Our coffee machine catches the last couple of droplets, after your cup is remove

Question

Our coffee machine catches the last couple of droplets, after your cup is removed on a shape to reduce plash of the coffee droplets.

These shapes are placed inside the spill reservoir.
The shape used by our coffee machine is shape nr 2 with the droplet falling at position 1, this works well but not perfect.
What shape would be even better for this purpose? And at which position should the droplet fall?

I've included some shapes I came up with but I'm open for other suggestions.

The image displays a 2D version of the shapes, the first 2 shapes are cone shaped and the last 2 would have to be open on the ends (or sides) to release the coffee into the spill reservoir.

Explanation / Answer

Obvouusly the highest number of splashes will be when the impact happens at right angle. This is because at such impact all the kinetic energy of the drop at one moment goes into the forces directed into different directions and tears the drop apart.

Conversely the least splashes will be when the drop impacts at narrow angle. In this case the drop continues sliding over the surface and keeps its integrity while kinetic energy slowly transforms into heat due to friction. The surface tension keeps the drop intact because all parts of the drop keep the same movement direction.

As such I would recommend a form which meets the drop in the most probable impact place with a sloppy angle. Evidently, such form should not be symmetric (because all even functions have zero derivative at zero).

The only variant of yours that has such property is the fourth. Additionally it changes the drop's velocity vector to the right direction so that any splaches that can occur will go to the right where it has a border that covers the greatest angle of possible splashes path out than all other variants.

The slope's curvature should be such that the amount of energy converted into heat was uniform along the drop's path and the pressure never exceed the surface tension. So when the drop has the highest speed, the angle between the forces acting on the drop (inertia and gravity combined) and the surface should be the smallest, but as long as the drop slows, the angle slows the angle should rise so to keep the component which is perpendicular to the surface constant.

The fourth picture roughly satisfies this criterion.

Thus the fourth variant in the best.

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