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Example: Mound with gravity.

This example is a mound of liquid sitting on a tabletop with gravity acting on it. The contact angle between the drop surface and the tabletop is adjustable, to simulate the different degrees to which the liquid wets the table. This example illustrates macros, variables, constraints with energy, and omitting faces from body surfaces.

The drop starts as a cube with one face (face 6 of the cube example) on the tabletop (the z = 0 plane). The most straightforward way to specify a contact angle is to declare face 6 to be constrained to stay on the tabletop and give it a surface tension different than the default of 1. But this leads to problems described below. The way the contact angle is handled instead is to omit face 6 and give the edges around face 6 an energy integrand that results in the same energy we would get if we did include face 6. If we let the interface energy density for face 6 be T, then we want a vectorfield w such that

  
        /             /
        |  T k . dS = | w . dl
        / face 6      / bdry of face 6

So by Green's Theorem, all we need is curl w = Tk, and I will use w = -Tyi. Here i j k are the standard unit basis vectors. In practice, I don't think about Green's Theorem as such; I just write down a line integral that sums up strips of surface.

I have chosen to parameterize the contact angle as the angle in degrees between the table and the surface on the interior of the drop. This angle can be adjusted by assigning a new value to the variable "angle" at runtime. I could have made WALLT the parameter directly, but then I wouldn't have had an excuse to show a macro.

mound skeleton The initial mound skeleton, with vertices and edges numbered.
Here is the datafile mound.fe:
// mound.fe
// Evolver data for drop of prescribed volume sitting on plane with gravity.
// Contact angle with plane can be varied.

PARAMETER angle = 90    // interior angle between plane and surface, degrees

gravity_constant 0  // start with gravity off

#define WALLT  (-cos(angle*pi/180))  // virtual tension of facet on plane
 
constraint 1   /* the table top */
formula: x3 = 0
energy:  // for contact angle
e1: -(WALLT*y)
e2: 0
e3: 0 

vertices
1   0.0  0.0 0.0  constraint 1  /* 4 vertices on plane */
2   1.0  0.0 0.0  constraint 1
3   1.0  1.0 0.0  constraint 1
4   0.0  1.0 0.0  constraint 1
5   0.0  0.0 1.0
6   1.0  0.0 1.0
7   1.0  1.0 1.0
8   0.0  1.0 1.0
9   2.0  2.0 0.0  fixed   /* for table top */
10  2.0 -1.0 0.0  fixed
11 -1.0 -1.0 0.0  fixed
12 -1.0  2.0 0.0  fixed

edges  /* given by endpoints and attribute */
1   1 2    constraint 1 /* 4 edges on plane */
2   2 3    constraint 1
3   3 4    constraint 1
4   4 1    constraint 1
5   5 6
6   6 7  
7   7 8 
8   8 5
9   1 5   
10  2 6  
11  3 7 
12  4 8
13  9 10   fixed  /* for table top */
14 10 11   fixed
15 11 12   fixed
16 12  9   fixed

faces  /* given by oriented edge loop */
1   1 10 -5  -9
2   2 11 -6 -10
3   3 12 -7 -11
4   4  9 -8 -12
5   5  6  7   8
7  13 14 15  16  density 0 fixed /* table top for display */

bodies  /* one body, defined by its oriented faces */
1   1 2 3 4 5   volume 1  density 1

read
re := refine edges where on_constraint 1
The mound itself was basically copied from cube.fe, but with face 6 deleted. The reason for this is that face 6 is not needed, and would actually get in the way. It is not needed for the volume calculation since it would always be at z = 0 and thus not contribute to the surface integral for volume. The bottom edges of the side faces are constrained to lie in the plane z = 0, so face 6 is not needed to keep them from catastrophically shrivelling up. We could have handled the contact angle by including face 6 with a surface tension equal to the interface energy density between the liquid and surface, but that can cause problems if the edges around face 6 try to migrate inward. After refinement a couple of times, interior vertices of the original face 6 have no forces acting on them, so they don't move. Hence it would be tough for face 6 to shrink when its outer vertices ran up against its inner vertices. The tabletop face, face 7, is entirely extraneous to the calculations. Its only purpose is to make a nice display. You could remove it and all its vertices and edges without affecting the shape of the mound. It's constraint 1 that is the tabletop as far as the mound is concerned. To see what happens with the bottom face present, load moundB.fe and do "run".

Now run Evolver on mound.fe. The command "re" defined at the end of the datafile is good to use first in order to refine some edges that need it. Refine and iterate a while. You should get a nice mound. It's not a hemisphere, since gravity is on by default with G = 1. If you use the G command to set "G 0" and iterate a while, you get a hemisphere. Try changing the contact angle, to 45 degrees (with the command "angle := 45"} or 135 degrees for example. You can also play with gravity. Set "G 10" to get a flattened drop, or "G -5" to get a drop hanging from the ceiling. "G -10" will cause the drop to try to break loose, but it can't, since its vertices are still constrained.


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