Multiple Scientific Applications
of Front Tracking Method

The Contact Angle

The contact angle plays the role of boundary conditions where more than three interfaces meet. Here we tried to establish a basic frame to study how the notions of the contact angle can be applied to describe a moving boundary node and how to implement them into the FronTier package.

Since flows in which the contact angle takes an important role involve the complicated free-boundary problem, knowing the contact angle is often the key to understanding the flow details at the vicinity of the contact line and to establishing a proper conservation equilibrium system.

Dynamic Contact Angle

The contact angle used to describe the intersection within the contact-line region under dynamic conditions is termed as the dynamic contact angle. Currently, there are two main methods to study the dynamic contact angle.

• One approach is to assume a constitutive relation between the static contact angle and the contact-line speed (Hocking 1995)
• Another approach is to assume that the microscopic contact angle is always equal to its static value, this is reasonable especially for flows with relatively low velocities (Neumann et al.)

Boundary Node Propagation

The contact angle exists only when more than three phase intersections occur at the same location. In FronTier, the interface points on such kind of contact line are either called CC-node (figure link below Left, Curve Crossing node) or B-node (Right, Boundary node). CC-nodes are interface points between three physical waves, and B-nodes are interface points between a physical wave and two other non-physical boundaries.

In FronTier, there are two existing main methods to study the B-node propagation. see figure below.

If we assumed that A, B, C, and D are all interface points on the physical wave. At each time step, all the new position of these points (i.e. A', B', C', and D') are computed by solving the wave propagation equations.

Since A is the boundary node which doesnot exactly apply the wave equation, the computed point A' can be assumed as just a temporary position. the final new position A" can be specified by the relative position of A' to the boundary line.

Three are two main ways in FronTier to specify the newly propagated position of a B-node:

• If A' is outside the boundary(as in figure above Left), then a straight line between A' and the closest interface point on the physical wave(i.e. B') is made. The intersection between this straight line and the boundary is the new position of the B-node.
• If A' is inside the physical boundary(as in figure above Right), there are several ways to extend the physical wave towards the boundary by using some existing nearby interface points.
  • make a straight line along A' and B' and the intersection between this line and the boundary.
  • make a circle by using three nearby points, A', B', C', the intersection between the circle and the boundary.
  • use Huyghens' principle of a propagating wave of light.

For a two dimensional problem, if we know the contact angle, the propagated B-node can be decided by drawing a straight line which passes through point B' and takes a value of the contact angle counterclockwisely away from the boundary line, and finding the intersection point with the boundary.

By using the method of the contact angles, the temporary point A' is not necessary and the new position of the B-node can be attained by using geometric method instead of solving any physical equation.

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Computational Results

There are many situations in which the contact angle plays a significant role, our simulations include: water drops on a horizontal plate or a sloping plate, a thin film problem where a gravity-driven flow runs into a solid plane.

Droplet Problems

The spreading of a water drop of volume V on a flat solid plate is a commonly used test case in studying the contact angle. we assumed that the surface of the plate is smooth enough so that we do not need to consider the roughness and surface resistance. Both the temperature and the pressure are under the standard environments.

The simulation is based on two dimensional 100 X 100 grids. Initially the water drop starts from a semi-circle and then spreads and changes its shape under the influence of the gravity. Because of the geometric symmetry only half of the semi-circle droplet was studied and the reflecting boundary condition is applied on the corresponding left boundary of the horizontal direction. The boundary condition for the solid wall(plate) is Neumann boundary.

At initial time, there is one B-node as shown in figure below and at this point the water drop, air and the solid wall intersect.

Since the droplet moves very slowly, we applied the static contact angle in our simulations. Figure below shows the computational results of a water drop spreading along a sloping plate at different time steps.

the following movie shows an elliptic water droplet falls from a certain height down into a sloping plate and then spreads around.

Thin Film Problems

We applied the above contact angle method to solve the B-node propagation in a thin film problem(See figure below). Water at 20^oC flows from the jet with the flow-rate of 1.7gr/s and then hits a sloping plate.

Neumann boundary is applied on the sloping plate and the flow-through Dirichlet boundary condition is applied on all other computational boundary. The computed velocity vector and density contour can be shown in the following movie link.

Several vortex structures are well captured and the plots show how the water hits and then flows over the sloping plate.

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