# Experiment:

We investigate the dynamics of an oil drop impacting an identical sessile drop sitting on a superamphiphobic surface. One example of such impacts:

Experiment: Impact of a hexadecane oil drop on another drop (of the same liquid) sitting on a superamphiphobic substrate. Experiment was done by Olinka Soto. Impact Weber number, We = 1.44, and the offset between the axes of the two drops is 0.48 times the equivalent radius of the drops.

On this page, I am presenting the code that we used to simulate the process shown in the above video. The results presented here are currently under review in Science Advances. For codes for all the cases included in the manuscript, please visit the GitHub repository. I did not upload all of them here to avoid repeatability.

# Numerical code

Id 1 is for the sessile drop, and Id 2 is mobile/impacting drop.

#include "grid/octree.h"
#include "navier-stokes/centered.h"
#define FILTERED // Smear density and viscosity jumps

To model non-coalescing drops, we use two different Volume of Fluid tracers (f1 and f2). For this, we use a modified version of two-phase.h. A proof-of-concept example is here.

#include "two-phaseDOD.h"
#include "tension.h"
#include "distance.h"

We use a modified adapt-wavelet algorithm available (here). It is written by César Pairetti (Thanks :)).

#include "adapt_wavelet_limited.h"

int MAXlevel = 12; // maximum level
#define MINlevel 5 // maximum level
#define tsnap (0.01) // timestep used to save simulation snapshot
// Error tolerances
#define fErr (5e-4) // error tolerance in VOF
#define K1Err (1e-4) // error tolerance in curvature of sessile drop
#define K2Err (1e-4) // error tolerance in curvature of mobile/impacting drop
#define VelErr (5e-3) // error tolerances in velocity
#define Mu21 (6e-3) // viscosity ratio between the gas and the liquid
#define Rho21 (1./770) // density ratio between the gas and the liquid
#define Zdist (3.) // height of impacting drop from the substrate

In the manuscript, offset parameter \chi is defined as: \displaystyle \chi = \frac{d}{2R} Here, d is the distance between the axes of two drops, and R is the equivalent radius of the drop. In the simulations, we input 2\chi (as the length scale is R).

#define Xoffset (0.50)
#define R2Drop(x,y,z) (sq(x-Xoffset) + sq(y) + sq(z-Zdist)) // Equation for the impacting drop
#define Ldomain 16 // Dimension of the domain

Boundary Conditions Back Wall is superamphiphobic and has the no-slip condition for velocity.

u.t[back] = dirichlet(0.);
u.r[back] = dirichlet(0.);
f1[back] = dirichlet(0.);
f2[back] = dirichlet(0.);
double tmax, We, Oh, Bo;

int main() {
tmax = 7.5;

Weber number is based on the impact velocity, U_0. \displaystyle We = \frac{\rho_lU_0^2R}{\gamma} Note that the impact Weber number we input is slightly less than the actual Weber number that we need. As the impacting drop falls, it also gains some kinetic energy. So, while analyzing the results, the Weber number should be taken at the instant, which one decides to choose as t = 0. These calculations get tricky as one goes to a higher \chi. However, the results do not change much as long as We \sim \mathcal{O}(1).

  We = 1.375; // We is 1 for 0.1801875 m/s <770*0.1801875^2*0.001/0.025>
init_grid (1 << MINlevel);
L0=Ldomain;

Navier Stokes equation for this case: \displaystyle \partial_tU_i+\nabla\cdot(U_iU_j) = \frac{1}{\hat{\rho}}\left(-\nabla p + Oh\nabla\cdot(2\hat{\mu}D_{ij}) + \kappa\delta_sn_i\right) + Bog_i The \hat{\rho} and \hat{\mu} are the VoF equations to calculate properties, given by: \displaystyle \hat{A} = (f_1+f_2) + (1-f_1-f_2)\frac{A_g}{A_l}

Ohnesorge number Oh: measure between surface tension and viscous forces. \displaystyle Oh = \frac{\mu_l}{\sqrt{\rho_l\gamma R}}

  Oh = 0.0216; // <0.003/sqrt(770*0.025*0.001)>

Bond number Bo: measure between Gravity and surface tension. \displaystyle Bo = \frac{\rho_lgR^2}{\gamma}

  Bo = 0.308; // <770*10*0.001^2/0.025>

Note: The subscript l denotes liquid. Also, the radius used in the dimensionless numbers is the equivalent radius of the drops \left(R = \left(3\pi V_l/4\right)^{1/3}\right). V_l is the volume of the two drops.

  fprintf(ferr, "tmax = %g. We = %g\n",tmax, We);
rho1 = 1.0; mu1 = Oh;
rho2 = Rho21; mu2 = Mu21*Oh;

