The Lattice Boltzmann Shield

Multiphase flow simulations — modeling oil and water, bubbles in liquid, droplets in air — face a persistent numerical challenge at the interface between phases. When two fluid domains approach contact, the thin film between them becomes computationally unresolvable. The grid can’t capture the physics at the nanometer scale where the film ruptures or doesn’t. The simulation crashes, produces artifacts, or forces premature coalescence.

Allen-Cahn phase-field models handle the interface as a diffuse zone — a smooth transition from one phase to the other, spread over a few grid cells. This avoids the need to track a sharp boundary. But the diffuse interface creates its own problem: when two interfaces approach each other, their diffuse zones overlap, and the model can’t distinguish “two interfaces close together” from “one merged interface.” The phases numerically merge before the physics says they should.

The fix is adaptive near-contact repulsion: an additional term in the Allen-Cahn equation that activates only when two interfaces are within a critical distance. The repulsion prevents premature merging by inserting a penalty for interface overlap. When the interfaces are far apart, the term is negligible. When they approach, it pushes back — a shield that preserves the thin film until the physics (not the numerics) dictates coalescence.

The adaptation is the key: the repulsion strength scales with the local interface separation. Too strong, and the phases never merge (unphysical). Too weak, and the numerical artifact returns. The adaptive scaling ties the repulsion to the resolved geometry, making the correction self-calibrating.