# Question on surface tension

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

ciao
Andrea

It’s good to see someone else struggled with the concept =)

The solution you came up with is quite similar to what I finally wrote in my report. Thank you for confirming this!

Sometimes the simplicity of the LBM and the inuitive approach that can be taken to explain how it functions scares me.

Try explaining the derivation of the Navier-stokes equations to a high school student