domain, exploiting the spherical symmetry. We initialize also he mass fractions in gas and in liquid phase, and we compute the effective initial radius from the volume fraction field.

```
event init (i = 0) {
fraction (f, circle(x,y,R0));
foreach() {
YL[] = f[]*YL0;
YG[] = (1. - f[])*YG0;
Y[] = YL[] + YG[];
}
effective_radius0 = cbrt (3.*statsf(f).sum);
}
```

We refine the domain according to the interface and the concentration field.

```
#if TREE
event adapt (i++) {
adapt_wavelet_leave_interface ({c,u.x,u.y}, {f},
(double[]){1e-3,1e-3,1.e-3}, maxlevel, minlevel, 1);
}
#endif
```

## Post-Processing

The following lines of code are for post-processing purposes.

### Compute Concentration

After the solution of the tracer diffusion step, the concentration is reconstructed from the mass fraction field. We do that in the *properties* event, which is executed right after *tracer_diffusion*.

```
event properties (i++) {
foreach()
c[] = YG[]*rho1/MW + f[]*1.;
}
```

### Output File

We write on a file the droplet radius and the concentration of the chemical species in a specific point of the domain, in time.

```
event output_data (i++) {
char name[80];
sprintf (name, "OutputData-%d", maxlevel);
static FILE * fp = fopen (name, "w");
double effective_radius = cbrt (3.*statsf(f).sum);
double radius_ratio = effective_radius / effective_radius0;
fprintf (fp, "%g %g %g %g\n",
t, effective_radius, radius_ratio,
interpolate(c, (1. + 0.2)*cos(M_PI/4), (1. + 0.2)*sin(M_PI/4) , 0) );
fflush (fp);
}
```

### Movie

We write the animation with the evolution of the concentration field and the gas-liquid interface.

```
event movie (t += 10.; t <= 600.) {
clear();
view (tx = -0.5, ty = -0.5);
draw_vof ("f", lw = 1.5);
squares ("c", min = 0., max = 1., linear = true);
save ("movie.mp4");
}
```

### Dump Files

We add the possibility to print dump files for post-processing of to restart the simulation.

```
#if DUMP
event snapshot (t += 100.,last) {
char name[80];
sprintf (name, "snapshot-%g", t);
dump (name);
}
#endif
```

## Results

```
reset
set xlabel "t Dmix/D_0^2"
set ylabel "(D/D_0)^2"
set size square
set key top right
set grid
plot "../data/epsteinplesset.sol" u ($1/4):2 w p ps 1.4 title "Theretical", \
"OutputData-8" u ($1/4):2 w l lw 2 title "LEVEL 8"
```

```
reset
set xlabel "t Dmix/D_0^2"
set ylabel "Concentration [mol/m^3]"
set size square
set key top right
set grid
plot "../data/epsteinplesset.sol" u ($1/4):3 w p ps 1.4 title "Theoretical", \
"OutputData-8" u ($1/4):4 w l lw 2 title "LEVEL 8"
```

## References

[epstein1950stability] |
Paul S Epstein and Milton S Plesset. On the stability of gas bubbles in liquid-gas solutions. |