# Droplets distribution

Establishment of a model for the water bombing from an aircraft

When a drop is too big, it gives birth to smaller droplets according to several breakup regimes, depending on the Weber number :

$We=\frac{aerodynamic force}{surface tension force}=\frac{\rho_G \| \vec{U_G}-\vec{V_p} \|^2 d_p}{\sigma}$

Breakup regimes

Another adimensional number plays a role in the atomization process : the Bond number

$Bo=\frac{gravity force}{surface tension force}=\frac{\rho_G g {d_p}^2}{\sigma}$

To remain stable, a droplet has to satisfy the following conditions :

• $We_{drop} \le We_{cr}$
• $Bo_{drop} \le Bo_{cr}$

When it does not, the drop - called the "mother" drop - gives birth to small droplets, called "children" droplets. To simplify the problem here, we chose to consider the division of the "mother" drop into two "children" droplets. The centre of mass C of the new-formed droplet will be randomly located on a sphere around the centre of mass M of the "mother" drop, the radius being equal to the mother drop's diameter :

Spherical coordinates

Our approach consisted in studying each droplet, evaluating its Weber and Bond numbers. Big at the beginning, all the drops are divided into smaller droplets, until the "children" droplets satisfy the stability criteria. Hence, at the end, the stable droplets are all the same size.

The droplets distribution is as below :

Droplets distribution right after the second atomization

The next step, now, is to calculate the movement equation of every single drop, in order to evaluate the mark on the ground.