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Disclaimer: I work in applied math and have limited background in Physics. Need a bit of help here.

Assuming I have a 2D droplet attached to some surface like so:enter image description here

This droplet is also experiencing gravity, which points directly downwards.

Assuming the droplet has a mass of $m$, gravity constant $g$ and has some reasonable volume profile (shape) that you may specify (this is used to determine contact area and perhaps to get surface tension if need be).

What is the critical bifurcation volume of the droplet so it would stick to the surface?

Or an easier question maybe: what is the adhesive forces between the droplet and the surface?

Even easier question: what is the adhesive force, assuming the contact area is of size $A$?

I'm guessing some roughness constant might need to be specified. So just specify this constant if need be. Thanks in advance!

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  • $\begingroup$ just trying to see if theres people here.... $\endgroup$ – Book Book Book Oct 23 '19 at 22:35
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    $\begingroup$ I think that you have made a false supposition that the droplet will separate at the interface of the two materials. The droplet will form a “neck” as it approaches the critical volume. Relating this critical volume to the sorts of parameters you are asking for will be a problem best done in 3D, associating a potential energy with each possible droplet shape, and then solving using the techniques of the Hamiltonian formalism. $\endgroup$ – Duncan Harris Oct 29 '19 at 1:05
  • $\begingroup$ Luckily for you the problem has been solved, so if this is a practical question you can always consult the literature on droplet shapes! $\endgroup$ – Duncan Harris Oct 29 '19 at 1:10
  • $\begingroup$ I'm myself curious of the answer, @Duncan Harris, do you perhaps have a position in mind? $\endgroup$ – kamilazdybal Nov 1 '19 at 21:56
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This may help to understand the origin of the adhesion force. I numerically solved the problem of dropping a drop from a solid wall using the system of Navier-Stokes equations and FEM. It was necessary to determine the parameter on which the flow in the drop depends. This parameter is $$W=\frac {\sigma}{d^2\rho g}$$ where $\sigma $ is the surface tension, $d$ is the radius of the drop, $\rho $ is the density, $g$ is the acceleration of gravity. If $W>1$, the drop adheres to the surface, if $W=1$, then the flow is transient, and if $W<1$, then the drop comes off under the influence of gravity. These three flow are shown in Fig. 1; above the figures, $W$ is indicated. Figure I tried to reproduce the following experiment with a drop of honey Figure 2 FEM allows us to describe large deformations, provided that the mesh is calculated at each step. It turns out this picture (shows the flow velocity in cylindrical coordinates for an axisymmetric drop) Figure 3

Here I took the small $W = 0.002$. It turned out such a change in the shape of the drop over time Figure 4

When the surface tension is sufficiently large, then surface instability occurs, as in this example, with $W = 0.1$ Figure 5

Figure 6

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  • $\begingroup$ absolutely amazing answer! $\endgroup$ – Book Book Book Nov 6 '19 at 23:02
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    $\begingroup$ Thank you! But I have not finished yet. I want to add a couple of animations. $\endgroup$ – Alex Trounev Nov 6 '19 at 23:41
  • $\begingroup$ just wondering, does the color stand for pressure in the diagrams? also gravity points right correct? $\endgroup$ – Book Book Book Nov 19 '19 at 22:21
  • $\begingroup$ I'm just trying to understand what awesome work you did here. what is the free surface boundary condition of this thing? did you have fluid outside of the droplet as well? $\endgroup$ – Book Book Book Nov 19 '19 at 23:26
  • $\begingroup$ where did the number W show up in your calculation? to what term was it attached to I'm guessing some gradient of velocity term (stress tensor)? $\endgroup$ – Book Book Book Nov 19 '19 at 23:29

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