Hi I’m modelling multiphase flow as suggested by Sukop in introduction to LB for geoscientists and engineers. It is achieved by incoporating attractive interaction forces between particles.

Now I want to illustrate a point on how surface tension due to attractive forces causes liquid to form a droplet. To do this I take the example of a square where due to differences in geometry a corner would be affected differently than a point along the edge. This would cause the form to change until equal forces are experienced along the interface i.e. a circle. Am I right to believe that a corner point would experience a greater force towards the center than a point along the edge would?

I’m not sure but I think you are right… and I buy your idea.

First let’s take the definition of the inter-particle force for ShanChen method for an easy case (I think in term of multi-compnents ShanChen method)
which is the force applied on the fluid_A at the position (x) is equal to:

F_A(x)= - rho_A(x)*G sum_i t_i rho_B(x+e_i) e_i

where i is the index for the lattice directions, e_i is the vector direction and rho_A and rhoB_B are respectively the density of the fluid inside the bubble and outside the bubble.
Ahh… I assume initial rho_A=rho_B and the square configuration you are talking about.
At the first time step I calculate F_A for to lattice points. One on the corner and the second one on the edge. I take the D2Q9 lattice.

If you look at the lattice node on the edge you see that you only have 3 neighbors of x on the edge that looks outside the bulk (where rho_B is not equal to zero) of your still square bubble. When you look at the case of x lattice node on the corner you have six lattice neighbors outside the bulk of the still square bubble. If you make the difference between F_A calculated at the two lattice node (edge and corner), I would say that the the force F_A on the corner should have a bigger magnitude that the one on the edge.
Three contribution from nearest neighbors drops (the one of the edge) and you still have three contribution for the corner.

I think that this game goes on until, as you said, your reach the equilibrium condition of the circular bubble shape … but I think that the kind of algebra that I described above is only valid at the first time step where you don’t have interfaces with a non zero thickens.

Am I writing bullshit here or what?
let me know what you think
Ciao Andrea