The Physics

Hele-Shaw Flow Simulation with Interactive Control using Complex Barycentric Coordinates

The Simulation

The Math

• Two plates separated by a small gap ℎ.• One/Two fluids trapped between the plates.• Constant injection/suction of rate 𝑄.

• The flow is described by the following setof equations [1]:

• With the boundary conditions:

2 𝑣 = −𝜕

𝜕𝑧𝑊

1 𝑊(𝑧) =𝑄

2𝜋log 𝑧 − 𝑠 + 𝑓(𝑧)

3 𝑅𝑒 𝑊 = 𝛾𝜅

• Uses the Cauchy-Green coordinates [2].

• At each iteration,

𝑓 𝑧 =

𝑗=0

𝑛

𝐶𝑗 𝑧 𝑓𝑗

The user interactively moves the singularity during the flow.

Exterior flow with obstacles.

The fluid is pumped from the skeleton of the bunny.

for 𝑧 ∈ 𝜕Ω

Real photograph by Antony Hall Our simulation

Funded by:

Our simulation starting from an interface of a real experiment.

References

[1] Gustafsson, Bjorn, and Alexander Vasil’ev. Conformal

and potential analysis in Hele-Shaw cells. Springer Science

& Business Media, 2006.

[2] Weber, Ofir, Mirela Ben‐Chen, and Craig Gotsman.

"Complex barycentric coordinates with applications to

planar shape deformation. Computer Graphics Forum. Vol. 28. No. 2. Blackwell Publishing Ltd, 2009.

o Solve equation (3) for 𝑓𝑗.

o Calculate the velocity from (2).o Update the vertices positions 𝑧𝑗.

Aviv Segall, Orestis Vantzos and Miri Ben-Chen