Velocity scale as the intertial-capillary velocity, \displaystyle U_\gamma = \sqrt{\frac{\gamma}{\rho_l R}}

  f1.sigma = 1.0; f2.sigma = 1.0;

run();
}

This event is specific to César’s adapt_wavelet_limited. Near the substrate, we refine the grid one level higher than the rest of the domain.

int refRegion(double x, double y, double z){
return (z < 0.128 ? MAXlevel+1 : MAXlevel);
}

## Initial Condition

event init(t = 0){
if(!restore (file = "dump")){

For Bond numbers > 0.1, assuming the sessile drop to be spherical is inaccurate. One can also see this in the experimental video above. So, we used the code (here) to get the initial shape of the sessile drop. We import an STL file.

    char filename[60];
sprintf(filename,"Sessile-Bo0.3080.stl");
FILE * fp = fopen (filename, "r");
if (fp == NULL){
fprintf(ferr, "There is no file named %s\n", filename);
return 1;
}
coord * p = input_stl (fp);
fclose (fp);
coord min, max;
bounding_box(p, &min, &max);
fprintf(ferr, "xmin %g xmax %g\nymin %g ymax %g\nzmin %g zmax %g\n", min.x, max.x, min.y, max.y, min.z, max.z);
origin((min.x+max.x)/2. - L0/2, (min.y+max.y)/2. - L0/2, 0.); // We choose (X,Y) of origin as the center of the sessile drop. And substrate at Z = 0.
refine(R2Drop(x,y,z) < sq(1.+1./16) && level < MAXlevel);
fraction(f2, 1. - R2Drop(x,y,z));
scalar d[];
distance (d, p);
while (adapt_wavelet_limited ((scalar *){f2, d}, (double[]){1e-6, 1e-6*L0}, refRegion).nf);
vertex scalar phi[];
foreach_vertex(){
phi[] = (d[] + d[-1] + d[0,-1] + d[-1,-1] +
d[0,0,-1] + d[-1,0,-1] + d[0,-1,-1] + d[-1,-1,-1])/8.;
}
boundary ((scalar *){phi});
fractions (phi, f1);
foreach () {
u.z[] = -sqrt(We)*f2[];
u.y[] = 0.0;
u.x[] = 0.0;
}
boundary((scalar *){f1, f2, u.x, u.y});
dump (file = "dump");

Note: I think distance.h is not compatible with mpi. So, I ran the file to import .stl file and generate the dump file at t = 0 locally. For this, OpenMP multi-threading can be used.

    return 1;
}
}

Gravity is added as a body forces. It would be nice to use something like reduced.h. But, I could not figure out how to do it with two different VoF tracers.

event acceleration(i++) {
face vector av = a;
foreach_face(z){
av.z[] -= Bo;
}
}

## Adaptive Mesh Refinement

event adapt(i++) {

We refine based on curvatures of the two drops along with the generally used VoF and velocity fields. This ensures that the refinement level along the interface is MAXlevel.

  scalar KAPPA1[], KAPPA2[];
curvature(f1, KAPPA1);
curvature(f2, KAPPA2);
adapt_wavelet_limited ((scalar *){f1, f2, KAPPA1, KAPPA2, u.x, u.y, u.z},
(double[]){fErr, fErr, K1Err, K2Err, VelErr, VelErr, VelErr},
refRegion, MINlevel);
}

## Dumping snapshots

event writingFiles (t = 0; t += tsnap; t <= tmax) {
dump (file = "dump");
char nameOut[80];
sprintf (nameOut, "intermediate/snapshot-%5.4f", t);
dump (file = nameOut);
}

## Log writing

event logWriting (i++) {
double ke = 0.;
foreach (reduction(+:ke)){
ke += 0.5*(sq(u.x[]) + sq(u.y[]) + sq(u.z[]))*rho(f1[]+f2[])*cube(Delta);
}
static FILE * fp;
if (i == 0) {
fprintf (ferr, "i dt t ke\n");
fp = fopen ("log", "w");
fprintf (fp, "i dt t ke\n");
fprintf (fp, "%d %g %g %g\n", i, dt, t, ke);
fclose(fp);
} else {
fp = fopen ("log", "a");
fprintf (fp, "%d %g %g %g\n", i, dt, t, ke);
fclose(fp);
}
fprintf (ferr, "%d %g %g %g\n", i, dt, t, ke);
}

## Running the code

Use the following procedure:

Step 1: Importing the stl file and generating the first dump file

#!/bin/bash
mkdir intermediate
qcc -fopenmp -O2 -Wall dropOnDropImpact.c -o dropOnDropImpact -lm
export OMP_NUM_THREADS=8
./dropOnDropImpact

Step 2: Follow the method described (here). Do not forget to use the dump file generated in the previous step.

# Output and Results

The post-processing codes and simulation data are available at: PostProcess

## The process

Impacting drop slides-off the sessile drop. It glides on the thin air layer. We model this non-coalescence effect by assuming two VoF tracers for the two drops. We = 1.50 and \chi = 0.25.