constant_2D.c

2D bubble with constant mass transfer rate
- References

2D bubble with constant mass transfer rate

We model a 2D dissolving bubble with constant mass transfer rate. A similar test can be found in Section 4.2 of Gennari et al.,2022.

The analytical solution for the radius reads: \displaystyle R(t) = R(0) + \frac{\dot{m}}{\rho_d}

This is the 2D version of the axisymmetric problem constant_axi.c.

#define F_ERR 1e-10
#define MASS_TRANSFER_RATE (-1e-3)

#include "grid/quadtree.h"
#define PHASE_CHANGE 1
#include "../navier-stokes/centered.h"
#include "../two-phase.h"
#include "tension.h"
#include "../diffusion.h"
#include "../phase_change_pure_species.h"
#include "../../adapt_wavelet_leave_interface.h"


#define RHOR 1000.
#define MUR 100.

#define WIDTH 10
#define TEND 1.

int LEVEL = 9;

We add the transport of a soluble tracer. But this is not relevant in this case, since the mass transfer rate is fixed.

scalar a_c[];
scalar * species = {a_c};
scalar m_a[];
scalar * m_list = {m_a};

int main()
{
  size (WIDTH);
  origin (-L0/2., -L0/2.);
  init_grid (64);

  rho1 = 1.;
  rho2 = rho1/RHOR;
  mu1 = 1.;
  mu2 = mu1/MUR;

  f.tracers = species;
  f.sigma = 1.;
  
  a_c.inverse = false;
  a_c.mol_mass = 1./RHOR; //molar mass
  a_c.diff = false; //we turn off diffusion of the tracer
  
  tt = 0.;

  TOLERANCE = 1e-4;
  DT = 1e-3;
  run();
}

We need outflow boundary conditions to let the liquid enter the domain as the bubble dissolves.

//Bcs
u.n[left] = neumann(0.);
p[left] = dirichlet(0.);
pf[left] = dirichlet(0.);

u.n[right] = neumann(0.);
p[right] = dirichlet(0.);
pf[right] = dirichlet(0.);

u.n[bottom] = neumann(0.);
p[bottom] = dirichlet(0.);
pf[bottom] = dirichlet(0.);

u.n[top] = neumann(0.);
p[top] = dirichlet(0.);
pf[top] = dirichlet(0.);

event init (t = 0) {
  refine (sq(x) + sq(y) < sq(4) && level < LEVEL);
  fraction (f, sq(x) + sq(y) - sq(1.));
}

We log the volume of the bubble

event log_simulation (i++) {
  double vol = 0.;
  foreach(reduction(+:vol))
    vol += (1. - f[])*dv();

  fprintf (stderr, "%g %g\n", t, vol);
  fflush (stderr);
}

event adapt (i++)
{
  double uemax = 1e-2;
  adapt_wavelet_leave_interface ({u.x,u.y}, {f}, (double[]){uemax,uemax,1e-3}, LEVEL, 5, 1);
}

event stop_simulation (t = TEND) {
  return 1;
}

References

[gennari2022]

Gabriele Gennari, Richard Jefferson-Loveday, and Stephen J. Pickering. A phase-change model for diffusion-driven mass transfer problems in incompressible two-phase flows. Chemical Engineering Science, 259:117791, 2022. [ DOI | http ]