Math Background

Reynolds number (Picture 1) describes the ratio of the inertial force and the viscous force.

As described in the picture, ρ is the density of the fluid, u is the characteristic velocity of the flow and L is the characteristic length, which in our cases is the volume enclosed by the surface aμ is the dynamic viscosity of the flow. As we can see from the definition, if Re<<1 that means that the viscous force is significantly larger than the inertial force.

If we let all the inertial term in Navier-Stokes equation vanishes, we get the equation that describes the Stokes flow. (Picture 2)

The velocity in the Stokes equation has a convenient solution, since the non-linearity has been canceled; it satisfies the condition of the Green’s function, which is a general solution to all these kind of linear differential equation.

The electric potential from a single point source can be described using Green’s Function (Picture 3): where x is the target point impacted by the source, y is the source point, q is the charge of the source point and c is -1 in this case. G, on the other hand, is the Green’s Function on 2D surface where p is defined as the length of the vector. We can use Riemann sum to expand it to a continuous surface; but to numerically evaluate it, we still approximate it with a finite sum (Picture 4). where j is just indexing the points we sample on the surface, σ is the charge density on the source and w is the weight per point contributing. We use an algorithm called Gaussian quadrature to compute the weight. While this is rather difficult to compute by hand, we can use computers to generate a weight for each point we choose to sample.

Suppose we sampled the same amount of the target points as the source points, we can generate a matrix φ that describes all the potentials. Furthermore, this formulation (Picture 5) gives us a way to compute the density if we know the potential simply by solving a linear equation.

There is one problem, still. As we only want to manipulate the boundary condition, if we let the source point and target points be both on the boundary, the Green’s function actually returns an undefined value given the distance is 0.

To prevent this, we instead turn to a double layer potential by understanding the surface as dipoles with a positive charge to the outside of the boundary. The potential is computed as picture 6 shown.

Where σ is the dipole moment and r is this distance. This is indeed the directional derivative of Green’s function on the direction of the normal vector.

Then, putting it into the integral form we have the equation shown in picture 7. Still, this breaks when x = y. But we are able to compute the limit (Picture 8). κ denotes the curvature.

As we switch to actual flow, the formula is a little bit different since the velocity potential u is not a scalar but a vector. But using the same idea, we have a similar formula (Picture 9), as Ω denotes the flow in the container and δ is on the boundary.

Using the same approximation we are able to compute the boundary charge density and approximate all flows inside. Therefore, with these methods, we are now able to simulate all kinds of periodic flow within any geometric shape.