# src/test/fall.c

# Impact of a viscoelastic drop on a solid

We solve the axisymmetric, incompressible, variable-density, Navier–Stokes equations with two phases and use the log-conformation method to include viscoelastic stresses. The curvature module is used to compute interface properties.

```
#include "axi.h"
#include "navier-stokes/centered.h"
#include "two-phase.h"
#include "log-conform.h"
#include "curvature.h"
```

The density and viscosity ratios are 1000. The Reynolds number based on the droplet diameter, velocity and viscosity is 5 and the Froude number is 2.26.

```
#define RHO_r 0.001
#define MU_r 0.001
#define RE 5.
#define FR 2.26
#define LEVEL 7
```

The dimensionless viscoelastic properties used are the ratio of the solvent to the total viscoelastic viscosity (polymeric plus solvent) and the Weissenberg number.

```
#define BETA 0.1
#define WI 1.0
scalar lambdav[], mupv[];
```

The drop comes from the right. We allow the fluid to get through that boundary.

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

The wall is at the left side. We apply a no-slip boundary condition and a non-wetting condition for the VOF tracer.

```
u.t[left] = dirichlet(0);
tau_qq[left] = dirichlet(0);
f[left] = 0.;
int main() {
```

The domain spans $[0:2.6]\times [0:2.6]$.

```
size (2.6);
init_grid (1 << LEVEL);
```

The densities and viscosities are defined by the parameters above.

```
rho1 = 1.;
rho2 = RHO_r;
mu1 = BETA/RE;
mu2 = MU_r/RE;
```

The viscoelastic fields will be set below.

```
mup = mupv;
λ = lambdav;
```

We set a maximum timestep. This is necessary for proper temporal resolution of the viscoelastic stresses.

```
DT = 2e-3;
run();
}
event init (t = 0) {
```

At a wall of normal $\mathbf{\text{n}}$ the component of the viscoelastic stress tensor $ta{u}_{{p}_{nn}}$ is zero. Since the left boundary is a wall, we set $ta{u}_{{p}_{xx}}$ equal to zero at that boundary.

```
scalar s = tau_p.x.x;
s[left] = dirichlet(0.);
```

The drop is centered on (2,0) and has a radius of 0.5.

` fraction (f, - sq(x - 2.) - sq(y) + sq(0.5));`

The initial velocity of the droplet is -1.

```
foreach()
u.x[] = - f[];
}
```

We add the acceleration of gravity.

```
event acceleration (i++) {
face vector av = a;
foreach_face(x)
av.x[] -= 1./sq(FR);
}
```

We update the viscoelastic properties. Only the droplet is viscoelastic.

```
event properties (i++) {
foreach() {
mupv[] = (1. - BETA)*clamp(f[],0,1)/RE;
lambdav[] = WI*clamp(f[],0,1);
}
boundary ({mupv, lambdav});
}
```

We adapt the solution at every timestep based on the interface and velocity errors.

```
#if TREE
event adapt (i++) {
adapt_wavelet ({f, u.x, u.y}, (double[]){1e-2, 5e-3, 5e-3},
maxlevel = LEVEL, minlevel = LEVEL - 2);
}
#endif
```

We track the spreading diameter of the droplet.

```
event logfile (i += 20; t <= 5) {
scalar pos[];
position (f, pos, {0,1});
fprintf (stderr, "%g %g\n", t, 2.*statsf(pos).max);
}
```

We generate a movie of the interface shape.

```
#include "view.h"
event viewing (i += 10) {
view (width = 400, height = 400, fov = 20, ty = -0.5,
quat = {0, 0, -0.707, 0.707});
clear();
draw_vof ("f", lw = 2);
squares ("u.x", linear = true);
box (notics = true);
mirror ({0,1}) {
draw_vof ("f", lw = 2);
squares ("u.y", linear = true);
box (notics = true);
}
save ("movie.mp4");
```

We can optionally visualise the results while we run.

```
#if 0
static FILE * fp = popen ("bppm","w");
save (fp = fp);
#endif
}
```

## Results

The time evolution of the maximum diameter is close to that reported by Figueiredo et al. (2016).

## References

[figueiredo2016] |
RA Figueiredo, CM Oishi, AM Afonso, IVM Tasso, and JA Cuminato. A two-phase solver for complex fluids: Studies of the Weissenberg effect. |